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Day 14 – June 10 – WBL Chapter 7 Circular Motion and Gravitation PC141 Intersession 2013Slide 1 Our discussion of motion up until now has been concerned entirely with translation, in which an object moves along a straight or curved line. In this chapter, we will introduce the concept of circular motion, in which an object follows a path with a constant radius of curvature. After introducing the rotational variables (angular displacement, angular velocity, etc.), we will develop equations to describe the evolution of these variables with time. Finally, we will discuss a rotational version of Newtons 2 nd law, and its application to gravitation.

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Day 14 – June 10 – WBL In order to quantify circular motion, we need to introduce new units of angular measure. Since a circle lies in a plane, we need 7.1 Angular Measure PC141 Intersession 2013Slide 2

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Day 14 – June 10 – WBL From the figure (and basic trigonometry…the same math that we used when describing vector components in chapter 3), we see that 7.1 Angular Measure PC141 Intersession 2013Slide 3

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Day 14 – June 10 – WBL Angular Measure PC141 Intersession 2013Slide 4

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Day 14 – June 10 – WBL We are all familiar with the most common unit for measuring angles – the degree (there are 360° in one full revolution). However, a much more useful unit is the radian (rad). This unit arises when one considers the relation between angular displacement and the arc length (s) over which a particle lying a distance r from the origin travels. 7.1 Angular Measure PC141 Intersession 2013Slide 5

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Day 14 – June 10 – WBL Angular Measure PC141 Intersession 2013Slide 6

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Day 14 – June 10 – WBL Problem #1: Car Tire Revolutions PC141 Intersession 2013Slide 7 WBL Ex 7.9 A car with a 65-cm-diameter wheel travels 3.0 km. How many revolutions does the wheel make in this distance? Solution: In class

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Day 14 – June 10 – WBL Angular Speed and Velocity PC141 Intersession 2013Slide 8

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Day 14 – June 10 – WBL Angular Speed and Velocity PC141 Intersession 2013Slide 9

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Day 14 – June 10 – WBL Actually defining the direction of angular motion is a bit tricky. Picture a few ants sitting on a turntable, which lies in the xy- plane. At any moment in time, each ant is moving in a different direction. However, all of them are rotating either clockwise or counterclockwise (Im pretty sure that turntables rotate clockwise, but I havent owned a turntable since 1989 and Im not willing to hit up a club just for the sake of making these notes). However, we cant really say that the motion is clockwise, because this depends on which side of the turntable were looking at. If we were to somehow get below the turntable and then look back up at it, we would claim that it rotates counterclockwise. 7.2 Angular Speed and Velocity PC141 Intersession 2013Slide 10

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Day 14 – June 10 – WBL Angular Speed and Velocity PC141 Intersession 2013Slide 11

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Day 14 – June 10 – WBL Angular Speed and Velocity PC141 Intersession 2013Slide 12

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Day 14 – June 10 – WBL This equation tells us that, while all particles in a rotating object have the same angular velocity, their tangential speeds depend on their distance from the origin. 7.2 Angular Speed and Velocity PC141 Intersession 2013Slide 13

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Day 14 – June 10 – WBL Period and Frequency Circular motion with constant angular speed results in a periodic repeating of the same motion – all particles in a rotating object simply travel along the same circular path over and over again. The time required to complete one rotation (or cycle) is called the period (T) of the motion. For example, the period of the Earths rotation about its axis is 24 hours. A related parameter is the frequency (f) of the circular motion. This is the number of revolutions in a given amount of time. the SI unit for frequency is s -1 (the inverse second), which is also called the Hertz (Hz). An object that completes 10 revolutions in 2 seconds has T = 0.2 s and f = 5 Hz. more or less (dont ask) 7.2 Angular Speed and Velocity PC141 Intersession 2013Slide 14

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Day 14 – June 10 – WBL Angular Speed and Velocity PC141 Intersession 2013Slide 15

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Day 14 – June 10 – WBL Problem #2: Merry-Go-Round PC141 Intersession 2013Slide 16 WBL Ex 7.27 A little boy jumps onto a small merry-go-round (radius of 2.00 m) in a park and rotates for 2.30 s through an arc length distance of 2.55 m before coming to rest. If he landed (and stayed) at a distance of 1.75 m from the central axis of rotation of the merry-go-round, what was his average angular speed and average tangential speed? Solution: In class

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Day 14 – June 10 – WBL Problem #3: Measuring the Speed of Light PC141 Intersession 2013Slide 17 The speed of light is measured as shown in the figure. A slotted wheel is rotated while light passes through the slots. The light travels to a mirror a distance L away and then returns to the wheel just in time to pass through the next slot. For this particular wheel, the radius is 5.0 cm and there are 500 slots around the edge. The mirror is L = 500 m from the wheel. The speed of light is measured at 3 x 10 8 m/s. What is the constant angular speed of the wheel, and what is the linear speed of a point on the edge of the wheel? Solution: In class

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Day 14 – June 10 – WBL Uniform circular motion occurs when an object moves at a constant speed in a circular path. Note that the object does not need to complete any full revolutions. Although the objects speed is constant, its velocity is not, since it is constantly 7.3 Uniform Circular Motion and Centripetal Acceleration PC141 Intersession 2013Slide 18 changing direction. Therefore, there must be an acceleration associated with this motion.

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Day 14 – June 10 – WBL Uniform Circular Motion and Centripetal Acceleration PC141 Intersession 2013Slide 19

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Day 14 – June 10 – WBL Uniform Circular Motion and Centripetal Acceleration PC141 Intersession 2013Slide 20

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Day 14 – June 10 – WBL Uniform Circular Motion and Centripetal Acceleration PC141 Intersession 2013Slide 21

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Day 14 – June 10 – WBL Centripetal Force cont It is important to note that the centripetal force is not a result of the uniform circular motion, it is the cause. If an object is moving along a circular path, there must be a reason. For example, a satellite orbiting the Earth maintains a circular path because of the gravitational force (which points inward, toward the center of the Earth). A rock that is swung at the end of a rope is subjected to a (radially inward) tension force along the rope. A car that travels around a curve experiences (radially inward) static frictional forces between the tires and the road. Should f s,max be reached, the car will skid in a straight, tangential line. If the rope breaks, the rock is flung along a straight, tangential line. If the Earth disappears, were all screwed the satellite will travel in a straight line. 7.3 Uniform Circular Motion and Centripetal Acceleration PC141 Intersession 2013Slide 22

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Day 14 – June 10 – WBL Problem #4: Uniform Circular Motion PC141 Intersession 2013Slide 23 WBL LP 7.9 In uniform circular motion, there is a… A …constant velocity B …constant angular velocity C …zero acceleration D …nonzero tangential acceleration

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Day 14 – June 10 – WBL Problem #5: Stunt Flying PC141 Intersession 2013Slide 24 WBL Ex 7.33 An airplane pilot is going to demonstrate flying in a tight vertical circle. To ensure that she doesnt black out at the bottom of the circle, the acceleration must not exceed 4.0g. If the speed of the plane is 50 m/s at the bottom of the circle, what is the minimum circle radius so that the 4.0g limit is not exceeded? Solution: In class

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Day 14 – June 10 – WBL Problem #6: Loop-the-Loop PC141 Intersession 2013Slide 25 WBL Ex 7.41 A block of mass m slides down an inclined plane into a loop-the-loop of radius r. a)Neglecting friction, what is the minimum speed the block must have at the highest point of the loop in order to stay in the loop? b)At what vertical height on the inclined plane (in terms of r) must the block be released if it is to have the required minimum speed at the top of the loop? Solution: In class

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Day 14 – June 10 – WBL Problem #7: Swinging a Rock PC141 Intersession 2013Slide 26 A string 1.00 m in length has a breaking strength of 50.0 N. A rock with mass 0.10 kg is tied to one end. The other end is held tightly, and the rock is swung in circular motion along a horizontal plane. What is the maximum angular velocity, in rpm, that the rock can be swung without breaking the string? Solution: In class

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