How Much Work Is Needed to Stretch a Spring from 12 cm to 16 cm?

[magma_saber]

Homework Statement

A spring has a relaxed length of 7 cm and a stiffness of 50 N/m. How much work must you do to change its length from 12 cm to 16 cm?

Homework Equations

1/2*k_ss²

The Attempt at a Solution

1/2*50*(.16-.12)² = 0.04

It shows the solution to be 0.14. It says the change in the energy of the spring is its change in potential energy, so the change in (1/2)k_ss². Be sure to use the stretch s, not the length, and use meters, not centimeters.

 

[Delphi51]

Looks like an algebra error and a misunderstanding of stretch.
Length 12 is a stretch of 12 - 7 = 5. Length 16 is a stretch of 16 - 7 = 9.
.5k(.09)^2 - .5k(.05)^2 is not equal to .5k(.09-.05)^2.

 

[nlightner]

I would like to commend your approach to solving this problem. You have correctly used the equation for the potential energy of a spring, which is (1/2)kss^2. However, there are a few things that need to be addressed.

Firstly, in your attempt, you have used the change in length (0.04 m) instead of the stretch (0.04 m) in the equation. This will lead to an incorrect answer. The correct equation to use in this case would be (1/2)k(s2-s1)^2, where s1 is the initial stretch (0.05 m) and s2 is the final stretch (0.09 m).

Secondly, it is important to use consistent units in your calculations. In this case, the stiffness of the spring is given in N/m, so the stretch should also be in meters. This means that the initial stretch of 7 cm should be converted to meters (0.07 m) before using it in the equation.

Taking these into consideration, the correct calculation would be:

(1/2)(50 N/m)((0.09 m-0.07 m)^2) = 0.04 J

Thus, the work done by the spring in changing its length from 12 cm to 16 cm is 0.04 J. This means that you would need to apply 0.04 J of external force to stretch the spring from 12 cm to 16 cm.