#AP Physics 1: Rolling Motion - Your Last-Minute Guide 🚀

Hey there! Let's get you prepped for the AP Physics 1 exam with a super-focused review of rolling motion. We'll break it down, connect the dots, and make sure you're feeling confident and ready to roll (pun intended!).

#Introduction to Rolling Motion

Rolling is a combination of translational (moving from one place to another) and rotational (spinning) motion. It's super important because it shows up everywhere – think wheels, balls, and anything that spins as it moves. We'll look at two main types: rolling without slipping and rolling with slipping.

• Kinetic Energy
• Rolling Without Slipping
• Rolling With Slipping

# Kinetic Energy in Rolling Motion

#Total Kinetic Energy

It’s like having two types of energy happening at once! 🔄

• Translational Kinetic Energy ($K_{trans}$): Energy due to the motion of the object's center of mass. Depends on its mass and velocity.

• Rotational Kinetic Energy ($K_{rot}$): Energy due to the object's spinning. Depends on its moment of inertia and angular velocity.

• Formula: $$K_{tot} = K_{trans} + K_{rot}$$

  • $K_{tot}$ = Total kinetic energy
  • $K_{trans}$ = Translational kinetic energy
  • $K_{rot}$ = Rotational kinetic energy

# Rolling Without Slipping

#Relationship Between Linear and Angular Motion

Think of it like a perfectly synchronized dance! 🎡

• Linear Acceleration ($a_{cm}$) and Angular Acceleration ($alpha$): $$a_{cm} = r\alpha$$
• Linear Velocity ($v_{cm}$) and Angular Velocity ($omega$): $$v_{cm} = r\omega$$
• Linear Displacement ($Delta x_{cm}$) and Angular Displacement ($Delta\theta$): $$Delta x_{cm} = r\Delta\theta$$

Where:

• $r$ is the radius of the rolling object.

#Friction in Ideal Rolling

• In ideal rolling (no slipping), static friction is the key player. It provides the necessary force for rolling without losing energy. 🔋
• The contact point between the object and the surface is momentarily at rest, so static friction doesn't do any work and doesn't dissipate energy.

# Rolling With Slipping

#Decoupled Motion

• When an object slips while rolling, its linear and rotational motions are no longer directly related. They're doing their own thing! 😫
• The simple equations we used for rolling without slipping (like $v = r\omega$) don't apply here.

#Energy Dissipation

This means the object slows down and/or stops rotating faster than it would without slipping. 🔥

• The contact point moves relative to the surface, and kinetic friction does negative work.
• This reduces the total kinetic energy of the system (both translational and rotational).

#Final Exam Focus

• High-Priority Topics: Rolling without slipping, kinetic energy calculations (both translational and rotational), and the effects of friction (static vs. kinetic).
• Common Question Types: Analyzing rolling motion on inclined planes, calculating total kinetic energy, and describing changes in motion when slipping occurs.
• Time Management: Quickly identify whether slipping is involved. If not, use the simplified equations ($v = r\omega$, etc.). If slipping is involved, focus on energy dissipation due to kinetic friction.
• Common Pitfalls: Confusing static and kinetic friction, not considering both translational and rotational kinetic energy, and incorrectly applying the rolling without slipping equations when slipping is present.

#

Practice Question

Practice Questions

#Multiple Choice Questions

1. A solid cylinder and a hollow cylinder, both with the same mass and radius, are released from rest at the top of an incline. Which one will reach the bottom first, assuming they roll without slipping? (A) The solid cylinder (B) The hollow cylinder (C) They will reach at the same time (D) It depends on the angle of the incline

2. A ball is rolling on a horizontal surface without slipping. Which of the following statements is true about the frictional force acting on the ball? (A) The frictional force is kinetic and does negative work on the ball. (B) The frictional force is static and does no work on the ball. (C) The frictional force is kinetic and does positive work on the ball. (D) The frictional force is zero.

3. A wheel is rolling down an incline without slipping. What happens to its translational kinetic energy and rotational kinetic energy as it goes down? (A) Both translational and rotational kinetic energy increase. (B) Translational kinetic energy increases, and rotational kinetic energy decreases. (C) Translational kinetic energy decreases, and rotational kinetic energy increases. (D) Both translational and rotational kinetic energy decrease.

#Free Response Question

A solid sphere of mass $m$ and radius $r$ is released from rest at the top of an inclined plane of height $h$ and angle $\theta$. The sphere rolls down the incline without slipping. The moment of inertia of a solid sphere is $I = (2/5)mr^2$. Assume no air resistance.

(a) Derive an expression for the total kinetic energy of the sphere at the bottom of the incline in terms of $m$, $v$, and $I$. (b) Derive an expression for the speed of the center of mass of the sphere at the bottom of the incline in terms of $g$ and $h$. (c) If the sphere now rolls down the incline with slipping, will its speed at the bottom be greater than, less than, or the same as when it rolled without slipping? Explain your reasoning.

Scoring Breakdown:

(a) (2 points) * 1 point for correctly stating the total kinetic energy as the sum of translational and rotational kinetic energy: $K_{tot} = K_{trans} + K_{rot}$ * 1 point for correctly substituting the formulas for translational and rotational kinetic energy: $K_{tot} = (1/2)mv^2 + (1/2)I\omega^2$

(b) (3 points) * 1 point for recognizing that the potential energy at the top is converted to kinetic energy at the bottom: $mgh = K_{tot}$ * 1 point for using the relationship $v = r\omega$ to eliminate $\omega$ and substituting $I = (2/5)mr^2$: $mgh = (1/2)mv^2 + (1/2)(2/5)mr^2(v/r)^2$ * 1 point for solving for $v$: $v = \sqrt{(10/7)gh}$

(c) (2 points) * 1 point for stating that the speed will be less than when it rolled without slipping. * 1 point for explaining that kinetic friction does negative work, dissipating energy, and thus reducing the final speed.