An object that possesses mechanical energy is able to do work. In fact, mechanical energy is often defined as the ability to do work. Any object that possesses mechanical energy – whether it is in the form of potential energy or kinetic energy – is able to do work. That is, its mechanical energy enables that object to apply a force to another object in order to cause it to be displaced.

Numerous examples can be given of how an object with mechanical energy can harness that energy in order to apply a force to cause another object to be displaced. A classic example involves the massive wrecking ball of a demolition machine. The wrecking ball is a massive object that is swung backwards to a high position and allowed to swing forward into building structure or other object in order to demolish it. Upon hitting the structure, the wrecking ball applies a force to it in order to cause the wall of the structure to be displaced. The diagram below depicts the process by which the mechanical energy of a wrecking ball can be used to do work.

A hammer is a tool that utilizes mechanical energy to do work. The mechanical energy of a hammer gives the hammer its ability to apply a force to a nail in order to cause it to be displaced. Because the hammer has mechanical energy (in the form of kinetic energy), it is able to do work on the nail. Mechanical energy is the ability to do work.

Another example that illustrates how mechanical energy is the ability of an object to do work can be seen any evening at your local bowling alley. The mechanical energy of a bowling ball gives the ball the ability to apply a force to a bowling pin in order to cause it to be displaced. Because the massive ball has mechanical energy (in the form of kinetic energy), it is able to do work on the pin. Mechanical energy is the ability to do work.

A dart gun is still another example of how mechanical energy of an object can do work on another object. When a dart gun is loaded and the springs are compressed, it possesses mechanical energy. The mechanical energy of the compressed springs gives the springs the ability to apply a force to the dart in order to cause it to be displaced. Because of the springs have mechanical energy (in the form of elastic potential energy), it is able to do work on the dart. Mechanical energy is the ability to do work.

A common scene in some parts of the countryside is a “wind farm.” High-speed winds are used to do work on the blades of a turbine at the so-called wind farm. The mechanical energy of the moving air gives the air particles the ability to apply a force and cause a displacement of the blades. As the blades spin, their energy is subsequently converted into electrical energy (a non-mechanical form of energy) and supplied to homes and industries in order to run electrical appliances. Because the moving wind has mechanical energy (in the form of kinetic energy), it is able to do work on the blades. Once more, mechanical energy is the ability to do work.

**The Total Mechanical Energy**

As already mentioned, the mechanical energy of an object can be the result of its motion (i.e., kinetic energy) and/or the result of its stored energy of position (i.e., potential energy). The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy. This sum is simply referred to as the total mechanical energy (abbreviated TME).

**TME = PE + KE**

As discussed earlier, there are two forms of potential energy discussed in our course – gravitational potential energy and elastic potential energy. Given this fact, the above equation can be rewritten:

**TME = PE _{grav}**

**+ PE**

_{spring}

**+ KE**

The diagram below depicts the motion of Li Ping Phar (esteemed Chinese ski jumper) as she glides down the hill and makes one of her record-setting jumps.

The total mechanical energy of Li Ping Phar is the sum of the potential and kinetic energies. The two forms of energy sum up to 50 000 Joules. Notice also that the total mechanical energy of Li Ping Phar is a constant value throughout her motion. There are conditions under which the total mechanical energy will be a constant value and conditions under which it will be a changing value. This is the subject of the work-energy relationship. For now, merely remember that total mechanical energy is the energy possessed by an object due to either its motion or its stored energy of position. The total amount of mechanical energy is merely the sum of these two forms of energy. And finally, an object with mechanical energy is able to do work on another object.

# Power

The quantity work has to do with a force causing a displacement. Work has nothing to do with the amount of time that this force acts to cause the displacement. Sometimes, the work is done very quickly and other times the work is done rather slowly. For example, a rock climber takes an abnormally long time to elevate her body up a few meters along the side of a cliff. On the other hand, a trail hiker (who selects the easier path up the mountain) might elevate her body a few meters in a short amount of time. The two people might do the same amount of work, yet the hiker does the work in considerably less time than the rock climber. The quantity that has to do with the rate at which a certain amount of work is done is known as the power. The hiker has a greater *power rating* than the rock climber.

Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation.

**Power = Work / time**

or **P = W / t**

The standard metric unit of power is the **Watt**. As is implied by the equation for power, a unit of power is equivalent to a unit of work divided by a unit of time. Thus, a Watt is equivalent to a Joule/second. For historical reasons, the *horsepower* is occasionally used to describe the power delivered by a machine. One horsepower is equivalent to approximately 750 Watts.

Most machines are designed and built to do work on objects. All machines are typically described by a power rating. The power rating indicates the rate at which that machine can do work upon other objects. Thus, the power of a machine is the work/time ratio for that particular machine. A car engine is an example of a machine that is given a power rating. The power rating relates to how rapidly the car can accelerate the car. Suppose that a 40-horsepower engine could accelerate the car from 0 mi/hr to 60 mi/hr in 16 seconds. If this were the case, then a car with four times the horsepower could do the same amount of work in one-fourth the time. That is, a 160-horsepower engine could accelerate the same car from 0 mi/hr to 60 mi/hr in 4 seconds. The point is that for the same amount of work, power and time are inversely proportional. The power equation suggests that a more powerful engine can do the same amount of work in less time.

A person is also a machine that has a *power rating*. Some people are more power-full than others. That is, some people are capable of doing the same amount of work in less time or more work in the same amount of time. A common physics lab involves quickly climbing a flight of stairs and using mass, height and time information to determine a student’s personal power. Despite the diagonal motion along the staircase, it is often assumed that the horizontal motion is constant and all the force from the steps is used to elevate the student upward at a constant speed. Thus, the weight of the student is equal to the force that does the work on the student and the height of the staircase is the upward displacement. Suppose that Ben Pumpiniron elevates his 80-kg body up the 2.0-meter stairwell in 1.8 seconds. If this were the case, then we could calculate Ben’s *power rating*. It can be assumed that Ben must apply an 800-Newton downward force upon the stairs to elevate his body. By so doing, the stairs would push upward on Ben’s body with just enough force to lift his body up the stairs. It can also be assumed that the angle between the force of the stairs on Ben and Ben’s displacement is 0 degrees. With these two approximations, Ben’s power rating could be determined as shown below.

Ben’s power rating is 871 Watts. He is quite a *horse*.

**Another Formula for Power**

The expression for power is work/time. And since the expression for work is force*displacement, the expression for power can be rewritten as (force*displacement)/time. Since the expression for velocity is displacement/time, the expression for power can be rewritten once more as force*velocity. This is shown below.

This new equation for power reveals that a powerful machine is both strong (big force) and fast (big velocity). A powerful car engine is strong and fast. A powerful piece of farm equipment is strong and fast. A powerful weightlifter is strong and fast. A powerful lineman on a football team is strong and fast. A *machine* that is strong enough to apply a big force to cause a displacement in a small mount of time (i.e., a big velocity) is a powerful machine.

**Check Your Understanding**

Use your understanding of work and power to answer the following questions. When finished, click the button to view the answers.

1. Two physics students, Will N. Andable and Ben Pumpiniron, are in the weightlifting room. Will lifts the 100-pound barbell over his head 10 times in one minute; Ben lifts the 100-pound barbell over his head 10 times in 10 seconds. Which student does the most work? ______________ Which student delivers the most power? ______________ Explain your answers.

**See Answer**

Ben and Will do the same amount of work. They apply the same force to lift the same barbell the same distance above their heads.

Yet, Ben is the most “power-full” since he does the same work in less time. Power and time are inversely proportional.

2. During a physics lab, Jack and Jill ran up a hill. Jack is twice as massive as Jill; yet Jill ascends the same distance in half the time. Who did the most work? ______________ Who delivered the most power? ______________ Explain your answers.

**See Answer**

Jack does more work than Jill. Jack must apply twice the force to lift his twice-as-massive body up the same flight of stairs. Yet, Jill is just as “power-full” as Jack. Jill does one-half the work yet does it one-half the time. The reduction in work done is compensated for by the reduction in time.

3. A tired squirrel (mass of approximately 1 kg) does push-ups by applying a force to elevate its center-of-mass by 5 cm in order to do a mere 0.50 Joule of work. If the tired squirrel does all this work in 2 seconds, then determine its power.

**See Answer**

The tired squirrel does 0.50 Joule of work in 2.0 seconds. The power rating of this squirrel is found by

P = W / t = (0.50 J) / (2.0 s) = **0.25 Watts**

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4. When doing a *chin-up*, a physics student lifts her 42.0-kg body a distance of 0.25 meters in 2 seconds. What is the power delivered by the student’s biceps?

**See Answer**

To raise her body upward at a constant speed, the student must apply a force which is equal to her weight (m•g). The work done to lift her body is

W = F * d = (411.6 N) * (0.250 m)

W = 102.9 J

The power is the work/time ratio which is (102.9 J) / (2 seconds) = **51.5 Watts** (rounded)

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5. Your household’s monthly electric bill is often expressed in kilowatt-hours. One **kilowatt-hour** is the amount of energy delivered by the flow of l kilowatt of electricity for one hour. Use conversion factors to show how many joules of energy you get when you buy 1 kilowatt-hour of electricity.

**See Answer**

Using conversion factors, it can be shown that 1 kilo-watt*hour is equivalent to **3.6 x 10 ^{6} Joules**. First, convert 1 kW-hr to 1000 Watt-hours. Then convert 1000 Watt-hours to 3.6 x 10

^{6}Watt-seconds. Since a Watt-second is equivalent to a Joule, you have found your answer.

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6. An escalator is used to move 20 passengers every minute from the first floor of a department store to the second. The second floor is located 5.20 meters above the first floor. The average passenger’s mass is 54.9 kg. Determine the power requirement of the escalator in order to move this number of passengers in this amount of time.

**See Answer**

A good strategy would involve determining the work required to elevate one average passenger. Then multiply this value by 20 to determine the total work for elevating 20 passengers. Finally, the power can be determined by dividing this total work value by the time required to do the work. The solution goes as follows:

W_{1 passenger} = F • d • cos(0 deg)

W_{1 passenger} = (54.9 kg • 9.8 m/s^{2}) • 5.20 m = 2798 J (rounded)

W_{20 passengers} = 55954 J (rounded)

P = W_{20 passengers} / time = (55954 J) / (60 s)

**P = 933 W**