Examples of Energy Transfer in a Closed System

A closed system is a type of system that does not allow the transfer of mass in or out of the system. These following examples of closed systems should help in understanding energy transfers, and it’s relation to work done.

Ball & Earth §

Take an closed System of a small ball falling down via gravity. The Earth is not a part of this system.

Gravity is an external force on the ball, because it is requires at least two objects in the system to be considered an internal force. I.e. we could just as easily use any other external force, as long as it has the same direction and magnitude as gravity.

Such a system has zero Potential Energy, because the ball is the only object in the system and potential energy is due to the configuration of multiple objects in a system.

If we look at this system in two configurations, we know for a fact that the system has gained energy, because there has been a change in position (displacement), due to the external force (gravity). Since the displacement is parallel to the external force, there has been positive work done on the system.

Furthermore, we know there has been constant acceleration, due to Newton’s second law, $a=\frac{F_{ext}}{m_{ball}}$. With that we can deduce the ball has had a change in velocity, and hence has a positive gain in kinetic energy.

The kinetic energy is a direct result of the work done on the system:

$$W=\Delta K$$

which is shows the Work-Energy Theorem.

Arrow and Target §

Take an example of an arrow, shot from a bow, hitting a stationary target. We consider the closed system to be only the arrow. The three configurations will be the arrow just before being fired from the bow, the arrow flying through the air, and the arrow hitting the target. For this example, let’s assume the air and Earth don’t exist, so we have no gravitational force nor air drag. Since the bow and target are not part of the system, we will simply show a change in position:

In configuration 1: (Instant bowstring is released)

• An external force (the bow) acts on the system.
• The arrow is not moving (yet).
• Since there are no conservative forces, the Potential Energy in the system is 0.
• The arrow velocity is zero, so the kinetic energy is also zero.
• Total energy of a system $E_{sy{s}_1}=\sum K_{obj}+U=0$

In configuration 2: (Instant arrow leaves bowstring)

• The bowstring’s external force has caused a displacement in the arrow, facing the same direction as the force, so $W=\vec{F}\cdot \vec{x}$ is positive.
• The contact force from the bowstring is no longer present.
• From the Work-Energy Theorem $W=\Delta K$, so the system now has a gain in kinetic energy, which translates to the arrow moving.

In configuration 3: (Instant arrow touches target wall)

• The target exerts an external force on the system, which is opposite to the displacement of the arrow.
• The arrow is still moving, so the system still has energy.

In configuration 4: (Instant arrow stops moving)

• The work done by the target on the system is negative, as the external force is opposite to the wall.
• This work removes energy from the system.
• Since $W=\Delta K$, negative work means a decrease in kinetic energy.
• The amount of work done by the target on the arrow is equal in magnitude to the amount of work done by the bow.
• The system has 0 energy.