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Uniform Circular Motion

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A B C Answer: B v Circular Motion ACT 1 A ball is going around in a circle attached to a string. If the string breaks at the instant shown, which path will the ball follow?

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Circular Motion

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For a particle in uniform circular motion, the velocity vector v remains constant in magnitude, but it continuously changes its direction. Uniform Circular Motion

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Acceleration in Uniform Circular Motion vv1vv1 vv2vv2 vvvv vv2vv2 vv1vv1 R RRRR centripetal acceleration a v a ave = v / t Acceleration toward center Acceleration is due to change in direction, not speed. Object turns “toward” center must be a force on object toward center.

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Centripetal Acceleration

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Centripetal Acceleration ACT Which motion has the largest centripetal acceleration?

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Preflight Consider the following situation: You are driving a car with constant speed around a horizontal circular track. On a piece of paper, draw a Free Body Diagram (FBD) for the car. How many forces are acting on the car? A) 1 B) 2 C) 3 D) 4 f W FNFN correct 16 F = ma = mv 2 /R a=v 2 /R R 0% 28% 44% 28% “Fn = Normal Force, W = Weight, the force of gravity, f = centripetal force.” “Gravity, Normal Force, Friction”

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Common Incorrect Responses Acceleration: F = ma –Centripetal Acceleration Force of Motion (Inertia not a force) –Forward Force, –Force of velocity –Speed Centrifugal Force (No such thing!) –Centripetal (really acceleration) –Inward force (really friction) Internal Forces (don’t count, cancel) –Car –Engine

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Net force may be provided by the tension in a string, the normal force, or friction, among other sources – as with any net force. Circular Motion Requires Net Force

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Preflights Consider the following situation: You are driving a car with constant speed around a horizontal circular track. On a piece of paper, draw a Free Body Diagram (FBD) for the car. The net force on the car is f W FNFN 16 A. Zero B. Pointing radially inward C. Pointing radially outward F = ma = mv 2 /R a=v 2 /R R correct 22% 67% 11% “Ever spin stuff in a bowl? It gravitates outward.” “Centripetal force is always pointing inward" “This is why many racetracks have banked roadways”

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Dip ACT Suppose you are driving through a valley whose bottom has a circular shape. If your mass is m, what is the magnitude of the normal force F N exerted on you by the car seat as you drive past the bottom of the hill A. F N mg v mg FNFN R F = ma F N - mg = mv 2 /R F N = mg + mv 2 /R a=v 2 /R correct 20

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While driving on a country road at a constant speed of 17.0 m/s, you encounter a dip in the road. The dip can be approximated by a circular arc with a radius of 65.0 m. What is the normal force exerted by the car seat on an 80.0 kg passenger at the bottom of the dip? Dip Example

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Bonnie sits on the outer rim of a merry-go-round with radius 3 meters, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every two seconds. –Klyde’s speed is: (a) (a) the same as Bonnie’s (b) (b) twice Bonnie’s (c) (c) half Bonnie’s Klyde Bonnie Bonnie travels 2 R in 2 seconds v B = 2 R / 2 = 9.42 m/s Klyde travels 2 (R/2) in 2 seconds v K = 2 (R/2) / 2 = 4.71 m/s Merry-Go-Round ACT 33

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Rounding a Corner A 1,200 kg car rounds a corner of radius r = 45.0 m. If the coefficient of friction between the tires and the road is s = 0.82, what is the maximum speed the car can have on the curve without skidding? Question: How does this result depend on the mass of the car?

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Banked Curves

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Revolving in a Circle An energetic father places his 20 kg child in a 5.0 kg cart to which is attached a 2.0 m long rope. He then holds the end of the rope and spins the cart and child in a circle, keeping the rope parallel to the ground. If the tension in the rope is 100 N, what is the cart’s tangential speed? Radial: Vertical: m = 25 kg r = 2 m T = 100 N v = ?

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50 Driving in your car with a constant speed of 12 m/s, you encounter a bump in the road that has a circular cross-section, as indicated in Figure 6–30. If the radius of curvature of the bump is 35 m, find the apparent weight of a 67-kg person in your car as you pass over the top of the bump. w N

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51 Referring to Problem 50, at what speed must you go over the bump if people in your car are to feel “weightless?”

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Centrifuge A centrifuge rotates at a rate such that the bottom of a test tube travels at a speed of 89.3 m/s. The bottom of the test tube is 8.50 cm from the axis of rotation. What is the centripetal acceleration a cp at the bottom of the test tube in m/s and in g (where 1 g = 9.81 m/s 2 )?

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