Question

A 210 g block is dropped onto a relaxed vertical spring that has a spring constant of k = 3.0 N/cm. The block becomes attached to the spring and compresses the spring 11 cm before momentarily stopping.

(a) While the spring is being compressed, what work is done on the block by the gravitational force on it?
(b) What work is done on the block by the spring force while the spring is being compressed?
(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?

Answer 1

(a)
`W_(p) = 0.23 Nm`

(b)
`W_(e) = 1.82 Nm`

(c) v= 4.42 m/s

(d) x = 20cm

Step-by-step explanation:

(a) The work done on the block by the gravitational force is:

`W_(p) = E_(p) = mgx`

where
`W_(p)`: is the work by the gravitational force,
`E_(p)`: potential energy, m: is the block's mass, g: gravitational acceleration and x: is the distance of compression

`W_(p) = mgx = 0.210 kg \cdot 9.81 (m)/(s^(2)) \cdot 0.11 m = 0.23 N\cdot m`

(b) The work done on the block by the spring is given by:

`W_(e) = E_(e) = (1)/(2) k\cdot x^(2)`

where
`W_(e)`: work by the spring force,
`E_(e)`: potential elastic energy, k: spring constant, x: distance of compression

`W_(e) = (1)/(2) 300 (N)/(m) \cdot (0.11m)^(2) = 1.82 N\cdot m`

(c) The speed of the block just before it hits the spring can be calculated by conservation of energy before and after the impact:

`E_(k) = E_(p) + E_(e)`

`(1)/(2) mv^(2) = mgx + (1)/(2) k\cdot x^(2)` (1)

`(1)/(2) mv^(2) = 0.23Nm + 1.82Nm`

`v = \sqrt (2 (0.23Nm + 1.82Nm))/(0.210kg) = 4.42 (m)/(s)`

(d) Using equation (1) we can determine the spring compression when the speed at impact is duplicated:

`(1)/(2) m(2v)^(2) = mgx + (1)/(2) k\cdot x^(2)`

`(1)/(2) k\cdot x^(2) + mgx - 2mv^(2) = 0` (2)

Solving the quadratic equation (2), we have the next spring compression (x):

`x = 0.20m = 20cm`

So, an increase in the speed at impact will also increase the spring compression.

I hope it helps you!