What is the Conservation of Energy of a Freely Falling Body and in a Simple Pendulum

Physics Topics cover a broad range of concepts that are essential to understanding the natural world.

What is the Law of Conservation of Energy?

Energy can be transformed (or changed) from one form to another. According to the law of conservation of energy : Whenever energy changes from one form to another, the total amount of energy remains constant. In other words, when energy changes from one form to another, there is no loss or gain of energy.

The total energy before and after transformation remains the same. Another definition of the law of conservation of energy is that: Energy can neither be created nor destroyed.

During the conversion of energy from one form to another, some energy may be wasted. For example, when electrical energy is converted into light energy in an electric bulb, then some electrical energy is wasted in the form of heat.

Although some energy may be wasted during conversion, but the total energy of the system remains the same. We will now take an example to understand the conservation of energy more clearly.

Conservation of Energy During the Free Fall of a Body

Suppose we have a ball of mass m and we raise it to a height h above the ground. The work done in raising the ball gives it a potential energy equal to m × g × h. Let us allow the ball to fall downwards. As the ball falls, its height h above the ground decreases and thus the potential energy also decreases. But as the ball falls, its velocity y constantly increases and, therefore, its kinetic energy \(\frac{1}{2}\) mv² also increases.

As the ball falls more and more, its potential energy is gradually converted into an equal amount of kinetic energy. But the sum of potential energy and kinetic energy of the ball remains the same at every point during its fall. When the ball just reaches the ground, its potential energy becomes zero (because h becomes zero) and its kinetic energy becomes the maximum (because v becomes the maximum).

At this stage, all the potential energy has been converted into kinetic energy. From this we conclude that the potential energy of ball has been changed into an equal amount of kinetic energy. There is no destruction of energy, and the total amount of energy remains constant. This is an example of the conservation of energy during the free fall of a body.

When a falling ball hits the ground, a sound (of hitting) is produced and the ground (where the ball hits) also gets heated slightly. This means that when a falling ball hits the ground, then some of its kinetic energy is converted into sound energy and heat energy. But the total energy (kinetic energy + sound energy + heat energy) remains the same. Thus, the law of conservation of energy is valid even after the ball hits the ground.

The conservation of energy during the free fall of a body will become more clear from the following data obtained in an experiment in which the potential energy (P.E.) and kinetic energy (K.E.) of a freely falling ball were calculated at different positions of its downward journey :

We can see from the data given in Figure that :

1. At position A, when the ball is at rest, it has 20 J of potential energy but zero kinetic energy. So, the total energy of the ball at position A is 20 + 0 = 20 J.
2.  At position B when the ball is falling, it has 15 J of potential energy and 5 J of kinetic energy. So, the total energy of the ball at position B is 15 + 5 = 20 J.
3. At position C when the ball has fallen by half the distance, it has 10 J of potential energy and 10 J of kinetic energy. So, the total energy of the ball at position C is 10 + 10 = 20 J.
4. At position D when the ball has fallen by more than half the distance, it has 5 J of potential energy and 15 J of kinetic energy. So, the total energy of the ball at position D is 5 + 15 = 20 J.
5. At position E when the ball is about to hit the ground, it has 0 J of potential energy and 20 J of kinetic energy. So, the total energy of the ball at position E is 0 + 20 = 20 J.

It is clear from the above -observations that as the ball falls downwards, its potential energy goes on decreasing but its kinetic energy goes on increasing. The decrease in potential energy of the freely falling ball at any point in its path appears as an equal increase in its kinetic energy.

So, the total energy (potential energy + kinetic energy) of the ball remains the same (20 joules) at every point during its free fall. Thus, the energy of a freely falling ball is conserved.

If, however, a ball is thrown upwards, then its kinetic energy goes on decreasing and its potential energy goes on increasing. The decrease in kinetic energy of the upward going ball at any point during its flight appears as an equal increase in its potential energy.

But the total energy (kinetic energy + potential energy) of a ball thrown upwards remains constant at every stage of its flight. In this way, the energy of a ball thrown upwards is also conserved. We will now discuss the case of a simple pendulum.

Conservation of Energy in a Simple Pendulum

A swinging simple pendulum is an example of conservation of energy. This is because a Swinging simple pendulum is a body whose energy can either be potential or kinetic, or a mixture of potential and kinetic, but its total energy at any instant of time remains the same.

Thus, a very simple illustration of the transformation of potential energy into kinetic energy, and of kinetic energy back into potential energy is given by a swinging simple pendulum (or an oscillating simple pendulum). This will become more clear from the following discussion.

A simple pendulum consists of a small metal ball (called bob) suspended by a long thread from a rigid support, such that the bob is free to swing back and forth when displaced (see Figure). Initially, the simple pendulum is at rest with its bob in the centre position (or mean position) A.

When the pendulum bob is pulled to one side to position B (to give it potential energy because of higher position of B with respect to position A), and then released, the bob starts swinging (moving back and forth) between positions B and C (see Figure).

1. When the pendulum bob is at position B (see Figure), it has only potential energy (but no kinetic energy).
2. As the bob starts moving down from position B to position A, its potential energy goes on decreasing but its kinetic energy goes on increasing.
3. When the bob reaches the centre position A, it has only kinetic energy (but no potential energy).
4. As the bob goes from position A towards position C, its kinetic energy goes on decreasing but its
   potential energy goes on increasing.
5. On reaching the extreme position C, the bob stops for a very small instant of time. So at position C, the bob has only potential energy (but no kinetic energy).

From the above discussion we conclude that at the extreme positions B and C of a swinging pendulum, all the energy of pendulum bob is potential, and at the centre position A, all the energy of the pendulum bob is kinetic. At all other intermediate positions, the energy of pendulum bob is partly potential and partly kinetic. But the total energy of the swinging pendulum at any instant of time remains the same (or conserved).

The pendulum bob keeps on oscillating (or swinging) for a considerable time but ultimately the oscillations die down and the pendulum stops oscillating. It comes to rest.

This is because the friction at the point of support of the pendulum and friction of air acting on the swinging bob converts the mechanical energy of the oscillating pendulum into heat energy gradually. This heat energy goes into the environment.

It should be noted that the body which does work loses energy and the body on which work is done, gains energy. For example, when we lift a stone from the ground and raise it to a height, we have to do some work on the stone. As a result of doing this work, we lose some energy from our body.

On the other hand, the stone which we raised, gains an equal amount of potential energy. Thus, the total energy remains the same. Now, when we kick a ball, we do some work. In doing this work we lose some energy from our body.

On the other hand, the ball gains an equal amount of kinetic energy and starts moving. Here also, the total energy of the system is conserved. And when we rub our hands vigorously against each other, we do work. The energy lost by our body in doing this work is transformed into heat energy.