Work In Pushing Against Spring and
Gravity

## Question:

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?
## Answer:

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*x^{2} where k is the spring constant and x is the
displacement. In this case that is 0.5 * 30 N/m
* 2^{2} m^{2} = 60 Joules. The potential
energy stored in the block due to its elevation change is 10 kg * 9.8 m/s^{2} * 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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