Work In Pushing Against Spring and Gravity


A 10 kg block rest on a smooth surface inclined 30 degrees from the horizontal. A long spring with spring constant k=30 N/m supports the block at the zero position. A horizontal force of 400 N is applied to the block. How much work has been done by all forces acting on the block when the block is at 2 meters up the plne from its starting point?


Let's get after this one using the work-energy theorem. Assume the mass of the spring is negligible compared to the block and also assume that no mechanical energy is lost to the surroundings due to friction. The block then has potential energy due to any elevation change, and if moving, kinetic energy. The spring has potential energy due to any displacement from its initial position. Since the block is initially at rest the initial kinetic energy is zero. We may choose the potential energy reference point such that the initial potential energy due to block elevation is zero and initial potential energy stored in the spring is zero. The work-energy theorem states that the total work done by all forces acting on the block will be equal to the change in kinetic energy of the block.

The work done by the 400 N force, Wf, since that force is constant, is the dot product of the horizontal force vector with the displacement vector up the plane. That is Wf=400*2*cos(30)=800*0.866=692.8 Joules. Part of that work went into stretching the spring, part into potential energy of the block due to elevation and part into giving the block some kinetic energy. The potential energy stored in a spring is 1/2*k*x2 where k is the spring constant and x is the displacement. In this case that is 0.5 * 30 N/m * 22 m2 = 60 Joules. The potential energy stored in the block due to its elevation change is 10 kg * 9.8 m/s2 * 2*sin(30) = 98 Joules. The difference between the 692.8 Joules supplied by the applied force and the 158 Joules stashed as potential energy, or 534.8 Joules must be the kinetic energy of the block at the 2 meter point. This is also the work done by all forces.

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