rotation-rolling-and-rigid-body-v3.pdf - Exercises on Rotation Rolling and Rigid Body Problems created by Raditya Difficulty level guide for PC1431

rotation-rolling-and-rigid-body-v3.pdf - Exercises on...

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Exercises on Rotation, Rolling, and Rigid Body Problems created by: Raditya Difficulty level guide for PC1431 students: S : PC1431 exam short questions difficulty level. L : PC1431 exam long questions difficulty level. H : Harder problems beyond PC1431 level. CH : Challenging/tricky problems. 1. [ S ] Point P in a rigid body is initially at coordinate (3 , 4) m, and is rotating counterclockwise about the origin with initial angular velocity of 1 rad/s, which increases at a constant rate of 1 rad/s 2 . Find the location of point P at t = 1 s. [ Ans: ( - 3 . 8 , 3 . 3)] 2. [ L ] The center of mass of a rigid body with mass 2 kg moves in the x direction with velocity v = 3 t - 2 t 2 . At time t = 0, it also begins to rotate clockwise in the xy plane with angular acceleration α = 2 t . At t = 3 s, a. Calculate the angular velocity of the rigid body. [ Ans: ω = 9 rad/s.] b. Calculate the velocity of a point P in the rigid body located 2 m above the center of mass. [ Ans: v P = 9 m/s in the positive x -direction.] c. The total kinetic energy of the rigid body is 90 J. Find its moment of inertia about the center of mass. [ Ans: I cm = 0 . 22 kg m 2 .] d. Find its moment of inertia about point P defined in part b. [ Ans: I P = 8 . 22 kg m 2 .] 3. [ S ] Consider the following figure: Suppose the radius of the ball is r , and the whole ball is moving to the right under pure rolling at a constant angular speed ω . Consider a point A which at time t = 0 is located at the point of contact between the surface and the ball. Take ˆ i to be a unit vector pointing to the right, and ˆ j to be a unit vector pointing downward . Find the velocity vector of point A as a function of time. [ Ans: ~v ( t ) = n [1 - cos( ωt )] ˆ i - sin( ωt ) ˆ j o ] 1
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4. Look at the figure below: A solid cylinder is being pulled by a horizontal force as shown above. There is friction between the cylinder and the plank, but there is no friction between the plank and the surface. Assuming the cylinder is rolling without slipping with respect to the plank, find: a. The acceleration of the cylinder ( a c ) and the plank ( a p )! [ Ans: a p = - F 3 M + m , a r = (4 M + m ) F (3 M + m ) m , where to the right is positive.] b. The minimum coefficient of static friction required to maintain this rolling without slipping condition! [ Ans: μ s , min = MF (3 M + m ) mg ] c. What happens if μ s < μ s , min that you calculated in part b? 5. [ L ] Look at the figure below: A yoyo of mass m , inner radius 2 R 3 , outer radius R , and moment of inertia I = xmR 2 is placed on top of an inclined plane as shown. The coefficient of static friction between the plane and cylinder is μ . A massless string is wound so many times inside the inner radius of the yoyo, and you pull the other end of the string as is shown in the above picture. a. Find the value of F so that the yoyo will be stationary (neither rotating nor moving), and find the minimum value of μ for this to be possible. [ Ans : F = 3 mg sin( θ ) and μ min = 2 tan( θ )] b. Find the range of values of F so that the yoyo is rolling without slipping with respect to the plane, assuming that μ is larger than the minimum value you found in part a. What is the significance of this assumption?
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  • Summer '04
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