PHYC10003 Chap.4 Impulse, Torque and Rotation
Impulse, Torque and Rotation
Impulse and torque extend familiar translational ideas into short collisions and rotational motion. Momentum change retains direction, so rebound problems demand signed velocities. Torque depends on lever arm, force and angle about a chosen axis. Rotational energy depends on angular speed and moment of inertia, whose value reflects how mass is distributed relative to that axis.
Impulse equals final momentum minus initial momentum, so a rebound crosses zero and produces a larger change than merely stopping from the same incoming speed. The area under a force-time graph gives signed impulse; dividing by contact duration gives average force but does not determine the peak. Torque requires an axis, a position vector, and a perpendicular force component or lever arm.
A force through the axis has zero torque, and equal forces can have different effects when their lever arms differ. Static equilibrium requires both force balance and torque balance. A couple shows why zero net force need not mean zero turning effect. Moment of inertia attaches to a specified axis and weights mass by squared distance from it, so identical total mass can resist angular acceleration differently.
Rotational kinetic energy depends on both inertia and angular speed squared. These relationships support checks based on direction, lever-arm limits, axis choice, and scaling. They also keep linear impulse, torque, and energy in their proper roles when a problem combines translation with rotation.
What this chapter covers
- 01
Impulse and momentum change
- 02
Average collision force
- 03
Torque about a chosen axis
- 04
Lever arm and force angle
- 05
Moment of inertia
- 06
Rotational kinetic energy
Worked example · free
Find average force during a rebound
- 1Choose right as positive and write final velocity as minus 1.5 m/s.
- 1Calculate momentum change as minus 468 kg m/s.
- 1Divide by 0.18 s to obtain minus 2600 N.
- 1Report 2.6 kN to the left.
Key terms
- Momentum
- Momentum is mass multiplied by velocity and therefore carries direction.
- Average Force
- Average force is the constant force that would deliver the same impulse over the stated duration.
- Lever Arm
- A lever arm is the perpendicular distance from an axis to a force's line of action.
- Rotational Energy
- Rotational energy is one half moment of inertia multiplied by angular speed squared.
- Angular Momentum
- Angular momentum is the rotational quantity changed by external torque accumulated over time.
Impulse, Torque and Rotation FAQ
Why is rebound impulse larger than stopping impulse?
Stopping changes momentum from its initial value to zero. Rebounding continues the change into the opposite direction, so final minus initial contains both speed magnitudes when the rebound velocity has the opposite sign. A signed velocity line makes that additional change visible before any multiplication by mass.
Can equal forces produce different torques?
Yes. Torque depends on distance from the axis and the perpendicular component of force. The same force produces more torque farther from the axis and zero torque if its line of action passes through the axis.
Why does moment of inertia depend on the axis?
Moment of inertia weights each mass element by squared distance from the chosen axis. Changing the axis changes those distances, so the same object can have different rotational resistance about different axes.
Exam move
For impulse, choose an axis and write a signed before-and-after momentum line before calculating a magnitude. Use separate drills for stopping, rebounding, and continuing in the same direction at a different speed. A rebound must place initial and final velocities on opposite sides of zero, so final minus initial contains both speed magnitudes.
Draw a simple force-time graph and obtain impulse from signed area; then divide by duration only when the question asks for average force. State explicitly that an average does not determine the peak without the force-time shape. For torque, mark the axis, position vector, force line, and intended positive rotational sense.
Decide whether the radius-angle form or perpendicular-distance form is clearer, and check that a force through the axis gives zero. Solve paired examples where equal forces act at different distances and where equal distances use different force angles. For equilibrium, write force balance and torque balance as independent conditions rather than expecting one to imply the other.
Move the torque axis to eliminate an unknown reaction and explain why this changes individual torques but not the physical equilibrium. Keep moment-of-inertia formulas symbolic until cancellation is complete, and name the axis attached to every inertia value. Compare equal-mass shapes to see why spreading mass farther from the axis increases rotational resistance.
Check squared angular-speed scaling by halving or doubling angular speed mentally. When energy changes, verify the sign of angular acceleration relative to angular velocity. Finish mixed problems by identifying whether the controlling principle is linear impulse, angular impulse, torque balance, rotational energy, or a combination, and keep the corresponding vector or scalar quantities distinct.
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