Question

A jack-in-the-box with a box of mass M and clown of mass

m = 285.0 g

rests on a horizontal table (see figure below). The clown is attached to the box by means of a lightweight, vertical spring with force constant

k = 1.00 ✕ 10² N/m.

Zara pushes down on the clown with an additional force of 5mg, further compressing the spring. When Zara removes her hand, the clown jumps upward, and the spring lifts the box off the table. What is the maximum value for the mass M of the box for this action to occur?

Answer 1

let mass of the box be M
mass of clown, m = 0.285 kg
spring constant, k = 10^2 N/m
force on the spring while pushing it down, F = 5mg
compression of the spring = x
so, 5mg = kx
x = 5mg/k
so when the spring is released, the spring goes to a maximum extenbsion of y
here from conservation of energy
0.5ky^2 + mgy = 0.5kx^2 - mgx
0.5ky^2 + mgy = 0.5k*25m^2g^2/k^2 - mg*5mg/k
0.5ky^2 + mgy = 7.5m^2g^2/k
0.5*100y^2 + 0.285*9.81*y - 7.5*0.285^2*9.81^2/100 = 0
50y^2 + 2.79585y - 0.968452875 = 0
solvinf for y
y = 0.113994 m

hence maximum force on the mass M when the spring pulls up is k*y = 11.399 N
hence Mg = 11.399
M = 1.162 kg is the maximum mass of box for which the box will be lifted off the floor when the spring is pushed by the force 5mg