If a particle or body is acted upon only by conservative forces energy is conserved. This means that the total kinetic and potential energy in the system remains constant, and does not change. Such a system has no friction forces acting on it, and as such is an idealized simplification for solving problems using energy calculations.

Common examples of conservative forces acting on a particle or body are gravitational forces and (elastic) spring forces.

Suppose a particle follows an arbitrary path, represented by the blue curve below.

The gravitational force acting on the particle is pointing down.

The arbitrary path traveled by the particle may be due to the presence of other forces also acting on the particle, but we do not need to consider them, since the work done by gravity is unaffected by them and can therefore be treated independently.

If Δ

where

A negative sign is present in the above equation because vertical displacement (Δ

The expression for

Consider a spring force acting on a particle, as shown below. The spring is attached to a wall at point

Note that the dashed line represents the equilibrium position of the spring, where the spring is unstretched. Once again, the possible presence of other forces acting on the particle is irrelevant to this discussion, since we are only focusing on the work done by the spring.

The work done by the spring (

where

For the above equation we are assuming we have a spring that obeys Hooke's Law.

Therefore, the work done by the spring depends only on the position of

The (total) work done on the particle by the various forces (conservative and non conservative) as it moves from position

Where:

The right side of the above equation represents the change in kinetic energy of the particle between

If an elastic spring force and gravity force are acting on the particle, we can write:

Substituting this equation into equation (1) we get

This can be rewritten as

This equation tells us that the sum of the kinetic energy (1/2

Equation (3) can be generalized as follows:

Where:

We define the potential energy for gravity as:

where the height

We define the potential energy for an elastic spring as:

where

One can make a choice whether to use the general equation (1) (which applies whether or not there is conservation of energy in the system), or equation (4) which applies only when there is conservation of energy in the system.

Equation (4) can also be applied to a system of particles that are only subjected to conservative forces. As a result we can write:

This equation tells us that the sum of the initial kinetic and potential energy in the system of particles is equal to the sum of the final kinetic and potential energy in the system of particles.

If we have a rigid body instead of a particle, the same basic analysis applies. We simply treat the center of mass of the rigid body as a particle, and apply the above procedure to find the work done by gravity. In other words, if we want to find the work done by gravity on the rigid body we look at the motion of its center of mass, and then apply the equation above for

Suppose the center of mass of a rigid body follows an arbitrary path, represented by the blue curve below.

The work done by gravity on the rigid body is dependent only on the change in vertical height Δ

If Δ

Note that it doesn’t matter at all how the rigid body moves. We only need to know the vertical displacement of its center of mass (as it moves from

If we replace the particle used in the previous case with a point on a rigid body (to which the end of the spring is attached), the work done by the spring on the rigid body is the same as for the work done on the particle (in the previous case). The figure below illustrates this situation.

The work done by the spring on the rigid body is dependent only on the amount the spring is stretched or compressed from its equilibrium (unstretched) position, as it moves from

The work done by the spring (

where

For the above equation we are assuming we have a spring that obeys Hooke's Law.

Note that it doesn’t matter at all how the rigid body moves. We only need to know the amount the spring is stretched or compressed from its equilibrium position (as it moves from

The (total) work done on the rigid body by the various forces (conservative and non conservative) as it moves from position

where

Now,

where the variables in

If an elastic spring force and gravity force are acting on the rigid body, we can write:

Substituting this equation into equation (5) we get

This can be rewritten as

This equation tells us that the sum of the kinetic energy (

Equation (7) can be generalized as follows:

Where:

We define the potential energy for gravity as:

where the height

We define the potential energy for an elastic spring as:

where

One can make a choice whether to use the general equation (5) (which applies whether or not there is conservation of energy in the system), or equation (8) which applies only when there is conservation of energy in the system.

Equation (8) also applies to a system of rigid bodies that are only subjected to conservative forces (meaning there is conservation of energy in the system). For example, frictionless pins or inextensible cords may connect the bodies. Consequently, the forces acting at the points of contact between the bodies contribute zero work. The reason for this is because these forces are equal and opposite on each pair of contacting bodies (Newton’s Third Law), and move through the same distance. Therefore they cancel out, and contribute zero work to the system.

The figure below shows a general pendulum, in which an arbitrary rigid body is swinging back and forth in a plane, about a pivot

Additional variables are defined as follows:

By geometry,

Since we are ignoring friction, the only forces acting on the pendulum are gravity and an (elastic) spring force. These forces are conservative, therefore we have conservation of energy in the system and we can apply equation (8):

Since

where

Similarly, for the lowest part of the swing:

The potential energy at the initial position is equal to the sum of the gravitational potential energy and the spring potential energy:

Similarly, for the lowest part of the swing:

Thus we can write the following equation for conservation of energy of the pendulum:

From this equation we can solve for

This is the angular velocity of the pendulum at the lowest point in the swing.

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