Question

Problem 4 A block of mass m slides at velocity vo across a horizontal frictionless surface toward a large curved movable ramp n and has a smooth circular frictionless face up which the block can easily slide. When the block slides up the ramp, it momentarily reaches a maximum height a shown in Figure II, and then slides back down the frictionless surface as shown in Figure III. face to the horizontal (a) Find the velocity of the ramp at the instantvo the block reaches its maximum height. (b) To what maximum height h does the center Figure I of mass of the block rise above its original height? (c) Determine the final velocity of the ramp and the final velocity of the block after the block returns to the level surface. Clearly indicate whether the block is moving to the right or to the left, relative to the horizontal surface. of Mass of Block mwmmm Figure Figure I

Answer 1

Mass of block is = m mass of the ramp is = 3 m initial velocity of the block is v_0 (a) when the block is reach at it's maximum height than the speed of the block will be zero. from energy conservation initial energy = final energy 1/2 m v_0^2 = m g h + 1/2 (3 m + m) v_i^2 1/2 m v_0^2 = m g h + 2 m v_i^2 2 v_i^2 = v_0^2/2 - g h v_i^2 = v_0^2/4 - g h/2 v_i =squareroot v_0^2/4 - gh/2 = 1/2 squareroot v_0^2 - 2 g h v_i = 1/2 squareroot = v_0^2 - 2 g h \r\n (b) The center of mass of the block rise up to a height h above its original height (c) from momentum conservation initial momentum = final momentum (3 m + m) v_i = 3 m v_f - m v_f' where v_f' be the final speed of the block. 4 m v_i = 3 m v_i - m v_f' 4 v_i = 3 v_f - v_f'