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# Mechanical Energy Study Guide (page 2)

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Updated on Sep 26, 2011

## Power

Any amount of work done can be performed in different intervals of time, and that will have an effect on how we perceive the total work done. How can we take this into account? Imagine you are taking the stairs up to the second level of a building and you take your time to go up. You are in no hurry, and if you run up the stairs, you will experience more tiredness. So, the shorter time you take to go up the stairs, the more your muscles work. So, how can you characterize this situation? We must define a new quantity: power.

There are other units that you might encounter for power, but the one referred to most often is horsepower (hp).

1 horsepower = 550 feet · pounds/seconds = 745.7 watts

Another way to determine power is through the expression of the work: For a constant force acting along the direction of the displacement, the previous definition becomes:

where the time rate of change of displacement defines the average speed:

and vo is the initial speed, where v is the final speed.

### Power

Power is proportional to the work performed on an object and inversely proportional to the time taken to perform the work:

P stands for power and Δt for the interval of time. The unit for power is called a watt and is equal to joules per second (W = J/s), in honor of James Watt, 1736-1819. One watt is the power needed to perform one joule of work in a time of 1 second.

#### Example

Consider a car of 1,500 kg starting from the stop light and accelerating for a distance of 25 m with an acceleration of 3.5 m/s2. Calculate the average power generated by the force producing the acceleration.

#### Solution

First, check to see that the quantities are expressed in 51. Next, list the known data.

m = 1,500 kg

a = 3.5 m/s2

d = 25 m

p = ?

We can use the second expression of the power because we can determine the accelerating force and the average speed using Newton's second law for the first one, and we can use the equations of motion for the second quantity.

F = m · a = 1,500kg · 3.5m/s2 = 5.2 · 103 N

Now, to find the average speed, although we are given displacement, we do not have the time, so we have to use the equations of motion (Lesson 3) to make the connection between average speed, acceleration, and displacement.

By raising to the second power, we have:

And we can also write the displacement as:

And then the expression for the average speed becomes:

And the average speed is:

We can now calculate the power done by the force of acceleration to be:

## Kinetic Energy

One result of doing work on objects, as we have seen in the examples before, might be motion, or it might be change in position. So, how are we characterizing the difference between these two situations— between motion and change in position? When an object is accelerated or decelerated because of work performed on it, we will measure the energy of motion known as the kinetic energy. Consider an accelerating force (net force) that changes the way an object moves. According to Newton's second law:

F = m · a

The force is acting over a distance d and along the displacement (α = 0°), then the work is:

W = F · d = m · a · d

From Lesson 3, we also know that:

v2 – vo2 = 2 · a · d

Then multiplying both sides with the mass m:

m · (v2vo2) = m · 2 · a · d

And then dividing by 2:

Or:

And again:

With the definition of kinetic energy, the expression of the work can also be written as:

KEKEoz = W

ΔKE = W

This formula is called the work-energy theorem and expresses the fact that when a net external force does work on an object, the work is equal to the change in kinetic energy. This equation also defines the unit for kinetic energy, which is a joule, because the two quantities—work and the change in kinetic energy—are not only proportional but equal. Note that the work-energy theorem does not apply to individual forces but only to net external forces.

### Kinetic Energy

The kinetic energy of an object of mass m and speed v is defined by:

## Gravitational Potential Energy

Consider a body being launched up an inclined plane, and suppose the friction is so small that we can neglect it altogether. If one calculates the work done to raise the object on the inclined plane, one has to consider only the work done by the force along the direction of the plane. That is — W · sin α where α is the angle the inclined plane makes with the horizontal, and W is the weight of the object (the minus shows that although they are on the same axis, the displacement and the force are in opposite directions). To avoid confusion, we will use the expression of the weight with respect to mass: W = m · g. Therefore, the force performing work is (see Figure 7.4):

m · g · sin α

And the displacement on the plane is h/sin α. Hence, the work done by the weight on the object is:

W = F · d = – m · g · sin α · (h/sin α) = – m · g · h

We can also write this expression as:

W = 0 – m · g · h

Or:

W = –(m · g · h – 0) = –(m · g · hm · g · 0)

This helps us define a new form of energy that is not related to motion, or to speed, but to the position from the ground—height (h).

We can now rewrite the expression of the work as the negative of the change in potential energy:

W = – (m · g · hm · g · ho)

W = –(PEPEo)

W = –ΔPE

In Figure 7.4, you can see that the same result can be obtained if you find the work done, by weight, on raising the object directly on the vertical:

W = m · g · h

In this case, the work performed is independent of the chosen path. This is called a conservative force. A conservative force is also defined as the force that determines zero work when it acts on a closed path. An example of a nonconservative force is friction.

### Gravitational Potential Energy

The gravitational potential energy of an object of mass m is defined by:

PE = m · g · h

where h is the distance from the surface of the earth that is considered zero potential energy.

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