IB · PHYSICS HL

Physics: Higher Level

Rigid body mechanics — HL Theme A

Name: ____________________Date: October 10, 2026

1.
Define torque, and state the two factors that determine the torque produced by a force acting on a rigid body.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: Defines torque as the turning effect of a force about a pivot or axis of rotation. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: States that torque depends on the magnitude of the force. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Develop this part of the answer: States that torque depends on the perpendicular distance from the pivot to the line of action of the force (the moment arm). Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Torque = force × perpendicular distance; only the component of distance perpendicular to the force's line of action contributes.
Marking points
- Defines torque as the turning effect of a force about a pivot or axis of rotation.
- States that torque depends on the magnitude of the force.
- States that torque depends on the perpendicular distance from the pivot to the line of action of the force (the moment arm).
Examiner tip: Torque = force × perpendicular distance; only the component of distance perpendicular to the force's line of action contributes.
2.
Marking analysis: A learner attempts the following task: “Define torque, and state the two factors that determine the torque produced by a force acting on a rigid body.” Their response addresses only this point: “Defines torque as the turning effect of a force about a pivot or axis of rotation.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Defines torque as the turning effect of a force about a pivot or axis of rotation. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that torque depends on the magnitude of the force. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: States that torque depends on the perpendicular distance from the pivot to the line of action of the force (the moment arm). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Defines torque as the turning effect of a force about a pivot or axis of rotation.
- Identifies the missing requirement: States that torque depends on the magnitude of the force.
- Identifies the missing requirement: States that torque depends on the perpendicular distance from the pivot to the line of action of the force (the moment arm).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
3.
A force of 15 N is applied perpendicular to a wrench at a distance of 0.20 m from a bolt. Calculate the torque produced.
[2 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Uses torque τ = Fr (force × perpendicular distance). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Obtains τ = 15 × 0.20 = 3.0 N m. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Torque is measured in newton-metres (N m), the same units as work/energy, but torque is a distinct physical quantity (a vector-like moment, not energy).
Marking points
- Uses torque τ = Fr (force × perpendicular distance).
- Obtains τ = 15 × 0.20 = 3.0 N m.
Examiner tip: Torque is measured in newton-metres (N m), the same units as work/energy, but torque is a distinct physical quantity (a vector-like moment, not energy).
4.
Marking analysis: A learner attempts the following task: “A force of 15 N is applied perpendicular to a wrench at a distance of 0.20 m from a bolt. Calculate the torque produced.” Their response addresses only this point: “Uses torque τ = Fr (force × perpendicular distance).” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Uses torque τ = Fr (force × perpendicular distance). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Obtains τ = 15 × 0.20 = 3.0 N m. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Uses torque τ = Fr (force × perpendicular distance).
- Identifies the missing requirement: Obtains τ = 15 × 0.20 = 3.0 N m.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
5.
Define moment of inertia, and state one factor (other than mass) that affects the moment of inertia of a rigid body about a given axis.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: Defines moment of inertia as a measure of an object's resistance to angular (rotational) acceleration about a given axis. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: States that moment of inertia depends on how the mass is distributed relative to the axis of rotation (mass further from the axis contributes more). Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Moment of inertia is the rotational analogue of mass in linear mechanics — it depends not just on how much mass there is, but on how far that mass is distributed from the axis.
Marking points
- Defines moment of inertia as a measure of an object's resistance to angular (rotational) acceleration about a given axis.
- States that moment of inertia depends on how the mass is distributed relative to the axis of rotation (mass further from the axis contributes more).
Examiner tip: Moment of inertia is the rotational analogue of mass in linear mechanics — it depends not just on how much mass there is, but on how far that mass is distributed from the axis.
6.
Marking analysis: A learner attempts the following task: “Define moment of inertia, and state one factor (other than mass) that affects the moment of inertia of a rigid body about a given axis.” Their response addresses only this point: “Defines moment of inertia as a measure of an object's resistance to angular (rotational) acceleration about a given axis.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Defines moment of inertia as a measure of an object's resistance to angular (rotational) acceleration about a given axis. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that moment of inertia depends on how the mass is distributed relative to the axis of rotation (mass further from the axis contributes more). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Defines moment of inertia as a measure of an object's resistance to angular (rotational) acceleration about a given axis.
- Identifies the missing requirement: States that moment of inertia depends on how the mass is distributed relative to the axis of rotation (mass further from the axis contributes more).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
7.
State the rotational form of Newton's second law, relating torque, moment of inertia and angular acceleration.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: States τ = Iα, where τ is the net torque, I is the moment of inertia, and α is the angular acceleration. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: States that this is the direct rotational analogue of F = ma in linear mechanics. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Every linear mechanics equation has a direct rotational analogue: F↔τ, m↔I, a↔α, v↔ω, and momentum p↔angular momentum L.
Marking points
- States τ = Iα, where τ is the net torque, I is the moment of inertia, and α is the angular acceleration.
- States that this is the direct rotational analogue of F = ma in linear mechanics.
Examiner tip: Every linear mechanics equation has a direct rotational analogue: F↔τ, m↔I, a↔α, v↔ω, and momentum p↔angular momentum L.
8.
Marking analysis: A learner attempts the following task: “State the rotational form of Newton's second law, relating torque, moment of inertia and angular acceleration.” Their response addresses only this point: “States τ = Iα, where τ is the net torque, I is the moment of inertia, and α is the angular acceleration.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States τ = Iα, where τ is the net torque, I is the moment of inertia, and α is the angular acceleration. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that this is the direct rotational analogue of F = ma in linear mechanics. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States τ = Iα, where τ is the net torque, I is the moment of inertia, and α is the angular acceleration.
- Identifies the missing requirement: States that this is the direct rotational analogue of F = ma in linear mechanics.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
9.
A rigid body has moment of inertia 4.0 kg m² about a fixed axis. A net torque of 12 N m is applied. Calculate its angular acceleration.
[2 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Uses τ = Iα. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Rearranges and substitutes α = 12/4.0 to obtain α = 3.0 rad s⁻². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Angular acceleration in these equations is always measured in radians per second squared, not degrees.
Marking points
- Uses τ = Iα.
- Rearranges and substitutes α = 12/4.0 to obtain α = 3.0 rad s⁻².
Examiner tip: Angular acceleration in these equations is always measured in radians per second squared, not degrees.
10.
Marking analysis: A learner attempts the following task: “A rigid body has moment of inertia 4.0 kg m² about a fixed axis. A net torque of 12 N m is applied. Calculate its angular acceleration.” Their response addresses only this point: “Uses τ = Iα.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Uses τ = Iα. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Rearranges and substitutes α = 12/4.0 to obtain α = 3.0 rad s⁻². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Uses τ = Iα.
- Identifies the missing requirement: Rearranges and substitutes α = 12/4.0 to obtain α = 3.0 rad s⁻².
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
11.
Define angular momentum, and state the principle of conservation of angular momentum.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: Defines angular momentum as L = Iω, the product of moment of inertia and angular velocity. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Develop this part of the answer: States that the total angular momentum of a system remains constant provided no external net torque acts on it. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Angular momentum conservation is the rotational analogue of linear momentum conservation — it applies whenever there is no external torque, exactly as linear momentum is conserved with no external force.
Marking points
- Defines angular momentum as L = Iω, the product of moment of inertia and angular velocity.
- States that the total angular momentum of a system remains constant provided no external net torque acts on it.
Examiner tip: Angular momentum conservation is the rotational analogue of linear momentum conservation — it applies whenever there is no external torque, exactly as linear momentum is conserved with no external force.
12.
Marking analysis: A learner attempts the following task: “Define angular momentum, and state the principle of conservation of angular momentum.” Their response addresses only this point: “Defines angular momentum as L = Iω, the product of moment of inertia and angular velocity.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Defines angular momentum as L = Iω, the product of moment of inertia and angular velocity. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that the total angular momentum of a system remains constant provided no external net torque acts on it. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Defines angular momentum as L = Iω, the product of moment of inertia and angular velocity.
- Identifies the missing requirement: States that the total angular momentum of a system remains constant provided no external net torque acts on it.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
13.
An ice skater spinning with arms outstretched has moment of inertia 4.0 kg m² and angular velocity 2.0 rad s⁻¹. She pulls her arms in, reducing her moment of inertia to 1.0 kg m². Calculate her new angular velocity, using conservation of angular momentum.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Uses conservation of angular momentum: I₁ω₁ = I₂ω₂. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes 4.0 × 2.0 = 1.0 × ω₂. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Obtains ω₂ = 8.0 rad s⁻¹. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: This classic example (a spinning skater pulling in their arms) shows that decreasing moment of inertia while conserving angular momentum must increase angular velocity.
Marking points
- Uses conservation of angular momentum: I₁ω₁ = I₂ω₂.
- Substitutes 4.0 × 2.0 = 1.0 × ω₂.
- Obtains ω₂ = 8.0 rad s⁻¹.
Examiner tip: This classic example (a spinning skater pulling in their arms) shows that decreasing moment of inertia while conserving angular momentum must increase angular velocity.
14.
Marking analysis: A learner attempts the following task: “An ice skater spinning with arms outstretched has moment of inertia 4.0 kg m² and angular velocity 2.0 rad s⁻¹. She pulls her arms in, reducing her moment of inertia to 1.0 kg m². Calculate her new angular velocity, using conservation of angular momentum.” Their response addresses only this point: “Uses conservation of angular momentum: I₁ω₁ = I₂ω₂.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Uses conservation of angular momentum: I₁ω₁ = I₂ω₂. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Substitutes 4.0 × 2.0 = 1.0 × ω₂. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Obtains ω₂ = 8.0 rad s⁻¹. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Uses conservation of angular momentum: I₁ω₁ = I₂ω₂.
- Identifies the missing requirement: Substitutes 4.0 × 2.0 = 1.0 × ω₂.
- Identifies the missing requirement: Obtains ω₂ = 8.0 rad s⁻¹.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
15.
State the condition(s) required for a rigid body to be in rotational equilibrium.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: States that the net (resultant) torque acting on the body about any point must be zero. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: States that for full (static) equilibrium, the net force must also be zero, in addition to the net torque. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Full static equilibrium requires both translational equilibrium (net force = 0) and rotational equilibrium (net torque = 0) simultaneously.
Marking points
- States that the net (resultant) torque acting on the body about any point must be zero.
- States that for full (static) equilibrium, the net force must also be zero, in addition to the net torque.
Examiner tip: Full static equilibrium requires both translational equilibrium (net force = 0) and rotational equilibrium (net torque = 0) simultaneously.
16.
Marking analysis: A learner attempts the following task: “State the condition(s) required for a rigid body to be in rotational equilibrium.” Their response addresses only this point: “States that the net (resultant) torque acting on the body about any point must be zero.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that the net (resultant) torque acting on the body about any point must be zero. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States that for full (static) equilibrium, the net force must also be zero, in addition to the net torque. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that the net (resultant) torque acting on the body about any point must be zero.
- Identifies the missing requirement: States that for full (static) equilibrium, the net force must also be zero, in addition to the net torque.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
17.
State the equation for rotational kinetic energy in terms of moment of inertia and angular velocity, and calculate the rotational kinetic energy of a wheel with moment of inertia 2.0 kg m² spinning at 5.0 rad s⁻¹.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States rotational kinetic energy = ½Iω². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes ½ × 2.0 × 5.0². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Obtains rotational kinetic energy = 25 J. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Rotational KE = ½Iω² is the direct rotational analogue of linear KE = ½mv², with I replacing m and ω replacing v.
Marking points
- States rotational kinetic energy = ½Iω².
- Substitutes ½ × 2.0 × 5.0².
- Obtains rotational kinetic energy = 25 J.
Examiner tip: Rotational KE = ½Iω² is the direct rotational analogue of linear KE = ½mv², with I replacing m and ω replacing v.
18.
Marking analysis: A learner attempts the following task: “State the equation for rotational kinetic energy in terms of moment of inertia and angular velocity, and calculate the rotational kinetic energy of a wheel with moment of inertia 2.0 kg m² spinning at 5.0 rad s⁻¹.” Their response addresses only this point: “States rotational kinetic energy = ½Iω².” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States rotational kinetic energy = ½Iω². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Substitutes ½ × 2.0 × 5.0². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Obtains rotational kinetic energy = 25 J. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States rotational kinetic energy = ½Iω².
- Identifies the missing requirement: Substitutes ½ × 2.0 × 5.0².
- Identifies the missing requirement: Obtains rotational kinetic energy = 25 J.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
19.
A uniform beam of weight 200 N and length 4.0 m is supported horizontally by a pivot at one end and a vertical cable at the other end. Calculate the tension in the cable, taking torques about the pivot.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Develop this part of the answer: States that the weight acts at the centre of the uniform beam, 2.0 m from the pivot. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Work through this mathematical step: Sets the sum of torques about the pivot to zero: cable tension × 4.0 = weight × 2.0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes T × 4.0 = 200 × 2.0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Obtains T = 100 N. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Taking torques about the pivot eliminates the unknown pivot reaction force from the equation entirely, since its moment arm about the pivot is zero — always choose the pivot as your reference point when it removes an unknown.
Marking points
- States that the weight acts at the centre of the uniform beam, 2.0 m from the pivot.
- Sets the sum of torques about the pivot to zero: cable tension × 4.0 = weight × 2.0.
- Substitutes T × 4.0 = 200 × 2.0.
- Obtains T = 100 N.
Examiner tip: Taking torques about the pivot eliminates the unknown pivot reaction force from the equation entirely, since its moment arm about the pivot is zero — always choose the pivot as your reference point when it removes an unknown.
20.
Marking analysis: A learner attempts the following task: “A uniform beam of weight 200 N and length 4.0 m is supported horizontally by a pivot at one end and a vertical cable at the other end. Calculate the tension in the cable, taking torques about the pivot.” Their response addresses only this point: “States that the weight acts at the centre of the uniform beam, 2.0 m from the pivot.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that the weight acts at the centre of the uniform beam, 2.0 m from the pivot. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Sets the sum of torques about the pivot to zero: cable tension × 4.0 = weight × 2.0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Substitutes T × 4.0 = 200 × 2.0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Obtains T = 100 N. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that the weight acts at the centre of the uniform beam, 2.0 m from the pivot.
- Identifies the missing requirement: Sets the sum of torques about the pivot to zero: cable tension × 4.0 = weight × 2.0.
- Identifies the missing requirement: Substitutes T × 4.0 = 200 × 2.0.
- Identifies the missing requirement: Obtains T = 100 N.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
21.
Two point masses, 2.0 kg and 3.0 kg, are attached to the ends of a light rigid rod, at distances of 0.40 m and 0.60 m respectively from a pivot at the rod's centre of rotation. Calculate the total moment of inertia of this system about the pivot.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States that for a system of point masses, I = Σmr². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates the contribution of the first mass: 2.0 × 0.40² = 0.32 kg m². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Calculates the contribution of the second mass: 3.0 × 0.60² = 1.08 kg m². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Obtains total moment of inertia = 0.32 + 1.08 = 1.40 kg m². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For a system of discrete point masses, moment of inertia is simply the sum of each individual mass multiplied by the square of its own distance from the axis — no single 'average' distance can be used instead.
Marking points
- States that for a system of point masses, I = Σmr².
- Calculates the contribution of the first mass: 2.0 × 0.40² = 0.32 kg m².
- Calculates the contribution of the second mass: 3.0 × 0.60² = 1.08 kg m².
- Obtains total moment of inertia = 0.32 + 1.08 = 1.40 kg m².
Examiner tip: For a system of discrete point masses, moment of inertia is simply the sum of each individual mass multiplied by the square of its own distance from the axis — no single 'average' distance can be used instead.
22.
Marking analysis: A learner attempts the following task: “Two point masses, 2.0 kg and 3.0 kg, are attached to the ends of a light rigid rod, at distances of 0.40 m and 0.60 m respectively from a pivot at the rod's centre of rotation. Calculate the total moment of inertia of this system about the pivot.” Their response addresses only this point: “States that for a system of point masses, I = Σmr².” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that for a system of point masses, I = Σmr². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Calculates the contribution of the first mass: 2.0 × 0.40² = 0.32 kg m². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Calculates the contribution of the second mass: 3.0 × 0.60² = 1.08 kg m². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Obtains total moment of inertia = 0.32 + 1.08 = 1.40 kg m². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that for a system of point masses, I = Σmr².
- Identifies the missing requirement: Calculates the contribution of the first mass: 2.0 × 0.40² = 0.32 kg m².
- Identifies the missing requirement: Calculates the contribution of the second mass: 3.0 × 0.60² = 1.08 kg m².
- Identifies the missing requirement: Obtains total moment of inertia = 0.32 + 1.08 = 1.40 kg m².
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
23.
A net torque of 8.0 N m acts on a rigid body for 5.0 s. Calculate the change in angular momentum produced, and state the equation used.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States that angular impulse = torque × time = ΔL. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes 8.0 × 5.0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Obtains ΔL = 40 kg m² s⁻¹. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Angular impulse (torque × time) is the direct rotational analogue of linear impulse (force × time), and both equal the resulting change in their respective momentum.
Marking points
- States that angular impulse = torque × time = ΔL.
- Substitutes 8.0 × 5.0.
- Obtains ΔL = 40 kg m² s⁻¹.
Examiner tip: Angular impulse (torque × time) is the direct rotational analogue of linear impulse (force × time), and both equal the resulting change in their respective momentum.
24.
Marking analysis: A learner attempts the following task: “A net torque of 8.0 N m acts on a rigid body for 5.0 s. Calculate the change in angular momentum produced, and state the equation used.” Their response addresses only this point: “States that angular impulse = torque × time = ΔL.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that angular impulse = torque × time = ΔL. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Substitutes 8.0 × 5.0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Obtains ΔL = 40 kg m² s⁻¹. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that angular impulse = torque × time = ΔL.
- Identifies the missing requirement: Substitutes 8.0 × 5.0.
- Identifies the missing requirement: Obtains ΔL = 40 kg m² s⁻¹.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
25.
A solid sphere of mass 2.0 kg and radius 0.10 m rolls without slipping along a horizontal surface at a linear speed of 3.0 m/s. Given that its moment of inertia about its central axis is I = (2/5)mr², and that for rolling without slipping v = ωr, calculate its total kinetic energy (translational plus rotational).
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Calculates the translational kinetic energy: ½mv² = ½ × 2.0 × 3.0² = 9.0 J. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Calculates the moment of inertia: I = (2/5) × 2.0 × 0.10² = 0.0080 kg m². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Uses ω = v/r = 3.0/0.10 = 30 rad s⁻¹. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Calculates the rotational kinetic energy: ½Iω² = ½ × 0.0080 × 30² = 3.6 J. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Obtains total kinetic energy = 9.0 + 3.6 = 12.6 J. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A rolling object always has both translational and rotational kinetic energy simultaneously — forgetting the rotational term (and treating it as if it were only sliding) is the most common error in these problems.
Marking points
- Calculates the translational kinetic energy: ½mv² = ½ × 2.0 × 3.0² = 9.0 J.
- Calculates the moment of inertia: I = (2/5) × 2.0 × 0.10² = 0.0080 kg m².
- Uses ω = v/r = 3.0/0.10 = 30 rad s⁻¹.
- Calculates the rotational kinetic energy: ½Iω² = ½ × 0.0080 × 30² = 3.6 J.
- Obtains total kinetic energy = 9.0 + 3.6 = 12.6 J.
Examiner tip: A rolling object always has both translational and rotational kinetic energy simultaneously — forgetting the rotational term (and treating it as if it were only sliding) is the most common error in these problems.
26.
Marking analysis: A learner attempts the following task: “A solid sphere of mass 2.0 kg and radius 0.10 m rolls without slipping along a horizontal surface at a linear speed of 3.0 m/s. Given that its moment of inertia about its central axis is I = (2/5)mr², and that for rolling without slipping v = ωr, calculate its total kinetic energy (translational plus rotational).” Their response addresses only this point: “Calculates the translational kinetic energy: ½mv² = ½ × 2.0 × 3.0² = 9.0 J.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Calculates the translational kinetic energy: ½mv² = ½ × 2.0 × 3.0² = 9.0 J. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Calculates the moment of inertia: I = (2/5) × 2.0 × 0.10² = 0.0080 kg m². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Uses ω = v/r = 3.0/0.10 = 30 rad s⁻¹. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Calculates the rotational kinetic energy: ½Iω² = ½ × 0.0080 × 30² = 3.6 J. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Obtains total kinetic energy = 9.0 + 3.6 = 12.6 J. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Calculates the translational kinetic energy: ½mv² = ½ × 2.0 × 3.0² = 9.0 J.
- Identifies the missing requirement: Calculates the moment of inertia: I = (2/5) × 2.0 × 0.10² = 0.0080 kg m².
- Identifies the missing requirement: Uses ω = v/r = 3.0/0.10 = 30 rad s⁻¹.
- Identifies the missing requirement: Calculates the rotational kinetic energy: ½Iω² = ½ × 0.0080 × 30² = 3.6 J.
- Identifies the missing requirement: Obtains total kinetic energy = 9.0 + 3.6 = 12.6 J.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
27.
A wheel completes one full rotation every 0.50 s. Calculate its angular velocity.
[2 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Uses ω = 2π/T. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes 2π/0.50 to obtain ω ≈ 12.6 rad s⁻¹. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Angular velocity relates to period in exactly the same way frequency does (ω = 2πf = 2π/T) — a shorter period always means a larger angular velocity.
Marking points
- Uses ω = 2π/T.
- Substitutes 2π/0.50 to obtain ω ≈ 12.6 rad s⁻¹.
Examiner tip: Angular velocity relates to period in exactly the same way frequency does (ω = 2πf = 2π/T) — a shorter period always means a larger angular velocity.
28.
Marking analysis: A learner attempts the following task: “A wheel completes one full rotation every 0.50 s. Calculate its angular velocity.” Their response addresses only this point: “Uses ω = 2π/T.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Uses ω = 2π/T. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Substitutes 2π/0.50 to obtain ω ≈ 12.6 rad s⁻¹. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Uses ω = 2π/T.
- Identifies the missing requirement: Substitutes 2π/0.50 to obtain ω ≈ 12.6 rad s⁻¹.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
29.
A solid disk and a thin hoop (ring) have the same mass and the same radius, and both rotate about their central axis. State which has the greater moment of inertia, and explain why, in terms of the distribution of mass.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Develop this part of the answer: States that the hoop has the greater moment of inertia. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
3. Develop this part of the answer: Explains that all of the hoop's mass is concentrated at the maximum distance (the radius) from the axis. Show which detail or principle supports it and how it addresses the command; equivalent supported wording is acceptable.
4. Work through this mathematical step: Explains that the disk's mass is distributed throughout its area, with some mass closer to the axis, so it contributes less to moment of inertia overall, since I = Σmr² weights mass by the square of its distance from the axis. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For the same mass and radius, moment of inertia depends entirely on how far that mass sits from the axis on average — concentrating mass at the rim (as in a hoop) always maximises it compared to spreading mass across the full disk.
Marking points
- States that the hoop has the greater moment of inertia.
- Explains that all of the hoop's mass is concentrated at the maximum distance (the radius) from the axis.
- Explains that the disk's mass is distributed throughout its area, with some mass closer to the axis, so it contributes less to moment of inertia overall, since I = Σmr² weights mass by the square of its distance from the axis.
Examiner tip: For the same mass and radius, moment of inertia depends entirely on how far that mass sits from the axis on average — concentrating mass at the rim (as in a hoop) always maximises it compared to spreading mass across the full disk.
30.
Marking analysis: A learner attempts the following task: “A solid disk and a thin hoop (ring) have the same mass and the same radius, and both rotate about their central axis. State which has the greater moment of inertia, and explain why, in terms of the distribution of mass.” Their response addresses only this point: “States that the hoop has the greater moment of inertia.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that the hoop has the greater moment of inertia. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Explains that all of the hoop's mass is concentrated at the maximum distance (the radius) from the axis. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Explains that the disk's mass is distributed throughout its area, with some mass closer to the axis, so it contributes less to moment of inertia overall, since I = Σmr² weights mass by the square of its distance from the axis. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that the hoop has the greater moment of inertia.
- Identifies the missing requirement: Explains that all of the hoop's mass is concentrated at the maximum distance (the radius) from the axis.
- Identifies the missing requirement: Explains that the disk's mass is distributed throughout its area, with some mass closer to the axis, so it contributes less to moment of inertia overall, since I = Σmr² weights mass by the square of its distance from the axis.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.