Work done by a spring force can be seen as the area of a triangle bounded by the

Question

Work done by a spring force can be seen as the area of a triangle bounded by the maximum and minimum positions and the maximum or minimum force as seen in the figure to the right. Thus the work done by the spring is W_sp = 1/2(F_top - F_bot)(x_top - x_bot) Show that this is equivalent to integral F rightarrow dr rightarrow for the spring. You know that at both the top and the bottom of the spring's motion, the mass is not moving, so its velocity is necessarily zero. What is the kinetic energy at both the top and bottom of the motion? Work transforms energy from one form to another, where did it go (or come from)? Compare it to the change in energy of the other two forms we have been examining. Be explicit in indicating whether the work done by the spring going from top to bottom of the mass' position is positive or negative. Similarly, the sign of the work (positive or negative) for the work done by the force of gravity is equally important.

Explanation / Answer

W = integral Fdr

= integral Fdx

= area of Fx plot

= 0.5 (Ftop-Fbottom)(xtop-xbottom)

Hence shown

Kinetic energy at top and bottom is zero because the velocity at top and bottom is zero

The energy comes from what eas stored as spring potential energy. It get converted into gravitational potential energy as the mass moves to the top. Kinetic energy change is zero

Work done by spring is positive because direction of spring force is upward and displacement of mass is also upward.

Work done by force of gravity is negative because direction of gravitational force is downward and displacement of mass is upward.

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