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Circular Motion & Centripetal Force
Period, Frequency, Speed, and Center-Directed Acceleration
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Circular motion happens whenever an object moves along a curved path at a fixed distance from a center. Even if the speed stays constant, the velocity is still changing because its direction keeps changing. That means the object is accelerating, and a net force must act on it. This idea explains the motion of planets, cars turning on roads, and objects tied to strings.
The inward force that keeps an object moving in a circle is called centripetal force. It always points toward the center of the circular path, while the velocity points tangent to the path. The required centripetal acceleration is a_c = v^2/r, so tighter turns or higher speeds need more inward force. In real situations, this force can come from tension, gravity, friction, or the normal force depending on the system.
Key Facts
- Centripetal acceleration: a_c = v^2/r
- Centripetal force: F_c = mv^2/r
- Using angular speed: v = rω
- Centripetal acceleration in angular form: a_c = rω^2
- Velocity in circular motion is always tangent to the path, while centripetal force points toward the center
- If the inward force disappears, the object moves in a straight line tangent to the circle
Vocabulary
- Circular motion
- Motion in which an object travels along a circular path around a fixed center.
- Centripetal force
- The net inward force that keeps an object moving in a circle.
- Centripetal acceleration
- The inward acceleration of an object in circular motion caused by the continuous change in velocity direction.
- Tangential velocity
- The instantaneous velocity of an object in circular motion, directed tangent to the circle.
- Radius
- The distance from the center of the circular path to the moving object.
Common Mistakes to Avoid
- Thinking centripetal force is a new separate force, when it is actually the name for the net inward force provided by tension, gravity, friction, or another real force. You must identify the actual physical source of the inward force in each problem.
- Drawing velocity toward the center, which is wrong because velocity is tangent to the circular path. The inward direction belongs to centripetal acceleration and centripetal force, not velocity.
- Using F = mv/r instead of F = mv^2/r, which misses the square on speed. This gives the wrong units and seriously underestimates the needed inward force.
- Assuming zero acceleration when speed is constant, which is wrong in circular motion because the direction of velocity is changing. Constant speed does not mean constant velocity.
Practice Questions
- 1 A 2.0 kg ball moves in a horizontal circle of radius 0.50 m at a speed of 4.0 m/s. Find its centripetal acceleration and the required centripetal force.
- 2 A car of mass 1200 kg rounds a flat curve of radius 60 m at 15 m/s. What centripetal force is needed to keep the car on the curve?
- 3 A ball on a string is whirled in a circle and the string suddenly breaks. Describe the direction the ball moves immediately after the break and explain why.