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Study Of Work, Power, and Energy

Work, Power, and Energy


When people move objects from one place to another, they are doing work. Lifting boxes, moving furniture, pushing lawnmowers, etc., all involve doing work.

Work is the product of the force and the displacement of an object in the direction of the force. Work is done only if an object is moved in the direction of the force.

The general Equation for work is W = F d

The unit for work is the Joule (J);  1 J = 1 N m

Example: A student pushes on an object with a force of 100 N. The object moves 2 m across the floor. Calculate the amount of work done.

G:  F = 100 N ; d = 2 m
F:  Work done to move the object.
E:  W = F d
S:  W = 100 N * 2 m
A:  W = 200 N m = 200 J

The study of work requires that the direction of the force acting on an object be taken into consideration. Only the component of the force acting in the direction of the motion does any actual work.

The component of the force, which does work on an object is in the direction of the motion experienced by the object. The calculation of this component force requires that you incorporate the cosine function into the work equation. Here is that equation:

Equation: W = F (cos A) d

The cosine function allows for the component force acting in the direction of the motion of an object to be calculated.

Example: A children’s radio flyer wagon is pulled along with the handle being at an angle of 45 degrees measured from the horizontal. The force applied to the handle is 40 N. Calculate the amount of work necessary to move the wagon 5 m from its original position.

G:  F = 40 N ; d = 5 m ; angle is 45 degrees
F:  Work done to move the wagon
E:  W = F (cos A) d
S:  W = 40 N (cos 45 deg) 5 m
A:  W = 105.1 N m = 105.1 J


Power is defined as the rate at which work is done. The equation is:

Equation:  P = W / t

The unit for power is the watt (W).  1 W = 1 J / s

A machine that does work at a rate of 1 J / s has a power of 1 W.  Because a J is a N m, a W = 1 N m / s

Example: An electric motor hoists a 1200-kg steel beam 8 m in 10 s. Calculate the power in watts. Then calculate the power in kilowatts.

G: m = 1200 kg ; d = 8 m ; t = 10 s
F: power in watts and in kilowatts.
E: P = W / t , and W = F d , and F = F(g) = weight = m g
S: F(g) = m g = 1200 kg * 9.8 m/s^2 = 11,760 N
    W = F d = 11,760 N * 8 m = 94,080 J
    P = W / t = 94,080 J / 10 s = 9,408 W
A: P = 9,408 W

Simple Machines: (Includes mechanical advantage and efficiency.)

All machines are combinations of the simple machines. There are six simple machines. They are the lever, the pulley, the inclined plane, the wheel and axle, the wedge, and the screw.

A bicycle is a complex machine made from a combination of simple machines. What are the simple machines used in a bicycle?

Machines are useful because they amplify our efforts and increase the amount of work we can do in a period of time. The ability of a machine to amplify our efforts is expressed in terms of mechanical advantage and efficiency.

The mechanical advantage of a machine is the ratio of the force exerted by the machine to the force applied to the machine. The equation is:

Equation: MA = F(resistance) / F(effort)

There is no unit involved here, because the units of force divide out.

Example: The effort applied to a lever is equal to 20 N of force. The resistance force against which the lever works is 80 N. Calculate the mechanical advantage of this machine.

G: F(effort) = 20 N ; F(resistance) = 80 N
F: the mechanical advantage of this machine.
E: MA = F(resistance) / F(effort)
S: MA = 80 N / 20 N
A: MA = 4

Ideal Mechanical Advantage is defined as the ratio of the effort distance to the resistance distance. The equation is:

IMA = d(effort) / d(resistance)

Example:  If a lever, set up to lift an object, is placed such that the end upon which a person pushes down is 3 m from the fulcrum and the object to be lifted is 1 m from the fulcrum, what is the Ideal Mechanical Advantage.

G: d(effort) = 3 m ; d(resistance)
F: The ideal mechanical advantage
E: IMA = d(effort) / d(resistance)
S: IMA = 3 m / 1 m
A: IMA = 3

Since the output work of a real machine (versus an ideal one) is always less than the input work, the efficiency of a real machine is expected to be less that 100%.  For example, automobiles have had efficiencies below 20%. Much of the energy in a gallon of gasoline ends up as heat, which is why the coolant and the oil are so necessary to the smooth operation of the vehicle.

The efficiency of a machine is the ratio of the output work to the input work. The equation is:

Equation: eff = [W(output) / W(input)] * 100 %   [Remember that multiplying by the 100 % value, the fraction produced by the division is multiplied by 100, and the % unit symbol is introduced into the measurement.]

Example: The work input into a machine is 110 J. The output work of the machine is 65 J. Determine the efficiency of the machine.

G: W(output) = 65 J ; W(input) = 110 J
F: The efficiency of the machine.
E: eff = [W(output) / W(input)] * 100 %
S: eff = [ 65 J / 110 J ] * 100 %
A: eff = 59 %

There is also another equation for efficiency. The eff = [MA / IMA) * 100%

Example: The mechanical advantage of a machine is found to be 5, while the ideal mechanical advantage of the machine is determined to be 9. Calculate the efficiency of this machine.

G: MA = 5, IMA = 9
F: The efficiency of this machine
E: eff = [MA / IMA) * 100%
S: eff = [ 5 / 9 ] * 100%
A: eff = 55.6 %

Energy: (Includes Potential and Kinetic Energy.)

Like work, energy is a measurable quantity. Energy is defined as the capacity to do work.

Work is the measure of transferred energy. Work is a means of measuring energy. We could say that it is a means of measuring the energy necessary to accomplish a specific task.

If you push or pull an object across a floor, energy is being transferred from you to the object and the floor. You have to provide a force to move the object. That force has to overcome any opposing force such as static and kinetic friction. That requires you use some of the energy you have taken in as food. The longer the distance over which you must sustain that force, the more work is done, and, likewise, the more energy is being transferred.

Some of the energy when objects are moved are in the forms of heat and on occasion light. Friction produces heat where two objects are in contact while moving past one another. In some instances light is generated by the contact. When a hammer strikes a piece of metal, it will not only generate heat at the point of contact, but it may also produce sparks.

It is not practical to measure the individual energies in an object moving across a floor or in the floor itself before and after the move occurs. It is, however possible to measure the work done in the process of moving the object, and the work done is the energy transferred (energy change) for the object floor system. Again remember that all measurements of work are measurements of the energy being transferred in each circumstance.

There are different types or kinds of energy. Frequently energy is defined as belonging to one of three categories. These categories of energy are described as gravitational, electromagnetic, and nuclear. These divisions may be broken down further into subcategories.

For example, thermal energy, mechanical energy, and light energy are terms often used when discussing energy. These terms help focus on only one particular aspect of a broader category of energy. Thermal energy is associated with the motion of particles, such as atoms or molecules in a substance. The speeds of the particles affect the kinetic energy of the particles and this kinetic energy is associated with the measure of thermal energy. When particles collide they transfer energy and it is the electrical force between the electrons that does the work and transfers the energy. The transfer of thermal energy belongs to the discussion of electromagnetic energy. However, in order to develop a sense of what thermal energy is, it is not necessary to study electromagnetic energy in depth. Typically, such a study is reserved for the chemistry and physics majors and those who are curious enough to pursue it on their own.

In this unit, gravitational potential energy and its relationship to kinetic energy is the focus of study. Potential energy is energy that is due to the position of an object. It is the energy of an object due to its position with respect to a reference position (a position defined as having 0 J). This reference or base level is arbitrarily chosen to be zero. If not specifically stated in a problem it is common practice to consider the ground to be the reference point. In other situations sea level might be the reference point, and so on.

Equation: GPE = m g h ; where m = mass, g = gravitational acceleration, and h = height.

Kinetic energy is due to the motion of an object. A moving object has mass and a velocity. The faster an object moves the more kinetic energy it has.

Equation: KE = (1/2) m v^2

Consider that energy is measured in terms of the work it does or can do. Energy has the same units as work. Energy and work are both scalar quantities. Regardless of the type or category to which it belongs, energy is always either potential or kinetic.

The following equations apply to work and energy.

W = /\PE = PE(f) - PE(i)
W = /\KE = KE(f) – KE(i)
/\PE = /\KE

Example: Calculate the potential energy of a 40 kg mass 20 m above the ground.

G: m = 40 kg, g = 9.81 m/s^2, h 20 m
F: The potential energy
E: PE = m g h
S: PE = 40 kg * 9.81 m/s^2 * 20 m
A: PE = 7,848 J

Example: Calculate the kinetic energy of a 100 kg mass traveling at 5 m/s.

G: m = 100 kg, v = 5 m/s
F: The kinetic energy
E: KE = (1/2) m v^2
S: KE = (1/2) * 100 kg * (5 m/s)^2
A: KE = 1,250 J

Conservation of Energy:  Energy can be stored and stored energy can be released. In the case of gravitational energy, energy can be stored in the form of placing an object at a greater height than it was originally located at. If that object is allowed to fall that energy that was stored now becomes kinetic energy. The kinetic energy of the object increases as the potential energy decreases. Not until the object is stopped, such as hitting the ground is the kinetic energy given up as heat, sound, or light.

The “Law of conservation of energy says that ‘energy can neither be created nor destroyed’.”

Consider an object at a certain height above the ground. The simple diagram below shows how the PE and KE values change as the object falls to the ground.

----- 10 m, PE = 1000 J, KE = 0 J
----- 5 m, PE = 500 J, KE = 500 J
----- 0 m, PE = 0 J, KE = 1000 J

The “Law of conservation of energy” can be expressed by the following equation.

Equation:  KE(i) + PE(i) = KE(f) + PE(f)

Example: A chunk of rock having a mass of 20 kg is dropped from a height of 10 m above the ground. (Here the ground is the reference (0 J) point.)

a. calculate the initial potential energy of the rock
b. calculate the final kinetic energy of the rock
c. calculate the final velocity of the rock just before impact with the ground.

Solution to part a:

G: m = 20 kg, g = 9.81 m/s^2, h = 10 m
F: The initial potential energy
E: PE = m g h
S: PE = 20 kg * 9.81 m/s^2 * 10 m
A: PE = 1,962 J

Solution to part b:

G: Initial KE = 0 J, Initial PE = 1,962 J
F: The final kinetic energy
E: KE(i) + PE(i) = KE(f) + PE(f)
S:  0 J + 1,962 J = KE(f) + 0 J
A: KE(f) = 1,962 J

Solution to part c:

G: m = 20 kg, KE(f) = 1,962 J
F: The final velocity of the object before impact
E: KE = (1/2) m v^2 ; v = (2 KE / m)^0.5
S: v = (2 * 1,962 J / 20 kg)^0.5
A: v = 14.0 m/s

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