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Unformatted text preview: Solution to ball on ramp problem Phys 1112 04/09/2009 Problem: A ball of uniform density is placed a distance ? up a ramp that makes an angle with the level surface beneath it. If the ball starts from rest and rolls down the ramp without slipping, how long does it take to reach the bottom? Solution: One way we can solve this problem by finding torques on the ball and their associated angular accelerations and then seeing how much the ball must roll to travel a distance ? . Putting this info together we can calculate the time it will take for the ball to make the required number of revolutions. From the rotational analogue to Newton's second law we know that the net torque on an object about a point is equal to the rate of change of its angular momentum about this point. ? = ? ? = ? ( ) ? = ? ? + ? ? Where I've used the product rule to expand ? ? . Since we're interested in torque about the center of mass of a ball the moment of inertia is constant which allows us to write, ? ? = 0 ?? ? = ? ? = ?? = ? So to find the angular acceleration of this ball about its CoM we need to find the net torque and moment of inertia about this point. To find the net torque, let's do a FBD. Force categories: N=normal force, W=weight, f=static friction. Subscripts: r=ramp, b=ball, e=earth....
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