A 360 g block is dropped onto a relaxed vertical spring that has a spring consta
ID: 1451197 • Letter: A
Question
A 360 g block is dropped onto a relaxed vertical spring that has a spring constant of k = 4.2 N/cm (see the figure). The block becomes attached to the spring and compresses the spring 17 cm before momentarily stopping. While the spring is being compressed, what work is done on the block by (a) the gravitational force on it and (b) the spring force? (c) What is the speed of the block just before it hits the spring? (Assume that friction is negligible.) (d) If the speed at impact is doubled, what is the maximum compression of the spring?
Explanation / Answer
mass of block m = 0.36 kg, dropped from a height say h, on to the vertical spriring of sring constant k = 4.2 N/m
k = 4.2 N/m, compression in the spring x = 0.17 m
a) work done by gravity on ball is = mgh = elastic potential energy developed in the spring= 1/2 kx^2
mgh = 1/2 kx^2 ==> h = 0.5*k*x^2/m*g
= 0.5*4.2*0.17^2/(0.36*9.8) = 0.017 m,
work done on the ball is W = mgh = 0.36*9.8*0.017=0.056 J
work done by spring on block = 1/2 kx^2 = 0.5*4.2*0.17^2 = -0.061 J
speed of the block just before it hits the block is mgh = 1/2 mv^2 ==> v = sqrt(2gh)= sqrt(2*9.8*0.017)= 0.577 m/s
1/2*mv^2 = 1/2*kx^2 ==> v1/v2 = x1/x2, v1/2v1 = x1/x2 ==> x2 = x1*2
f the speed at impact is doubled,the maximum compression of the spring also doubled.