Lesson 1: Rolling Without Slipping: The Condition (v = rω)

1. What does 'rolling without slipping' actually mean in simple terms?

It means the object acts like a gear on a track. The surface of the wheel does not slide or "rub" against the ground. For every inch the wheel rotates, the object moves exactly one inch forward. Think of Velcro: the point touching the ground "sticks" momentarily, allowing the rest of the wheel to pivot over it.

2. If a wheel is rolling without slipping, is the bottom point actually stopped?

Yes. This is the hardest concept to visualize. At the specific instant the bottom of the tire touches the road, its velocity relative to the ground is zero.

• Why: The center moves forward at velocity v. The bottom of the wheel spins backward at velocity v (due to rotation).
• Result: Forward v + Backward v = 0. This is why static friction (not kinetic) acts on rolling wheels.

3. "How do I convert linear velocity (v) to angular velocity (ω) correctly?"

• Use the "Bridge Equation": v = rω.
• The Trap: Ensure r is in meters (not diameter!) and ω is in radians per second (not RPM).
• Tip: If you are given diameter, immediately divide by 2. If given RPM, multiply by 2π/60.

4. Why is the Instantaneous Axis of Rotation located at the contact point?

Because at that exact moment, the entire wheel is effectively pivoting around that stationary bottom point (like a pole vaulter pivoting on the planted pole).

The Cheat Code: You can calculate Kinetic Energy easily using K = ½ I(point) · ω² (using the Moment of Inertia at the contact point) instead of summing translational and rotational energies separately.

5. Does the condition v = r · ω apply if the wheel is slipping or skidding?

No.

• Slipping (Burnout): The wheel spins too fast for the motion (v < rω).
• Skidding (Braking on ice): The wheel moves but doesn't spin enough (v > rω).
• The equation v = rω is a constraint—it is only true when there is perfect traction.

Lesson 2: Total Kinetic Energy: Rolling Down an Incline

6. Why do we need a separate formula for Rotational Kinetic Energy?

Because energy is required to make the object spin, not just move. If you have a block and a ball of the same mass moving at the same speed, the ball has more total energy because it is also spinning.

1. Translation: Energy of moving from A to B (½ mv²).
2. Rotation: Energy of spinning in place (½ Iω²).

7. Does static friction do any Work on an object rolling without slipping?

No.

The Physics: Work = Force × Distance. Since the point of contact has zero velocity (it is momentarily stationary), the distance it moves while the force is applied is zero. Therefore, static friction forces the object to roll, but it does not take energy away from the system (unlike kinetic friction, which generates heat).

8. How do I set up the Total Kinetic Energy equation for a rolling object?

• Add them together: K_total = K_trans + K_rot K_total = ½ mv² + ½ Iω²
• Pro Tip: Substitute ω = v/r immediately to solve for velocity.

9. Sphere vs. Cylinder vs. Hoop: Which one reaches the bottom of the ramp first?

• The object with the smallest Moment of Inertia (I) wins.
• The Winner: The Solid Sphere.
• The Why: A Hoop (I = mr²) puts a lot of its energy into spinning (because the mass is far from the center), leaving less energy for translational speed. The Sphere (I = 2/5 mr²) is easy to spin, so it uses more energy for linear speed.

Lesson 3: What is Torque? The Key to Understanding Rotational Motion

10. What is the actual difference between Torque (τ) and Force (F)?

• Force tries to move an object in a line (push/pull).
• Torque tries to twist an object around an axis.

Analogy: Force is how hard you press the door handle; Torque is how effective that push is at opening the door (based on how far you are from the hinge).

11. If the Net Force on an object is zero, does that mean the Net Torque is zero too?

No.

Example: A steering wheel. You pull down with the left hand and push up with the right hand. The forces cancel (Net Force = 0, so the wheel doesn't fly off the column), but the Torques add up, causing it to spin. This is called a "Force Couple."

12. How do I know when to use sin(θ) or cos(θ) for the Torque equation?

• Always use sin(θ) if θ is the angle between the position vector (r) and the Force vector (F).
• The Rule: Torque cares about the perpendicular force.
• τ = r · F · sin(θ)
• If you push straight at the hinge (parallel), sin(0) = 0, so Torque is zero.

13. How do I solve the 'Ladder leaning against a wall' equilibrium problem?

This is a 3-step system:

1. Sum of Forces (x): Friction at floor = Normal force from wall.
2. Sum of Forces (y): Normal force from floor = Gravity (mg).
3. Sum of Torques: Choose the pivot at the bottom of the ladder (this cancels out friction and normal force there) and set Clockwise Torques = Counter-Clockwise Torques.

14. Why is Torque considered a vector? Why does it point along the axis?

It follows the Right Hand Rule. If you curl your fingers in the direction of the spin, your thumb points along the axis. This vector direction tells us the axis of the rotation, not the direction the object moves.

Lesson 4: Newton’s Second Law for Rotation

Focus: Dynamics, Moment of Inertia, and massive pulleys.

15. What is Moment of Inertia (I) and how is it different from just Mass?

• Mass tells you how hard it is to push something. Moment of Inertia tells you how hard it is to spin something.
• It depends on mass AND location. Mass further from the axis (like a rim) is much harder to spin than mass near the axis (like a coin).

16. Which direction does friction point when a ball rolls down a ramp?

Up the ramp.

Reasoning: To roll down, the ball must start rotating forward (clockwise). To cause a clockwise spin, you need a force that creates torque in that direction. Friction acts at the bottom surface; pointing it up the ramp creates that necessary twist around the center.

17. Which 'Moment of Inertia' formula do I use for weird shapes?

Most exams provide them, but memorize these three tiers:

1. Hoop/Point Mass: I = MR² (Hardest to spin)
2. Cylinder/Disk: I = ½ MR²
3. Solid Sphere: I = 2/5 MR² (Easiest to spin)

18. How do I relate linear acceleration 'a' to angular acceleration 'α'?

Use the Bridge Equation: a = rα.

This is the key to connecting Newton's Law (F = ma) with the Torque Law (τ = Iα). You usually solve one equation for α, substitute it into the other using a/r, and solve.

19. How do I handle a pulley problem if the pulley has mass and is not massless?

Treat the pulley as a separate object with its own torque equation.

The Change: The tension in the rope is not the same on both sides (T1 ≠ T2). The difference in tension creates the torque that spins the pulley: (T2 - T1) · r = I · α

20. How does the Parallel Axis Theorem actually work and when should I avoid using it?

• It allows you to find I for a rotation point that isn't the center.
• Formula: I_new = I_cm + Md² (where d is the distance from the center of mass to the new pivot).
• Constraint: You can ONLY use it if the axes are parallel, and one of them is the Center of Mass. You cannot jump from "Pivot A" to "Pivot B" directly; you must go via the Center of Mass.

Lesson 5: Angular Momentum (L = Iω)

Focus: Defining the quantity of rotation.

21. What is Angular Momentum (L) in simple English?

It is the "quantity of rotation" or the "oomph" of the spin. It tells you how difficult it would be to stop the object from spinning. Just as a semi-truck has more momentum than a bike at the same speed, a spinning merry-go-round has more Angular Momentum than a spinning coin.

22. If an object is moving in a straight line, can it still have Angular Momentum?

Yes. This is confusing!

Angular Momentum is always measured relative to a pivot point.

If a car drives past you, it has angular momentum relative to you because its position angle is changing.

Formula: L = mvr sin(θ).

Lesson 6: Conservation of Angular Momentum

23. A figure skater pulls their arms in: How do I calculate the new speed?

• Use Conservation of Angular Momentum: L_initial = L_final. I₁ω₁ = I₂ω₂
• When arms come in, Moment of Inertia (I) decreases (mass is closer to center). To keep L constant, Angular Velocity (ω) must increase.

24. A bullet hits a pivoted rod and sticks: Is Energy or Momentum conserved?

• Kinetic Energy: NOT Conserved. It's an inelastic collision (heat/sound loss).
• Linear Momentum: NOT Conserved. The pivot exerts an external force to hold the rod in place.

Angular Momentum: CONSERVED. The pivot force is at the axis, so it causes zero torque. Zero external torque means Angular Momentum is conserved.

25. Why is Angular Momentum conserved in the skater example if she does work pulling her arms in?

• Conservation: L is conserved because no external torque (like someone pushing her) is acting.
• Energy: Kinetic Energy actually increases. The work she does with her muscles to pull her arms inward against the centrifugal force is converted into extra rotational kinetic energy.