Calculate the total energy of the system if both potential energies are zero at x = 0.?

Question

A child's pogo stick (Fig P5.75) stores energy in a spring (k = 2.60 104 N/m). At position A (x1 = -0.100 m), the spring compression is a maximum and the child is momentarily at rest. At position B (x = 0), the spring is relaxed and the child is moving upward. At position C, the child is again momentarily at rest at the top of the jump. Assume that the combined mass of child and pogo stick is 25.0 kg.

Fig P5.75 http://www.webassign.net/sercp/p5-75.gif

(a) Calculate the total energy of the system if both potential energies are zero at x = 0.
J
(b) Determine x2.
m
(c) Calculate the speed of the child at x = 0.
m/s
(d) Determine the value of x for which the kinetic energy of the system is a maximum.
mm
(e) Obtain the child's maximum upward speed.
m/s

Answer 1

For gravitational potential energy, the change when moving a height H is (M)(g)(H), where:
M is the mass
g is the acceleration due to gravity (about 9.8 m/s^2 at the earth's surface)

The kinetic energy is given by:
Ek = (1/2)(M)(V^2), where
V is the velocity

The potential energy of a spring is given by:
Es = (1/2)(Ks)(D^2), where
Ks is the spring constant
D is the displacment.

We are dealing with an "ideal" system which means that no energy is lost to friction. Thus the sum of all the energies is constant.

Es + Eg + Ek = Et

where Et is the constant total energy for this system.

At point A, the child is at rest, so the velocity V is 0 and the kinetic energy is 0. The gravitational potential energy is (-0.1)(M)(g), and the rest of the energy is the potential energy of the compressed spring, which has been compressed 0.1m

We know the spring displacement and spring constant, and the mass of the system, and so can compute the spring potential energy. Add in the gravitational energy, which we can also compute, and we have the total energy.

At point C, the child is again at rest so the kinetic energy is 0. The spring is completely relaxed so its potential energy is 0 too. That says that all of the energy is the potential energy due to gravity, which is given by (x2)(M)(g).

We have the total energy and so we can compute x2.

At point B, the potential energy of gravity is 0 and, since the spring is relaxed, the potential energy of the spring is also zero. This means that all the energy is the kinetic energy.

We know the total energy and the equation for kinetic energy so we can compute first V^2 and then V, the speed of the child at x = 0.

Now comes the interesting part. The value of x for which the kinetic energy is a maximum is the value at which the child's speed is maximum. Since there is no friction, it doesn't matter if we use the upward speed or the downward speed.

Let's think in terms of downward speed because it is a bit easier.

Gravity is pulling down on the child with a force equal to the weight of the combined mass. Starting at point C, the child starts falling due to its weight. It acceleration is the constant g until it reaches x = 0, where it starts compressing the spring.

From then on the spring pushes up on the combined mass. That force is given by the spring equation:

F = -(Ks)(D)

At x = 0, D = 0 so the force up starts out at 0. As the spring is compressed, the force keeps increasing. As it does, the downward acceleration decreases. But the velocity continues to increase as long as the force of the spring up is less than the weight of the child.

Eventually, the spring compresses enough that the spring force is exactly equal to the weight of the combined mass.

That is: (Ks)(x3) = (M)(g)

We have Ks, M, and g and so can compute x3.

As the child continues going down, the force of the spring becomes larger than the weight and starts slowing the child down. So x3 is the position of maximum velocity.

Knowing x3, we can compute the spring potential energy there and the gravitational potential energy. What's left over is the kinetic energy so we can compute the velocity at x3 from the kinetic energy.