Periodic Motion Energy

A pendulum bob swinging to and fro on the end of a string. There are only two forces acting upon the pendulum bob. Gravity (an internal force) acts downward and the tensional force (an external force) pulls upwards towards the pivot point. The external force does not do work since at all times it is directed at a 90-degree angle to the motion. (Review a previous page to convince yourself that F•d•cosine angle = 0 J for the force of tension.)

As the pendulum bob swings to and fro, its height above the tabletop (and in turn its speed) is constantly changing. As the height decreases, potential energy is lost; and simultaneously the kinetic energy is gained. Yet at all times, the sum of the potential and kinetic energies of the bob remains constant. The total mechanical energy is 6 J. There is no loss or gain of mechanical energy, only a transformation from kinetic energy to potential energy (and vice versa). This is depicted in the diagram below.

Definition and Graphic Provided by Physics Classroom

The glider will be pulled to the right of its equilibrium position and be released from rest (position A). As mentioned, the glider then accelerates towards position C (the equilibrium position). Once the glider passes the equilibrium position, it begins to slow down as the spring force pulls it backwards against its motion. By the time it has reached position E, the glider has slowed down to a momentary pause before changing directions and accelerating back towards position C. Once again, after the glider passes position C, it begins to slow down as it approaches position A. Once at position A, the cycle begins all over again ... and again ... and again.

The kinetic energy possessed by an object is the energy it possesses due to its motion. It is a quantity that depends upon both mass and speed. The equation that relates kinetic energy (KE) to mass (m) and speed (v) is

KE = ½•m•v2

The faster an object moves, the more kinetic energy that it will possess. We can combine this concept with the discussion above about how speed changes during the course of motion. This blending of the concepts would lead us to conclude that the kinetic energy of the mass on the spring increases as it approaches the equilibrium position; and it decreases as it moves away from the equilibrium position.

More on the energy of an oscillating mass on a spring

Kinetic energy is only one form of mechanical energy. The other form is potential energy. Potential energy is the stored energy of position possessed by an object. The potential energy could be gravitational potential energy, in which case the position refers to the height above the ground. Or the potential energy could be elastic potential energy, in which case the position refers to the position of the mass on the spring relative to the equilibrium position. For our vibrating air track glider, there is no change in height. So the gravitational potential energy does not change. This form of potential energy is not of much interest in our analysis of the energy changes. There is however a change in the position of the mass relative to its equilibrium position. Every time the spring is compressed or stretched relative to its relaxed position, there is an increase in the elastic potential energy. The amount of elastic potential energy depends on the amount of stretch or compression of the spring. The equation that relates the amount of elastic potential energy (PEspring) to the amount of compression or stretch (x) is

PEspring = ½ • k•x^2

where k is the spring constant (in N/m) and x is the distance that the spring is stretched or compressed relative to the relaxed, unstretched position.

When the air track glider is at its equilibrium position (position C), it is moving it's fastest (as discussed above). At this position, the value of x is 0 meter. So the amount of elastic potential energy (PEspring) is 0 Joules. This is the position where the potential energy is the least. When the glider is at position A, the spring is stretched the greatest distance and the elastic potential energy is a maximum. A similar statement can be made for position E. At position E, the spring is compressed the most and the elastic potential energy at this location is also a maximum. Since the spring stretches as much as compresses, the elastic potential energy at position A (the stretched position) is the same as at position E (the compressed position). At these two positions - A and E - the velocity is 0 m/s and the kinetic energy is 0 J. So just like the case of a vibrating pendulum, a vibrating mass on a spring has the greatest potential energy when it has the smallest kinetic energy. And it also has the smallest potential energy (position C) when it has the greatest kinetic energy. These principles are shown in the animation below.

Definition and graphic provided by Physics Classroom