Question

A block of mass 0.500 𝑘𝑔 is pushed against a horizontal spring of negligible mass until the spring is compressed a distance 𝑥 as shown in the figure. The spring constant is 450 𝑁/𝑚. When it is released, the block travels along a frictionless, horizontal surface to point B, at the bottom of a vertical circular track of radius 𝑅=1.00 𝑚, and continues to move up the track. The speed of the block at the bottom of the track is 𝑣𝐵=12.0 𝑚/𝑠, and the block experiences an average frictional force of 7.00 N while sliding up at point C.

a. Calculate the compression distance 𝑥 of the spring?

b. Find the work done by the frictional force on the block as it slides from B to C. (Hint: 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒=2𝜋𝑅)

c. Assume that the block never falls off as it reaches point C, what speed do you predict for the block at the top of the track?

Answer 1

a)

$$0.5mv_b^2=0.5kx^2\\(0.5)(12)^2=(450)x^2\\x=0.4\ m$$

b)

$$W=\pi RF=\pi(1)(7)=22.0\ J$$

c)

$$0.5mv_b^2=0.5mv_c^2+mg(2R)+W\\ (0.5)(12)^2=0.5v_c^2+4(0.5)(9.8)(1)+22\\v_c=7.80\frac{m}{s}$$