If a spring is compressed an additional 4.0 cm after initially being compressed 2.0 cm, how much more work is done in the second compression?

To determine how much more work is done in the second compression of the spring, we can use the formula for the work done on a spring, which is given by the expression ( W = \frac{1}{2} k x^2 ), where ( W ) is the work done, ( k ) is the spring constant, and ( x ) is the compression of the spring.

Initially, the spring is compressed by 2.0 cm. Therefore, the work done in this initial compression can be calculated as follows:

[ W_1 = \frac{1}{2} k (0.02)^2 ]

Now, when the spring is further compressed an additional 4.0 cm, the total compression becomes 6.0 cm. The work done during this additional compression is:

[ W_2 = \frac{1}{2} k (0.06)^2 ]

To find the additional work done, we can calculate ( W_2 - W_1 ):

1. Work done for 6.0 cm compression:

[ W_2 = \frac{1}{2} k (0.06)^2 = \frac{1}{2} k (