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Understanding Uniform Circular Motion

This chapter explains the concept of uniform circular motion, including definitions, equations, and examples. Learn about centripetal acceleration, tangential velocity, and how to calculate the period and frequency of circular motion.

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Understanding Uniform Circular Motion

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  1. Chapter 4 Motion in two and three dimensions

  2. 4.7: Uniform Circular Motion

  3. Circular Motion - Definition Path of Motion – Circle. Uniform Circular Motion (UCM) – constant speed v. r: radius of circle Period T (s): time for a complete rotation v If n rotations happen in time t: T=t/n. v Frequency f (s-1, Hertz-Hz): number of rotations per unit of time - f=n/t. r v v T=1/f, f=1/T, Tf=1

  4. Tangential (Linear) Velocity C = 2πr = vT or

  5. Exercises Set 1 1. A car’s tire rotates at 1200 RPM. a. What is the frequency? f = 1200/60s = 20 Hz b. What is the period? T= 1/f = 1/20 Hz = 0.05s c. What is the speed (R= 0.15 m) v = 2πRf = 2 x 3.14 x 0.15m x 20 Hz = 18.85 m/s

  6. Ex2 Problem: The earth is 1.50 x1011 m (93 million miles) from the sun. What is its speed in m/s (neglecting the motion of the sun through the galaxy)? sec

  7. Angular Measure The position of an object can be described using polar coordinates—r and θ—rather than x and y. The figure at left gives the conversion between the two descriptions.

  8. Uniform Circular Motion and Centripetal Acceleration A careful look at the change in the velocity vector of an object moving in a circle at constant speed shows that the acceleration is toward the center of the circle.

  9. 4.7: Uniform Circular Motion As the direction of the velocity of the particle changes, there is an acceleration!!! CENTRIPETAL (center-seeking) ACCELERATION Here v is the speed of the particle and r is the radius of the circle.

  10. 4.7.1. A ball is whirled on the end of a string in a horizontal circle of radius R at constant speed v. By which one of the following means can the centripetal acceleration of the ball be increased by a factor of two? a) Keep the radius fixed and increase the period by a factor of two. b) Keep the radius fixed and decrease the period by a factor of two. c) Keep the speed fixed and increase the radius by a factor of two. d) Keep the speed fixed and decrease the radius by a factor of two. e) Keep the radius fixed and increase the speed by a factor of two.

  11. 4.7.1. A ball is whirled on the end of a string in a horizontal circle of radius R at constant speed v. By which one of the following means can the centripetal acceleration of the ball be increased by a factor of two? a) Keep the radius fixed and increase the period by a factor of two. b) Keep the radius fixed and decrease the period by a factor of two. c) Keep the speed fixed and increase the radius by a factor of two. d) Keep the speed fixed and decrease the radius by a factor of two. e) Keep the radius fixed and increase the speed by a factor of two.

  12. Exercises Set 4 2) A car is traveling along a circular path of 50 m radius at 22 m/s. a. What is the car’s acceleration? b. How much time to complete a circuit?

  13. Example Set 5 3) Satellite, radius 1.3 x 107 m, g = 2.5 m/s2. a) What is the speed? b) Period?

  14. Ex. 6 n = 0.56 x 60 = 33.44 rev/min

  15. 4.7: Centripetal acceleration, proof of a = v2/r Note

  16. Sample problem, top gun pilots Because we do not know radius R, let’s solve for R from the period equation for R and substitute into the acceleration eqn. Speed v here is the (constant) magnitude of the velocity during the turning. To find the period T of the motion, first note that the final velocity is the reverse of the initial velocity. This means the aircraft leaves on the opposite side of the circle from the initial point and must have completed half a circle in the given 24.0 s. Thus a full circle would have taken T 48.0 s. Substituting these values into our equation for a, we find We assume the turn is made with uniform circular motion. Then the pilot’s acceleration is centripetal and has magnitude a given by a =v2/R. Also, the time required to complete a full circle is the period given by T =2pR/v

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