A 190 g block is dropped onto a relaxed vertical spring that has a spring consta

Question

A 190 g block is dropped onto a relaxed vertical spring that has a spring constant of k = 2.3 N/cm (see the figure). The block becomes attached to the spring and compresses the spring 13 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

here,

mass ,m = 0.19 kg

spring constant , K = 2.3 N/cm = 230 N/m

x = 0.13 m

a)

the work done by the gravitational force , Wg = m * g * x

Wg = 0.19 * 9.81 * 0.13 J

Wg = 0.24 J

b)

the work done on block by the spring force , Ws = - 0.5 * k * x^2

Ws = - 0.5 * 230 * 0.13^2

Ws = - 1.94 J

c)

let the initial kinetic energy be KE

KE = Ws - Wg

0.5 * m * v^2 = 1.94 - 0.24

0.5 * 0.19 * v^2 = 1.94 - 0.24

v = 4.23 m/s

d)

if the speed of impact is doubled

v' = 4.23 * 2 = 8.46 m/s

let the compression in the spring be x'

potential energy stored in the spring = initial kinetic energy + gravitational energy lost

0.5 * k * x'^2 = 0.5 * m* v^2 + m * g * x'

0.5 * 230 * x'^2 = 0.5 * 0.19 * 8.46^2 + 0.19 * 9.81 * x'  

x' = 0.25 m

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