PHYS2001_Ch. 5

PHYS2001_Ch. 5 - Ch. 5 Uniform Circular Motion An object...

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Ch. 5 – Uniform Circular Motion An object traveling at constant speed on a circular path is undergoing Uniform Circular Motion (UCM). Example : Spinning a ball on a string. Since the speed is constant in UCM, it’s sometimes useful to talk about the period of the motion, which is the time it takes to complete 1 revolution. r v The distance of one revolution is just the circumference of the circle: 2 π r v r t 2 velocity distance = = This time is the period , T : v r T 2 = Notice, units are [ s ].
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Example : How long does it take a plane traveling at a constant speed of 110 m/s to fly once around a circle with a radius of 2850 m? Once around is just the period: v r T π 2 = m/s 110 m) 2850 ( 2 = min. 2.72 s 163 = = Look at UCM from above again. r v The speed is constant, but is the velocity constant??? NO!!! The velocity changes direction as the ball moves around in a circle. Since the velocity is changing direction, the ball must be accelerating! What is the direction of the acceleration? D It points inward , toward the center of the motion! a c This acceleration is called the Centripetal Acceleration : a c
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How do we calculate the centripetal acceleration? We calculate it the same way we would for any acceleration. We need to find the change in the velocity of the ball over time. Let’s see how the velocity of the ball changes as it moves from one point to another along the circle: r θ v t t f What is the change in the velocity, v = v f - v o ? v v v v v v Notice that the direction of v is in toward the center of the circular motion. Now look at the section of arc completed by the ball as it moves for a time t = t f - t o s r r r s = s But in the limit that t → 0 , s = s r s = ' vt = So I end up with two similar triangles: r r vt v v v r vt v v =
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r v t v 2 = Rearrange this, and we get: But this is just the acceleration! r v a c 2 = a c is the Centripetal Acceleration Since there is an acceleration, Newton’s 2 nd Law tells us there must be a force. What direction would this force point??? It would point in the same direction as the acceleration, thus the force pulls in toward the center of the circular motion. This is called the Centripetal Force: c c ma F = r mv F c 2 =
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Consider the ball spinning on a string again. There is no force pulling the ball to the outside. The only forces acting on the ball are its weight and the tension force. W T If the tension is made large enough, then the weight becomes negligible. The sensation you have of being “thrown to the outside” when your body undergoes circular motion is a “centrifugal force effect”. They are not real forces, but fictitious. When you move in a circle, you are accelerating, and thus you are not in an inertial reference frame .
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PHYS2001_Ch. 5 - Ch. 5 Uniform Circular Motion An object...

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