Suppose a round rigid body (hoop, sphere, disk) with mass m, radius R, and radiu

Question

Suppose a round rigid body (hoop, sphere, disk) with mass m, radius R, and radius of gyration kG rolls without sliding toward a small obstruction ("bump"). The obstruction has a height h above the flat, even surface. In this problem, you receive step-bv-step instructions how to use conservation of angular momentum to find the minimum velocity vG that the body must have in order to roll over the obstruction at A The linear impulse-momentum diagram is given above at the right There is no impulse due to a normal force because the body has lifted off the surface. Whv has the impulse due to the weight been ignored? Find the angular momentum about point A just before (Igw plus the moment of mvG) and immediately after (IG w0* plus the moment of mvG*). This quantity is conserved, so equate the two expressions. Note that immediately after the collision, A is the ICOR for the body. Use that to relate vG* to w* and use "rolling without sliding" to relate vG and w. Finally, although energy is not conserved from start to finish, it is conserved from immediately after the impact on. To just clear the obstruction, the kinetic energy immediately after impact must equal the potential energy as the body clears the obstruction (in other words, the velocity is zero as the body reaches its highest point). Use that to solve for the minimum value of vG. Use your expression from (d) to find the minimum velocity for a disk of mass m = 2.5 kg, radius R = 0.3 m and an obstruction height h = 0.05 m

Explanation / Answer

Firstly, energy is conserved throughout. The ball is rolling without slipping which means the point of contact of the ball is stationary which in turn means that friction is not doing any work. If friction is not doing work, there is no loss of energy.

Secondly, the impulse due to weight is not ignored. It is counteracted by the normal reaction impulse. Once the ball is air-borne, the impulse due to obstruction and due to normal reaction become zero. Then the only force acting is weight (air drag and all are assumed to be absent). Impulse due to friction is zero since at the time of impact, the ball is not accelerating which means friction is zero.

For rolling without slipping, v = Rw.

I hope the concept is a little clearer now. You will be able to solve the problem easily now.

Cheers!

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