LECTURE 11CIRCULAR MOTION

Instructor: Kazumi Tolich

Lecture 11

¨ Reading chapter 6-5¤ Uniform circular motion¤ Centripetal and tangential acceleration of non-uniform circular motion¤ Dynamics of circular motion

n Centripetal forcen Unbanked and banked curvesn Loop-the-Loop

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Centripetal force and acceleration

¨ Centripetal force, 𝐹"#, is any form of force (normal force, tension, gravity, etc) applied toward the center of the circle that keeps an object in a circular motion.

¨ Centripetal acceleration, 𝑎"#, is an acceleration of an object caused by a centripetal force.

¨ The magnitude of a net centripetal force required for an object with a mass 𝑚 going around in a circular path with a radius 𝑟 with a uniform speed 𝑣 is given by

(𝐹"#

�

�

= 𝑚𝑎"# = 𝑚𝑣+

𝑟

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Velocity and acceleration of a circular motion

¨ The direction of the velocity is always tangential to the path, perpendicular to the circle’s radius.

¨ An object under a uniform circular motion moves at constant speed and has a centripetal acceleration.

¨ An object in a circular motion with varying speed has both centripetal and tangential accelerations.

𝐚-.-/0 = 𝐚"# + 𝐚-

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Quiz: 1

Why don’t satellites fall into Earth?

Q. Why don’t satellites fall into Earth because of Earth’s gravity?

A. They do.The tangential velocity of the satellite is fast enough so that the distance the satellite falls towards Earth, and the distance it travels in the tangential direction in a given time follow the satellite’s orbit.

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Example: 1

¨ Igor is an engineer in a spacecraft orbiting Earth at an altitude ℎ = 520 km with a constant speed 𝑣 = 7.6 km/s. Igor’s mass is 𝑚 = 79 kg.a) What is his acceleration?b) What (centripetal) gravitational force does

Earth exert on Igor?

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Quiz: 28

Example: 2

¨ A bicyclist travels at a constant speed of 𝑣 = 9.00 m/s in a circle of radius 𝑟 = 25.0 m on a flat ground. The combined mass of the bicycle and rider is 𝑚 = 85.0 kg.

a) Calculate the magnitude of the force of friction exerted by the road on the bicycle.

b) If the coefficient of static friction between the tires and this road is 𝜇4 = 1.0, what is the maximum speed the bicycle can go before skidding?

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Quiz: 3 & 410

Example: 3

¨ The radius of curvature of the track at the top of a loop-the-loop on a roller-coaster ride is 𝑟. The mass of the roller-coaster is 𝑚.a) Calculate the speed of the coaster, 𝑣, as a function

of the normal force on the coaster by the track, 𝑁.

b) What is the minimum speed at which the coaster must be going at the top of the loop-the-loop to barely make it?

¨ Loop-the-loop fails when you do not have enough speed at the top.¤ http://www.youtube.com/watch?v=tzQJNeqiGG4

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