AP Physics C: Mechanics Questions on Non-Uniform Circular Motion

📚 Quick Study Guide

• 📐 Angular Velocity ($\omega$): The rate of change of angular displacement. Measured in rad/s. Formula: $\omega = \frac{d\theta}{dt}$
• ⏱️ Angular Acceleration ($\alpha$): The rate of change of angular velocity. Measured in rad/s². Formula: $\alpha = \frac{d\omega}{dt}$
• ↔️ Tangential Velocity ($v_t$): The linear speed of a point on a rotating object. Formula: $v_t = r\omega$
• 🎢 Tangential Acceleration ($a_t$): The linear acceleration of a point on a rotating object. Formula: $a_t = r\alpha$
• towards_the_center Centripetal Acceleration ($a_c$): Acceleration directed towards the center of the circle, responsible for changing the direction of velocity. Formula: $a_c = \frac{v^2}{r} = r\omega^2$
• 💪 Net Force: In non-uniform circular motion, the net force has both radial (centripetal) and tangential components. $\vec{F}_{net} = \vec{F}_c + \vec{F}_t$
• 💡 Key Concept: The tangential acceleration is responsible for changing the speed of the object, while centripetal acceleration changes the direction. In non-uniform motion, both exist and may vary with time.

Practice Quiz

1. A particle moves in a circle of radius $r$ with a speed that increases at a constant rate. At a certain instant, the acceleration makes an angle of $45^\circ$ with the radius. What is the tangential acceleration at that instant?
   1. $a_t = a_c$
   2. $a_t = \sqrt{2} a_c$
   3. $a_t = \frac{a_c}{\sqrt{2}}$
   4. $a_t = 2a_c$
2. A mass $m$ is attached to a string and whirled in a vertical circle. The speed of the mass is not constant. Which of the following statements is true at the bottom of the circle?
   1. The tension in the string is equal to $mg$.
   2. The tension in the string is greater than $mg$.
   3. The tension in the string is less than $mg$.
   4. The tension in the string is equal to zero.
3. A roller coaster car is moving through a vertical loop. At what point in the loop is the normal force on the car the smallest?
   1. At the bottom of the loop
   2. At the top of the loop
   3. Halfway up the loop
   4. The normal force is constant throughout the loop
4. An object is moving in a circular path with increasing speed. Which of the following vectors best represents the direction of the net force on the object?
   1. A vector pointing towards the center of the circle.
   2. A vector tangent to the circle in the direction of motion.
   3. A vector pointing towards the center with a tangential component in the direction of motion.
   4. A vector pointing away from the center of the circle.
5. A particle is moving in a non-uniform circular motion. If the angular velocity is given by $\omega(t) = 2t^3 + t$, what is the angular acceleration at $t=2$ s?
   1. 12 rad/s²
   2. 24 rad/s²
   3. 48 rad/s²
   4. 50 rad/s²
6. A small block is placed on a rotating turntable. The turntable starts from rest and accelerates with a constant angular acceleration. Which of the following will increase the tangential acceleration of the block?
   1. Decreasing the angular acceleration.
   2. Decreasing the radius of the block from the center.
   3. Increasing the mass of the block.
   4. Increasing the angular acceleration.
7. A car accelerates uniformly from rest on a circular track. What happens to the centripetal acceleration as the car's speed increases?
   1. It decreases linearly.
   2. It remains constant.
   3. It increases linearly.
   4. It increases quadratically.