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Lecture 16: Rotational Motion

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Questions of Yesterday 1) You are going through a vertical loop on roller coaster at a constant speed. At what point is the force exerted by the tracks on you (and the cart you are in) the greatest? a) at the highest point b) at the lowest point c) halfway between the highest and lowest point d) the force is equal over the whole loop 2) You are on a merry-go-round moving at constant speed. If you move to the outer edge of the merry-go-round, what happens to the net centripetal force keeping you on the merry-go-round? a) it increases b) it decreases c) it stays the same d) there is no net centripetal force acting on you

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Rotational Motion: Angular Quantities srsr = Angular Position: = f - i Angular Displacement: av = f - i t f - t i t = Average Angular Velocity: lim t -> 0 = t Instantaneous Angular Velocity: av = f - i t f - t i t = Average Angular Acceleration: lim t -> 0 = t Instantaneous Angular Acceleration: Motion with Constant : = 0 + t = 0 t + 1/2 t 2 2 = 0 2 + 2

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Angular and Linear Quantities r titi srsr = tftf Displacement: Direction of linear velocity v of an object moving in a circular path is always TANGENT to the path ss vT=rvT=r Tangential Speed: aT=raT=r Tangential Acceleration:

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Centripetal Acceleration r vfvf Centripetal Acceleration always points towards the CENTER of the circle v f - v i t f - t i a av = vivi -vi-vi vfvf vv

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Centripetal Force F = ma If an object is accelerating what do know about (think Newton’s 2nd law)? Can an object be moving in a circular path if no forces are acting on? If an object is undergoing constant speed circular motion what direction is the net force acting on the object? mv 2 r F c = ma c =

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Centripetal Acceleration What if your tangential speed is NOT constant? r vfvf vivi -vi-vi vfvf vv r Acceleration has both tangential and centripetal components! v2rv2r ac=ac= a = (a c 2 + a T 2 ) 1/2 vv vcvc vTvT aT=raT=r

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Centripetal Force In what direction is the net force if an object is undergoing circular motion and changing its tangential speed? -vi-vi vfvf vv a acac aTaT FF FTFT FCFC F T = ma T mv 2 r F c = ma c = F = ma Just like linear motion (∑F x = ma x, ∑F y = ma y )… must split vector equation in the perpendicular components!!

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Practice Problem An air puck of mass 0.5 kg is tied to a string and allowed to revolve in a circular radius of 1.0 m on a frictionless horizontal table. The other end of the string passes through a hole in the center of the table and a mass of 1.0 kg is tied to it. The suspended mass remains in equilibrium while the puck on the tabletop revolves. What is the tension in the string? What is the horizontal force acting on the puck? What is the speed of the puck?

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Practice Problem Tarzan (m = 100 kg) tries to cross a river by swinging from a 10- m-long vine. His speed at the bottom of the swing (as he just clears the water) is 8.0 m/s. Tarzan doesn’t know that the vine has a breaking strength of 1500 N. Does he make it safely across the river? If not, what is the maximum speed he can have to make it? If Tarzan continues swinging on the vine what is the highest point he reaches? What is the tension in the vine at this highest point? What is the net force on Tarzan at this point?

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Planetary Motion Why do the planets revolve around the sun, and the moon revolve around the Earth? Is there a net force acting on the planets and moons? How do you know? What is the direction of the force?

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Gravitational Force Force of attraction between any two objects in the Universe. Gravitational force causes…. Objects in free fall near the Earth’s surface to accelerate towards the Earth the moon to orbit the earth & the planets to orbit the sun An astronaut to be able to jump higher on the Moon than on Earth

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Gravitational Force If gravity is an attractive force why doesn’t the moon crash into the Earth? The moon is constantly falling towards Earth The planets are constantly falling towards the sun

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Gravitational Force Newton’s Law of Gravitational Force m 1,m 2 = mass of objects attracting each other r = distance between the objects Universal gravitational constant = G = 6.67*10 -11 N*m 2 /kg 2 F g = G m1m2m1m2 r2r2 Gravitational Force between two objects is felt equally by both objects F g of m 1 exerted by m 2 = F g of m 2 exerted by m 1

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Gravitational Force What if you have many objects near each other? The net gravitational force felt on an object is equal to the sum of the gravitational forces exerted by all the surrounding objects ∑F gE = F gSE + F gME S E M

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Practice Problem Objects with masses of 200 kg and 500 kg are separated by 0.500 m. Find the net gravitational force exerted by these objects on a 50.0 kg object placed midway between them. At what position can the 50.0 kg object be placed so as to experience a net force of zero?

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Questions of the Day You are riding on a Ferris wheel moving at constant speed. 1a) At what point is the net force acting on you the greatest? a) the top b) the bottom c) halfway between top and bottom d) the force is the same over the whole motion 1b) Is the net force doing work on you? a) YES b) NO 2) If the mass of the moon were doubled, what would happen to its centripetal acceleration? a) it would increase b) it would decrease c) it would stay the same

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Practice Problem A 0.500-kg pendulum bob passes through the lowest part of its path at a speed of 5.00 m/s. What is the tension in the pendulum cable at this point if the pendulum is 100.0 cm long? When the pendulum reaches its highest point, what angle does the cable make with the vertical? What is the tension in the pendulum cable when the pendulum reaches its highest point? What is the net force acting on the pendulum at this point? What is the direction of the acceleration?

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