Potential Energy

Conservative and Dissipative Forces

Before moving on to the second type of energy, let us look at conservative and dissipative forces.  Recall that we defined the work to be

 

 

What would happen if we then traversed a second path, only this time going from x2 to x1?  In some of the cases, we would find that the total work done would become zero.  This is the hallmark of a conservative force.  Specifically, we say that a force is conservative if

 

  1. It is independent of the path of the body and depends only on the starting point and ending point,
  2. It is completely reversible, and
  3. When the starting and ending points are the same, the total work is zero.

 

If the force is not conservative, it is called a dissipative force.

 

Example:

The force of gravity is a conservative force.  Let a box of mass m be moved around the path shown below and acted on only by gravity.

 

 

Can break the work up into 4 pieces W12, W23, W34, W41.  Then

 

 

Definition of Potential Energy

Whenever a force is conserved, we can define a potential energy for that force as follows:

 

 

(12.1)

 

This is interpreted physically as the potential energy due to the force at position x2 relative to x1.

 

Example:

What is the gravitational potential energy of an object of mass m at a distance h above the ground?

 

 

Define the ground to be at x1 = 0.  Then since the force is constant

 

 

Work-Energy Theorem

            Now let’s use (11.2) to find an important relationship between work and energy.  Let us define Wext to be the work done by all non-conservative forces.  We call all of these forces the external forces.  We can see from the definition of the potential energy that the work done to change altitude is just the opposite of the change in potential energy.  So, using (11.2) and generalizing this to all potential energies, we can write

 

 

where the sum is over the change in potential energies for each conservative force.  Moving the potential energy to the right side of the equation,

 

 

(12.2)

 

This is the work-energy theorem, which states that the total external work done on a system must equal the change in total mechanical energy, where the total mechanical energy is defined to be E = K + U.  Notice that if there are no external forces, then the mechanical energy is constant and we can write

 

 

Example:

A cannon shoots a shell straight up with an initial velocity of 50 m/s.  How high does the shell go before falling back to earth, neglecting air resistance?  Solve this using forces and energy conservation.

 

 

Forces:

Using the equations for position, we have

 

y(t) = y0 + v0t - ½gt2

 

To find the maximum, need to find where the time when the velocity is 0

 

 

Thus,

 

 

Energy Conservation:

Since there are no non-conservative forces acting on the shell, the total energy must remain constant.  The highest point will be reached when the kinetic energy is zero.  Thus

 

Gravitational Potential Energy

Recall that we calculated the work done by a non-uniform gravitational field and by a spring earlier.  At that time, we saw that the work done by a gravitational field was just

 

 

Now recall the definition of the potential energy of a force.  It is related to the work done by a force by the relation

 

dW = -dU

 

It is standard to define the zero potential point for an inverse square field to be at infinity.  Then the work supplied by gravity in bringing an object of mass m to a distance R from a mass M is just (setting r0 = and r' = R)

 

 

and the potential energy at that point, relative to infinity, is

 

 

Notice that the potential energy is negative.  Thus, we say that gravity creates a potential energy well.

Potential Energy of a Spring

Similarly, the work done by the spring as it is stretched a distance x is

 

W = -½kx2

 

where we are now taking the unstretched position to be x = 0.  Then the potential energy with respect to the unstretched position is

 

U = ½kx2

 

Example:

A 0.1 kg ball sits on a spring gun which is aimed upward.  If the spring is compressed 0.05 m and has a spring constant of 250 N/m, how high does the ball go when the gun is fired?

 

Using the work-energy theorem, we see that

 

 

Thus,

 

 

Complex Motions

By combining the linear and rotational motion, we can describe fairly complex motion in a simple manner.

 

Example:

A solid 5 kg cylinder of radius 0.25 m rolls without slipping down a 3 m long frictionless ramp inclined at an angle of 30°.  What is its speed when it reaches the bottom?

 

 

Since there are no dissipative forces, we see that

 

 

Power

So far, we have only considered the amount of work that has been done, but we have not asked how rapidly did it occur.  The rate at which work is performed (or energy is transferred) is called the power.  In a manner analogous with the definition of velocity, we can define the average power to be

 

 

If we let the interval Dt go to zero, the we get the instantaneous power

 

 

(12.3)

 

From dimensional analysis, we see that the unit of power is a joule per sec.  This unit has its own name, and is called a Watt (W).  If the force is constant, we can rewrite (12.3) as

 

 

(12.4)

 

This is the power supplied by the force F to the object.  Again, if we are considering rotational motion, the rotational power for a constant torque is given by

 

 

Example:

A mass of 100 kg is pushed along the floor at a constant rate of 2 m/s.  If the coefficient of sliding friction is 0.25, at what rate is work being done in order to keep it in motion?

 

 

Here we are interested in the power supplied by the force F.  By Newton's third law, we see that it is equal and opposite to the force of friction.  Thus