Question

207 APISICS C MECHANICS TRE RES I Nole: Figure not drawn to scale. block travels down the inclined plane and makes a smooth transition onto a horizontal surface. While traveling on the horizontal surface, the block collides with and attaches to an ideal spring of spring constant k. There is negligible friction between the block and both the inclined plane and the horizontal surface, and the spring has negligible mass. Express all algebraic answers for parts (a). (b), and (e) in terms of m, h, and physical constants, as appropriate. 2. A block of mass m starts at rest at the top of an inclined plane of height h, as shown in the figure above. The . Derive an expression for the speed of the block just before it collides with the spring ii. Is the speed halfway down the incline greater than, less than, or equal to one-half the speed at the bottom of the inclined plane? Greater than Less than Equal to Justify your answer (b) Derive an expression for the maximum compression of the spring. (o) Determine an expression for the time from when the block collides with the spring to when the spring The block is again released from rest at the top of the incline, and when it reaches the horizontal surface it is reaches its maximum compression. moving with speed o. Now suppose the block experiences a resistive force as it slides on the horizontal surface. The magnitude of the resistive force F is given as a function of speed vby F B, where B is a positive constant with units of kg/m. i. Write, but do NOT solve, a differential equation for the speed of the block on the horizontal surface as a function of time t before it reaches the spring. Express your answer in terms of m, h, k, p, v, and physical constants, as appropriate. ii. Using the differential equation from part (d)i, show that the speed of the block v(t) as a function of time I can be written in the form vi) 납+m.wherejste speed at t=0. 2017 The College Board Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE

Answer 1

Mass m, plant height h, spring constant k speed of block that before collision with spring = u from conservation of energy mgh = 0.5 mu^2 u = squareroot 2gh speed half way down u_1 = squareroot 2gh/2 = squareroot gh half speed at bottom = squareroot 2gh/2 = squareroot gh/2 so speed half way down > half speed at bottom max compression = x_0 So mgh = 1/2 k x_0^2 = > x_0 = squareroot 2 mgh/k time taken for maximum compression of spring = T now x_0 = squareroot 2 mgh/k but time period of spring mass system T' = squareroot M/K also T = T'/4, hence T = 1/4 squareroot M/K speed at bottom of inline = v_0 resistive force F = beta v^2 so on horizontal surface at distance x from plank m x = -beta v^2 = -beta x^2 = > m v = -beta v^2 = > integral m dv/v^2 = -integral beta dt = > m