WORK AND ENERGY
Work in physics is defined as the product of a force and the distance an object moves while the force is being applied to it. The force and distance must be in the same direction. The equation is:
W = FscosӨ
Where W is the work done, F is the magnitude of the applied force, s is the distance moved during application of the force, and Ө is the angle between the direction of the force and the direction of the distance moved.
Example
Find the work done by a woman pulling a suitcase by exerting a force of 94 N at an angle of 25 degrees with the horizontal for a distance of 35 m.
Find the work done if the force is parallel to the horizontal.
The basic unit of work is the joule(J) and is equal to the product of one newton and one meter. In the English system of measurement, the unit of work is the foot pound, the product of a force of one pound acting through a distance of one foot.
In order for work to be done, there must be a force acting and a distance moved. If the force is zero, the work done is zero. Also, if the distance moved is zero, the work done is zero. If the angle between the direction of the force and the distance moved is 90o, the work done is zero since the force and distance must be in the same direction. If the force and distance are in opposite directions, the work done by the force is negative since the cos180o is negative one.
The work-energy theorem states that work done on an object by a net force results in a change in the kinetic energy of the object. This means that positive work done on an object will result in an increase in the kinetic energy of the object and negative work will result in a decrease in the object's kinetic energy.
Kinetic energy is defined as energy of motion and is calculated using the formula:
KE = ½MV2
Where KE is the kinetic energy, M is the mass of the moving object, and V is the speed of the moving object. The kinetic energy of an object can change due to changes in either the mass or speed but we will usually be considering a change in speed. This leads to the work-energy equation:
W = ΣFscosӨ = ½MVf2 - ½MVo2
Where W is the work done, ΣF is the net force, s is the distance moved, Ө is the angle between the net force and the distance moved, M is the mass of the object, Vf is the final speed of the object and Vo is the initial speed of the object.
Example
A 0.045 kg golf ball is hit with a club that gives it an initial velocity of 41.0 m/s. Find the work done on the golf ball.
If the ball and the club are in contact for a distance of 0.100 m, find the average force exerted on the ball by the club.
Work Done By the Force of Gravity
When an object falls from a height ho to a height hf, the equation used to calculate the work done by gravity is:
W = Mg(h0 - hf)
Where W is the work done, M is the mass of the falling object, g is the acceleration due to gravity, hf is the final height, and ho is the initial height. Mg is the force acting on the object and the difference in height is the distance it moved. Since they act in the same direction, the cosine of the angle between them is one and does not affect the magnitude of the work. If an object initially goes up and then comes down, we need only consider the difference in initial height and final height. The total path followed does not affect total work done if we ignore friction.
Gravitational potential energy is equal to the work done in raising the mass that has this energy to the height associated with this energy. Since the force that must be applied to move the mass upward at a constant velocity is equal to its weight, its
GPE is equal to mgh where m is its mass, g is the acceleration due to gravity, and h is the height to which it is raised.
Remember that the potential energy created here belongs to the object and the earth as a system although we will often talk about potential energy of the object alone, not the system.
Example
Find the gravitational potential energy of a 55 kg person at the top of the SearsTower, 443 m above the ground.
The gravitational force is a conservative force. Conservative is defined as either of two ways:
(1) A force is conservative if the work done is independent of the path taken by the object between its initial and final positions.
(2) A force is conservative when it does no net work on an object moving around a closed path so that it ends its motion at the same place it started.
Some forces are not conservative. Friction is a nonconservative force since the amount of work done by a frictional force depends on the path taken by the object. Since in most normal situations work is done by both conservative and nonconservative forces, the equation for work done by a net force is:
W = Wc + Wnc
Where W is the total work done, Wc is the work done by a conservative force, and Wnc is the work done by a nonconservative force.
Example
A 16 kg sled is pulled along a horizontal surface with a horizontal force of 24 n. Starting from rest, the sled reaches a speed of 2.0 m/s in a distance of 8.0 m. Find the coefficient of kinetic friction between the sled and the surface.
With this equation, if gravity is the only conservative force acting on the object, then we can write:
Wc = PE0 - PEf
And from the work-energy theorem:
Wnc + PE0 - PEf = KEf - KEo
which gives us:
Wnc = ΔKE + ΔPE
And tells us that any work done by an outside force on a system results in a change in the potential energy of a system, the kinetic energy of a system, or both.
Example
Rocket Man starts from rest on the ground and is propelled straight up. At a height of 16 m he has a speed of 5.0 m/s.
His mass including his propulsion unit remains approximately constant at 136 kg. Find the work done on him and his propulsion unit.
If Wnc is zero, then we have a truly conservative system and any change in potential energy leads to an equal and opposite change in kinetic energy.
The principle of conservation of mechanical energy arises from the work energy theorem. If an outside force does work on a system then the total energy of the system must change. The equation is :
Wnc = Ef - Eo
which indicates that work done by a nonconservative force is equal to the change in total energy of the system. This change can occur either in the potential or kinetic energy of the system. It can also occur in both.
If there is no work done on the system by a nonconservative force, then the total energy of the system does not change and, in fact, is equal to a constant value.In this case energy can be changed from kinetic to potential or potential to kinetic in a fashion that keeps the total of the two constant.
E = KE + PE
is the equation for this relationship. An increase in KE will result in an equal decrease in PE and an increase in PE will result in an equal decrease in KE.
Example
A gymnast is swinging on a high bar. The distance between his waist(center of mass) and the bar is 1.1 m. If his speed at the top of his swing is zero and his gain in speed is due entirely to his change in gravitational potential energy, find his speed at the bottom of his swing.
Another concept associated with work and the transfer of energy is power. Power is defined as the rate at which work is done. The equation for the calculation of power from its definition is:
P = ΔW/Δt
Where P is the average power, ΔW is the work done, and Δt is the time required to do the work. The SI unit of power is the watt which equals one joule per second. 1000 watts is called a kilowatt and other metric prefixes are used to express very large or small numbers of watts(mega, giga, milli, etc.).
Power is also used to express the rate of energy consumption. A light bulb rated at 60 watts uses 60 joules of electrical energy every second.
Example
Find the average power required to accelerate a car from rest to 20.0 m/s in 5.6 s along a level stretch of road if the weight of the car is 9000 n.
The rate of energy production of an automobile is generally given in an English system power unit, the horsepower which is approximately 746 watts. We expect more horsepower to mean greater change in velocity but if we are accelerating a more massive load, a smaller acceleration will result. The equation relating force, power and velocity comes from the basic definition of power.
P = ΔW/Δt
P = FaveS/Δt
P = FaveV
Remember that these values are average or constant values.
Other forms of energy are conserved as well as mechanical energy. It turns out that the total amount of energy that exists before an event is equal to the total amount of energy that exists after the event. However, some of the energy may have changed from one form to another.
Consider the case of a candy bar that is eaten by a hiker. The candy bar contains chemical potential energy which is taken in by the hiker. The hiker's body converts that chemical potential energy into heat and kinetic energy. The heat is lost to the atmosphere and the kinetic energy can be changed into gravitational potential energy as the hiker climbs. If we add the heat energy and gravitational potential energy together we get the total energy originally in the candy bar.
This is an example of the Law of Conservation of Energy which states:
ENERGY CAN BE NEITHER CREATED NOR DESTROYED, BUT CAN ONLY BE CONVERTED FROM ONE FORM TO ANOTHER.
So the total energy does not change.
The one notable exception to this occurs during nuclear reactions when matter and energy are converted according to Einstein's equation E = mc2.
When work is done by a variable force, the equations become a little more complicated. If the force varies in a linear fashion with distance, an average force can be used to find the work done.
Consider a spring being stretched where the magnitude of the force = kx, with k as the spring constant in newtons per meter and x is the distance stretched. When x = 0, F = 0 and when x=xmax, Fmax = kxmax. The equation for the work done becomes:
W = Favexmax
Since Fave = (0 + Fmax)/2,
W = ½(Fmaxxmax)
W = ½(kxmax2)
This is the equation for the energy stored in a stretched spring if we do not stretch it beyond its elastic limit.
If the equation governing the relationship between the force and the distance over which it is applied becomes more complex, it becomes necessary to use more advanced forms of mathematical analysis(calculus, etc.).
The amount of work can be approximated using a graph of force vs distance. The area bounded by the X axis and the graph of the force function represents the work done. If you count the blocks and partial blocks in that area you get a good approximation for the work done.
W = 9.00 X .0278 = .25J for one block.
There are 242 blocks under the curve.
Total work = 242 X .25 = 60.5J