# 1) Work definition for constant and variable forces. 2) What is the equations for the spring force and spring energy. 3) how to derive the Kinetic Energy work theorem from Newton's 2nd law and...

1) Work definition for constant and variable forces.

2) What is the equations for the spring force and spring energy.

3) how to derive the Kinetic Energy work theorem from Newton's 2nd law and the definition of work.

Please explain in details.

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### 1 Answer

1. A force F is applied on an object and object displaced from its initial position either in direction of force or in some other direction. Since displacement and force both are vector quantity so we define work W a scala quantity as F. S ( dot product of F and s the displacement).

`W=F.S` Joules

If F is variable force then we define displacement vector say ds ( very small displacement in direction of force). We define a dot product of F and ds , sum it in limit of considered displacement say `s_0,and s_1` .

`W=int_(s_0)^(s_1)F.ds`

We can define W, by considering component of F in direction of displacement if displacement is not in direction of force.

2. Let spring constant is K . Let force F Newton is applied to stretch by x meter .Then work done W (potential energy) :

`W=(1/2)(kx)x=1/2kx^2 J`