Which point of a wheel rolling without slipping is momentarily stationary against the ground?

The point touching the ground. Forward motion and rotation cancel there.

Where between two unequal masses does their centre of mass lie?

Nearer the heavier mass. The heavier mass contributes more to the average position.

If a × b points upward, where does b × a point?

Downward. Reversing the order reverses direction without changing magnitude.

What happens to this particle’s angular momentum about O under the upward net force?

Its outward component increases. The net torque points outward, increasing angular momentum in that direction.

A spoon’s weight and support force are equal and opposite, but their vertical lines differ. Will it remain balanced?

No. The forces cancel, but their turning effects do not.

How does each particle’s contribution change when its distance from the axis doubles?

It becomes four times larger. Distance is squared, while the particle’s mass stays unchanged.

Two doors have different moments of inertia. Equal nonzero net torques act about their hinges. Which has the greater angular acceleration magnitude?

The door with lower inertia. The same torque causes greater angular acceleration when moment of inertia is smaller.

With negligible outside torque, what happens to spin speed when the skater pulls inward?

It increases. Smaller moment of inertia requires faster spin to conserve angular momentum.

Teach Systems of Particles and Rotational Motion in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather a steel spoon, paper and a pencil. Use a household door.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

Rolling wheels move forward while their points move at different speeds.
A flywheel reduces changes in a machine’s spin rate.
A skater spins faster by pulling the arms inward.

What to teach

How to teach it

Describing motion

Wheel velocity arrows show translation and rotation combining into rolling without slipping.

1. Translation, rotation and rolling

A rigid body keeps distances between its points unchanged.

Show: Look at the diagram. Here, v means the centre’s ground speed. Compare the bottom arrows.
Ask: How will the rolling wheel’s contact point move against the ground?
Say: The bottom velocities cancel, leaving the contact point momentarily still.
Used in: Rolling wheels carry loads with less rubbing than sliding would.

If your child says…“A solid object’s points must all move alike.”

Ask: Do the arrows at the top and bottom combine in the same way?
What’s true: Keeping distances fixed allows different points to have different velocities.

Unequal masses tumble while their centre of mass follows a curved path.

2. Centre of mass and its motion

Mass distribution locates it; net external force changes its velocity.

Show: Find CM in the diagram. Follow its path below; Mg means total weight.
Ask: Will tumbling make the centre of mass zigzag?
Say: No. It lies nearer the heavier mass. Ignoring air resistance, gravity makes its path a parabola.
Used in: Tracking the centre of mass simplifies predictions for tumbling objects.

If your child says…“The centre of mass is always halfway.”

Ask: Is the marked centre halfway along, or nearer the heavier particle?
What’s true: For two unequal masses, it lies nearer the heavier mass.

Vector arrows show opposite cross-product directions when their order reverses.

3. Vector product and angular velocity

Cross products give perpendicular directions; angular velocity describes turning.

Show: Look at arrows A and B in the diagram. Curl your right fingers from A towards B, then reverse. Compare the out and in symbols.
Ask: When you curl from B towards A, where will your right thumb point?
Say: Reversing the order reverses the cross product’s perpendicular direction.
Used in: Cross products help engineers find torque directions when designing rotating shafts.

If your child says…“Swapping vectors changes nothing.”

Ask: Where does your right thumb point after you reverse the order?
What’s true: Swapping the vectors reverses the cross product.

Turning and balance

Position and upward momentum give outward angular momentum; upward force gives outward torque.

4. Torque and angular momentum

Torque changes angular momentum about the same reference point.

Show: In the diagram, follow position r, then momentum p. L means angular momentum. Compare the force.
Ask: What will this force change about L?
Say: Position crossed with momentum gives outward angular momentum. The upward force produces outward torque, increasing that angular momentum.
Used in: A lever’s longer effort arm reduces the force needed to lift its load.

If your child says…“Straight-line motion has no angular momentum.”

Ask: Does the particle’s upward path pass through O?
What’s true: A particle moving past an origin has angular momentum when its path misses that origin.

A steel spoon rests across one extended finger.

5. Equilibrium and centre of gravity

Balance requires both forces and their turning effects to cancel.

Show: Rest a steel spoon across one finger. After the prediction, shift your supporting finger until the spoon balances.
Ask: Will balancing the upward and downward forces alone prevent tipping?
Say: For balance, the forces and their turning effects must cancel.
Used in: A car’s axles support different loads when its centre of gravity is off-centre.

If your child says…“If forces cancel, everything balances.”

Ask: What happens if your finger supports the spoon away from its balance point?
What’s true: Equal opposite forces can still turn an object when their lines differ.

$I_1=2(1)(0.20)^2=0.08\,\mathrm{kg\,m^2}$

$I_2=2(1)(0.40)^2=0.32\,\mathrm{kg\,m^2}$

6. Moment of inertia

Mass farther from the chosen axis resists spin changes more.

Show: Write these lines for two 1 kg particles. Say: I₁ and I₂ are their combined inertias before and after doubling perpendicular distances.
Ask: What is I₂ divided by I₁?
Say: The ratio is four because distance is squared.
Used in: A flywheel’s large moment of inertia reduces changes in spin rate.

If your child says…“Twice the distance means twice the inertia.”

Ask: What happens when you square the doubled distance?
What’s true: Doubling the distance makes that mass’s contribution four times larger.

Changing spin

A slightly open household door with its hinges and handle visible.

7. Rotational kinematics and dynamics

Angular acceleration measures spin changes caused by net torque.

Show: Open a door slightly. Keep fingers away from the hinges and the gap between the door and frame. Push squarely halfway between hinges and handle. Repeat at the handle from the same opening, pushing equally gently. An adult catches each swing.
Ask: Where will the same push speed it up faster?
Say: At the handle, because farther pushes give more torque.
Used in: An engine supplies power against friction to keep a shaft spinning steadily.

If your child says…“To find angular acceleration, divide torque by mass.”

Ask: Would moving the door’s mass closer to its hinges change how easily it turns?
What’s true: Divide net torque by moment of inertia, which includes mass distribution around the axis.

Skater mass distributions show inward movement and faster spin with unchanged angular momentum.

8. Conservation of angular momentum

Redistributing mass can change spin speed while angular momentum stays constant.

Show: Compare the arm positions and spin arrows in the diagram.
Ask: How can pulling inward increase spin speed?
Say: With negligible outside torque, angular momentum stays constant. Pulling inward reduces inertia, increasing spin speed.
Used in: A skater pulls inward to spin faster without needing an external torque.

If your child says…“Without an outside torque, spin speed cannot change.”

Ask: If moment of inertia decreases, can unchanged spin speed keep their product constant?
What’s true: With negligible outside torque, angular momentum stays constant even when inward movement increases spin speed.

A real problem

Spin helped stabilise Aryabhata’s orientation.

Aryabhata launched on 19 April 1975.

Its scientific work included studying cosmic X-rays, the upper atmosphere and the Sun.

It spun about its axis of greatest moment of inertia.

ISRO NASA

A wooden spinning top rotates upright on a tabletop.

Concept Lab · Common misconception

The tempting mistake in Systems of Particles and Rotational Motion

Open standalone misconception page →

Does force alone decide how easily a door turns?

Wrong

The same push turns the door equally well anywhere.

Why it feels right. I focus on how hard I push.

Right

Turning effect also depends on the perpendicular lever arm.

Worked example

Push 0.2 m from hinge2 N·m

Push 0.8 m from hinge8 N·m

The force is 10 N in both trials. Four times the lever arm gives four times the torque.

A perpendicular 10 N push 0.8 m from the hinge gives 8 N·m of torque.
At 0.2 m it gives 2 N·m.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

Which point of a wheel rolling without slipping is momentarily stationary against the ground?

The point touching the ground. Forward motion and rotation cancel there.
Where between two unequal masses does their centre of mass lie?

Nearer the heavier mass. The heavier mass contributes more to the average position.
If a × b points upward, where does b × a point?

Downward. Reversing the order reverses direction without changing magnitude.
What happens to this particle’s angular momentum about O under the upward net force?

Its outward component increases. The net torque points outward, increasing angular momentum in that direction.
A spoon’s weight and support force are equal and opposite, but their vertical lines differ. Will it remain balanced?

No. The forces cancel, but their turning effects do not.
How does each particle’s contribution change when its distance from the axis doubles?

It becomes four times larger. Distance is squared, while the particle’s mass stays unchanged.
Two doors have different moments of inertia. Equal nonzero net torques act about their hinges. Which has the greater angular acceleration magnitude?

The door with lower inertia. The same torque causes greater angular acceleration when moment of inertia is smaller.
With negligible outside torque, what happens to spin speed when the skater pulls inward?

It increases. Smaller moment of inertia requires faster spin to conserve angular momentum.

15 more questions with answers, 3 with pictures

Teach Systems of Particles and Rotational Motion in eight steps.

Why it matters