AP Physics – Uniform Circular Motion and Centripetal Acceleration

AP Physics – Uniform Circular Motion and Centripetal Acceleration

AP Physics: Uniform Circular Motion & Centripetal Acceleration – Exam-Ready Study Guide

What This Is

Uniform circular motion (UCM) describes an object moving at a constant speed along a circular path. Even though the speed is constant, the direction of the velocity changes continuously, meaning the object is accelerating—this is called centripetal acceleration. This concept is crucial on the AP exam because it connects kinematics, forces, and Newton’s laws. Real-world example: A car turning around a roundabout at a steady 20 mph is in UCM—its speed doesn’t change, but its velocity (direction) does, requiring a net force (friction from the road) to keep it moving in a circle.

Key Terms & Concepts

• Uniform Circular Motion (UCM): Motion at constant speed along a circular path. Speed is constant, but velocity changes direction.
• Period (T): Time for one complete revolution (units: seconds). T = 1/f, where f = frequency (revolutions per second).
• Frequency (f): Number of revolutions per second (units: Hz or s?¹). f = 1/T.
• Centripetal Acceleration (a?): Acceleration directed toward the center of the circle, causing the change in velocity direction. a? = v²/r, where v = linear speed, r = radius.
• Centripetal Force (F?): The net force causing centripetal acceleration. F? = m·a? = m·v²/r (Newton’s 2nd Law). Not a new type of force—it’s the resultant of forces like tension, friction, or gravity.
• Linear Speed (v): Distance traveled per unit time along the circular path. v = 2?r/T or v = 2?rf.
• Tangential Velocity: The velocity vector at any point on the circle, tangent to the path (perpendicular to the radius).
• Centrifugal "Force": Not a real force! It’s the apparent outward force felt in a rotating reference frame (e.g., you sliding outward in a turning car). In an inertial frame, it’s just inertia resisting the centripetal force.
• Banked Curves: Roads or tracks tilted at an angle to help provide the centripetal force needed for turning (e.g., NASCAR tracks). The normal force has a horizontal component that contributes to F?.
• Vertical Circles: Objects moving in a vertical loop (e.g., roller coasters). At the top, F? = T + mg (tension + gravity); at the bottom, F? = T – mg.

Step-by-Step: Solving UCM Problems

1. Draw a free-body diagram (FBD):
2. Identify all forces acting on the object (e.g., tension, friction, gravity, normal force).
3. Label the center of the circle and draw the centripetal acceleration (a?) toward it.
4. Resolve forces toward/away from the center:
5. Only forces along the radius contribute to F?. Ignore tangential forces (they change speed, not direction).
6. Apply Newton’s 2nd Law for circular motion:
7. ?F = m·a? = m·v²/r (toward the center).
8. Write equations for the net force in the radial direction.
9. Solve for the unknown:
10. Use kinematic equations if needed (e.g., v = 2?r/T to relate speed and period).
11. Check units and reasonableness:
12. Ensure forces are in Newtons (N), radii in meters (m), and speeds in m/s.
13. Example: A 1000-kg car turning at 10 m/s with r = 50 m should have F? = 2000 N (reasonable for friction).

Common Mistakes

• Mistake: Confusing centripetal force with a new type of force. Correction: Centripetal force is the net force causing circular motion (e.g., friction, tension, or gravity). It’s not a separate force on the FBD.

• Mistake: Using speed (v) instead of velocity (vector) in calculations. Correction: Speed is constant in UCM, but velocity changes direction. Acceleration depends on v²/r, not just v.

• Mistake: Forgetting that centripetal acceleration is always toward the center. Correction: Even if an object is at the top of a vertical circle, a? points down (toward the center).

• Mistake: Ignoring gravity in vertical circle problems. Correction: At the top of a loop, F? = T + mg (both forces point toward the center). At the bottom, F? = T – mg.

• Mistake: Misapplying F? = m·v²/r to non-circular motion. Correction: This formula only applies to uniform circular motion (constant speed, fixed radius).

AP Exam Insights

• FRQ Hotspots:
• Force analysis in vertical circles (e.g., minimum speed to stay in a loop).
• Banked curves (combining normal force and friction to find max speed).
• Comparing centripetal and tangential acceleration (e.g., a spinning disk slowing down).
• Multiple-Choice Traps:
• Centrifugal force is often included as a distractor—it’s not real in an inertial frame!
• Questions may give period (T) but ask for speed (v)—remember v = 2?r/T.
• Direction of acceleration: Always toward the center, even if the object is moving "up" in a vertical circle.
• Tricky Distinction:
• Centripetal acceleration (a?) vs. tangential acceleration (a?):
  • a? changes direction of velocity (UCM).
  • a? changes speed (non-UCM, e.g., a car speeding up around a curve).

Quick Check Questions

1. Multiple Choice: A 2-kg object moves in a circle of radius 0.5 m at 4 m/s. What is the centripetal force acting on it? (A) 4 N (B) 16 N (C) 32 N (D) 64 N Answer: (D) 64 N. F? = m·v²/r = 2·(4)²/0.5 = 64 N.

2. Short FRQ: A car travels around a banked curve (no friction) at the design speed of 20 m/s. The curve has a radius of 80 m. (a) What is the angle-of the bank? (b) If the car’s speed doubles, what happens to the centripetal force required? Answer: (a) ? = tan?¹(v²/(r·g)) = tan?¹(20²/(80·9.8))-27°. (b) F? quadruples (since F?-v²).

3. Multiple Choice: A ball on a string is swung in a vertical circle. At the top of the circle, the tension in the string is: (A) Greater than the ball’s weight. (B) Equal to the ball’s weight. (C) Less than the ball’s weight. (D) Zero. Answer: (C) Less than the ball’s weight. At the top, F? = T + mg, so T = F? – mg. If F? = mg, T = 0 (minimum speed).

Last-Minute Cram Sheet

1. Centripetal acceleration: a? = v²/r (toward center).
2. Centripetal force: F? = m·v²/r (net force toward center).
3. Linear speed: v = 2?r/T or v = 2?rf.
4. Period (T): Time for one revolution; T = 1/f.
5. Vertical circles: Top-F? = T + mg; bottom-F? = T – mg.
6. Banked curves: tan? = v²/(r·g) (no friction).
7. Centrifugal force is fake! (Only inertia in a rotating frame.)
8. Speed is constant, but velocity changes direction-acceleration exists!
9. Minimum speed for vertical loop: v = ?(r·g) (at the top, T = 0).
10. Units check: F? in N, v in m/s, r in m, T in s.