Question

Part A The spring has a stiffness k 50 lb/ft and an unstretched length of 2 ft. As shown, it is confined by the plate and wall using cables so that its length is 1.5 ft. A 4.1-lb block is given a speed vA when it is at A, and it slides down the incline having a coefficient of kinetic friction fk -0.2. It strikes the plate and pushes it forward 0.25 ft before stopping. Neglect the mass of the plate and spring. (Figure 1) Determine the speed of the block at A Express your answer to three significant figures and include the appropriate units. Figure 1 of1 ft A 3.98 Submit Previous Answers Request Answer X Incorrect; Try Again;5 attempts remaining 3 ft 1.5 ft Provide Feedback

Answer 1

Consider the following free-body diagram of the block and spring. N 3 ft µwcosO wsin 0 wcos0 0.25 ft 1.25 ft- Write the expression for the frictional force acting on the block as follows: f, = H,N f, = H,w cos e Write the expression for the net force acting on the block in the direction of motion as follows: Net force on the block = wsin 0 - uwcos 0

Write the expression for the work done by block to compress the spring by 0.25 ft as follows: W, = Net force on the block x distance moved by block = (wsin e- 4w cos 0)x(3+0.25) -(41-02-41-25 3 =| 4.1x-0.2 x 4.1x. |× 3.25 = (2.46 – 0.66)x 3.25 W, = 5.85 lb-ft Write the expression for the change in potential energy of the spring as follows: U„ =-k(x? -x}) Here, x, is the initial compression and x, is the final compression. x50x (0.75² –0.52) sp 2 =-x50x0.3125 2 = 7.81 lb· ft Since, spring force will work in opposite direction of the motion of the block. Thus, spring potential energy will have negative sign.

Write the expression for the initial kinetic energy of the block as follows: 4.1 2 32.2 T = 0.064v; Write the expression for the final kinetic energy of the block as follows: 1( 4.1 2 32.2 T, = 0 Write the expression for the work-energy theorem as follows: T, +£U-2 = T; 0.064v +W, - U,, = 0 0.064v, + 5.85 – 7.81=0 7.81-5.85 %3D 0.064 v = 30.625 v = 5.53 ft/s Thus, speed of the block at point A is 5.53 ft/s