How Far Will the Spring Compress When a Box Falls on It?

[DeadFishFactory]

Homework Statement

A 2.14 kg block is dropped from a height of 43.6 cm onto a spring of force constant 18.6 N/cm. Find the maximum distance the spring will be compressed.

I know that the mass is 2.14 kg, the force constant of the spring is K=1860 N/m.

I assume the final velocity, the final height of the block after being dropped, and the final distance the spring will be compressed are 0.

Homework Equations

U+K=0
v^2=v(0)^2+2AS

The Attempt at a Solution

To find the initial velocity, I use the constant acceleration equation:

0=V(0)^2-19.7(.436); V(0)=2.92m/s

Since I know (or assume) a conservative force, I use the equation:

U+K=0

mgh+(1/2)mv(0)^2+(1/2)kx^2=0

2.14(-9.8)(.436)-.5(2.14)(2.92)^2+(1/2)(1860)x^2=0

x=.139 m = 13.9 cm

The book got 11.1 cm. Can anyone tell me what I did wrong and how to do the problem correctly? Must help is appreciated. Thanks!

 

[Doc Al]

The initial (and final) velocity is zero. What you found is the speed at which the block hits the spring. That would be OK, but then you messed up the energy conservation.

It will be much easier if you just stuck with energy conservation from start to finish. Hint: the total distance dropped is not just 43.6 cm.

 

[m2003]

Your solution seems to be correct. The book may have used a slightly different value for the gravitational acceleration (9.81 m/s^2 instead of 9.8 m/s^2) or may have rounded off intermediate calculations, leading to a slightly different final answer. As a scientist, it is important to understand that there can sometimes be slight variations in answers due to different assumptions or rounding off of values. As long as your approach and calculations are correct, your answer should be considered valid. It may be helpful to double check your calculations and see if there are any minor differences that could account for the discrepancy.