Work: the transfer of mechanical energy
Article objectives
Work: the transfer of mechanical energy
The concept of work
The mass contained in a closed system is a conserved quantity, but if the system is not closed, we also have ways of measuring the amount of mass that goes in or out. The water company does this with a meter that records your water use.
Likewise, we often have a system that is not closed, and would like to know how much energy comes in or out. Energy, however, is not a physical substance like water, so energy transfer cannot be measured with the same kind of meter. How can we tell, for instance, how much useful energy a tractor can “put out” on one tank of gas?
The law of conservation of energy guarantees that all the chemical energy in the gasoline will reappear in some form, but not necessarily in a form that is useful for doing farm work. Tractors, like cars, are extremely inefficient, and typically 90% of the energy they consume is converted directly into heat, which is carried away by the exhaust and the air flowing over the radiator. We wish to distinguish the energy that comes out directly as heat from the energy that serves to accelerate a trailer or to plow a field, so we define a technical meaning of the ordinary word “work” to express the distinction:
\(text{Definition of work}\)
Figure a: Work is a transfer of energy.
Work is the amount of energy transferred into or out of a system, not counting energy transferred by heat conduction.
The conduction of heat is to be distinguished from heating by friction. When a hot potato heats up your hands by conduction, the energy transfer occurs without any force, but when friction heats your car's brake shoes, there is a force involved. The transfer of energy with and without a force are measured by completely different methods, so we wish to include heat transfer by frictional heating under the definition of work, but not heat transfer by conduction. The definition of work could thus be restated as the amount of energy transferred by forces.
Figure b: The tractor raises the weight over the pulley, increasing its gravitational potential energy.
Figure c: The tractor accelerates the trailer, increasing its kinetic energy.
Figure d: The tractor pulls a plow. Energy is expended in frictional heating of the plow and the dirt, and in breaking dirt clods and lifting dirt up to the sides of the furrow.
The examples in figures b-d show that there are many different ways in which energy can be transferred. Even so, all these examples have two things in common:
- A force is involved.
- The tractor travels some distance as it does the work.
In b, the increase in the height of the weight, Δ y, is the same as the distance the tractor travels, which we'll call d. For simplicity, we discuss the case where the tractor raises the weight at constant speed, so that there is no change in the kinetic energy of the weight, and we assume that there is negligible friction in the pulley, so that the force the tractor applies to the rope is the same as the rope's upward force on the weight. By Newton's first law, these forces are also of the same magnitude as the earth's gravitational force on the weight. The increase in the weight's potential energy is given by FΔ y, so the work done by the tractor on the weight equals Fd, the product of the force and the distance moved:
$$W = Fd $$
In figure c, the tractor's force on the trailer accelerates it, increasing its kinetic energy. If frictional forces on the trailer are negligible, then the increase in the trailer's kinetic energy can be found using algebra to find the potential energy due to gravity. Just as in figure b, we have
$$W = Fd $$
Does this equation always give the right answer? Well, sort of. In figure d, there are two quantities of work you might want to calculate: the work done by the tractor on the plow and the work done by the plow on the dirt. These two quantities can't both equal Fd. Most of the energy transmitted through the cable goes into frictional heating of the plow and the dirt. The work done by the plow on the dirt is less than the work done by the tractor on the plow, by an amount equal to the heat absorbed by the plow. It turns out that the equation W=Fd gives the work done by the tractor, not the work done by the plow. How are you supposed to know when the equation will work and when it won't? We will restrict ourselves to examples in which W=Fd gives the right answer; essentially the reason the ambiguities come up is that when one surface is slipping past another, d may be hard to define, because the two surfaces move different distances.
Figure e: The baseball pitcher put kinetic energy into the ball, so he did work on it. To do the greatest possible amount of work, he applied the greatest possible force over the greatest possible distance.
We have also been using examples in which the force is in the same direction as the motion, and the force is constant. (If the force was not constant, we would have to represent it with a function, not a symbol that stands for a number.) To summarize, we have:
Rule for calculating work (simplest version)
The work done by a force can be calculated as
$$W = Fd $$
if the force is constant and in the same direction as the motion. Some ambiguities are encountered in cases such as kinetic friction.
Example 1: Mechanical work done in an earthquake
◊ In 1998, geologists discovered evidence for a big prehistoric earthquake in Pasadena, between 10,000 and 15,000 years ago. They found that the two sides of the fault moved 6.7 m relative to one another, and estimated that the force between them was 1.3x\(10^{17}\) N. How much energy was released?
◊ Multiplying the force by the distance gives 9x\(10^{17}\) J. For comparison, the Northridge earthquake of 1994, which killed 57 people and did 40 billion dollars of damage, released 22 times less energy.
Example 2: The fall factor
Surprisingly, the climber is in more danger at 1 than at 2. The distance d is the amount by which the rope will stretch while work is done to transfer the kinetic energy of a fall out of her body.
Counterintuitively, the rock climber may be in more danger in figure g/1 than later when she gets up to position g/2.
Along her route, the climber has placed removable rock anchors (not shown) and carabiners attached to the anchors. She clips the rope into each carabiner so that it can travel but can't pop out. In both 1 and 2, she has ascended a certain distance above her last anchor, so that if she falls, she will drop through a height h that is about twice this distance, and this fall height is about the same in both cases. In fact, h is somewhat larger than twice her height above the last anchor, because the rope is intentionally designed to stretch under the big force of a falling climber who suddenly brings it taut.
To see why we want a stretchy rope, consider the equation F=W/d in the case where d is zero; F would theoretically become infinite. In a fall, the climber loses a fixed amount of gravitational energy mgh. This is transformed into an equal amount of kinetic energy as she falls, and eventually this kinetic energy has to be transferred out of her body when the rope comes up taut. If the rope was not stretchy, then the distance traveled at the point where the rope attaches to her harness would be zero, and the force exerted would theoretically be infinite. Before the rope reached the theoretically infinite tension F it would break (or her back would break, or her anchors would be pulled out of the rock). We want the rope to be stretchy enough to make d fairly big, so that dividing W by d gives a small force.1
In g/1 and g/2, the fall h is about the same. What is different is the length L of rope that has been paid out. A longer rope can stretch more, so the distance d traveled after the “catch” is proportional to L. Combining F=W/d, W∝ h, and d∝ L, we have F∝ h/L. For these reasons, rock climbers define a fall factor f=h/L. The larger fall factor in g/1 is more dangerous.
Machines can increase force, but not work.
Figure h shows a pulley arrangement for doubling the force supplied by the tractor. The tension in the left-hand rope is equal throughout, assuming negligible friction, so there are two forces pulling the pulley to the left, each equal to the original force exerted by the tractor on the rope. This doubled force is transmitted through the right-hand rope to the stump.
Figure h: The pulley doubles the force the tractor can exert on the stump.
It might seem as though this arrangement would also double the work done by the tractor, but look again. As the tractor moves forward 2 meters, 1 meter of rope comes around the pulley, and the pulley moves 1 m to the left. Although the pulley exerts double the force on the stump, the pulley and stump only move half as far, so the work done on the stump is no greater that it would have been without the pulley.
The same is true for any mechanical arrangement that increases or decreases force, such as the gears on a ten-speed bike. You can't get out more work than you put in, because that would violate conservation of energy. If you shift gears so that your force on the pedals is amplified, the result is that you just have to spin the pedals more times.
No work is done without motion.
It strikes most students as nonsensical when they are told that if they stand still and hold a heavy bag of cement, they are doing no work on the bag. Even if it makes sense mathematically that W=Fd gives zero when d is zero, it seems to violate common sense. You would certainly become tired! The solution is simple. Physicists have taken over the common word “work” and given it a new technical meaning, which is the transfer of energy. The energy of the bag of cement is not changing, and that is what the physicist means by saying no work is done on the bag.
There is a transformation of energy, but it is taking place entirely within your own muscles, which are converting chemical energy into heat. Physiologically, a human muscle is not like a tree limb, which can support a weight indefinitely without the expenditure of energy. Each muscle cell's contraction is generated by zillions of little molecular machines, which take turns supporting the tension. When a particular molecule goes on or off duty, it moves, and since it moves while exerting a force, it is doing work. There is work, but it is work done by one molecule in a muscle cell on another.
Positive and negative work
When object A transfers energy to object B, we say that A does positive work on B. B is said to do negative work on A. In other words, a machine like a tractor is defined as doing positive work. This use of the plus and minus signs relates in a logical and consistent way to their use in indicating the directions of force and motion in one dimension. In figure i, suppose we choose a coordinate system with the x axis pointing to the right. Then the force the spring exerts on the ball is always a positive number. The ball's motion, however, changes directions. The symbol d is really just a shorter way of writing the familiar quantity Δ x, whose positive and negative signs indicate direction.
Figure i: Whenever energy is transferred out of the spring, the same amount has to be transferred into the ball, and vice versa. As the spring compresses, the ball is doing positive work on the spring (giving up its KE and transferring energy into the spring as PE), and as it decompresses the ball is doing negative work (extracting energy).
While the ball is moving to the left, we use d<0 to represent its direction of motion, and the work done by the spring, Fd, comes out negative. This indicates that the spring is taking kinetic energy out of the ball, and accepting it in the form of its own potential energy.
As the ball is reaccelerated to the right, it has d>0, Fd is positive, and the spring does positive work on the ball. Potential energy is transferred out of the spring and deposited in the ball as kinetic energy.
In summary: Rule for calculating work (including cases of negative work)
The work done by a force can be calculated as $$W = Fd $$
if the force is constant and along the same line as the motion. The quantity d is to be interpreted as a synonym for Δ x, i.e., positive and negative signs are used to indicate the direction of motion. Some ambiguities are encountered in cases such as kinetic friction.
There are many examples where the transfer of energy out of an object cancels out the transfer of energy in. When the tractor pulls the plow with a rope, the rope does negative work on the tractor and positive work on the plow. The total work done by the rope is zero, which makes sense, since it is not changing its energy.
It may seem that when your arms do negative work by lowering a bag of cement, the cement is not really transferring energy into your body. If your body was storing potential energy like a compressed spring, you would be able to raise and lower a weight all day, recycling the same energy. The bag of cement does transfer energy into your body, but your body accepts it as heat, not as potential energy. The tension in the muscles that control the speed of the motion also results in the conversion of chemical energy to heat, for the same physiological reasons discussed previously in the case where you just hold the bag still.
One of the advantages of electric cars over gasoline-powered cars is that it is just as easy to put energy back in a battery as it is to take energy out. When you step on the brakes in a gas car, the brake shoes do negative work on the rest of the car. The kinetic energy of the car is transmitted through the brakes and accepted by the brake shoes in the form of heat. The energy cannot be recovered. Electric cars, however, are designed to use regenerative braking. The brakes don't use friction at all. They are electrical, and when you step on the brake, the negative work done by the brakes means they accept the energy and put it in the battery for later use. This is one of the reasons why an electric car is far better for the environment than a gas car, even if the ultimate source of the electrical energy happens to be the burning of oil in the electric company's plant. The electric car recycles the same energy over and over, and only dissipates heat due to air friction and rolling resistance, not braking. (The electric company's power plant can also be fitted with expensive pollution-reduction equipment that would be prohibitively expensive or bulky for a passenger car.)
Figure j: Left: No mechanical work occurs in the man's body while he holds himself motionless. There is a transformation of chemical energy into heat, but this happens at the microscopic level inside the tensed muscles. Right: When the woman lifts herself, her arms do positive work on her body, transforming chemical energy into gravitational potential energy and heat. On the way back down, the arms' work is negative; gravitational potential energy is transformed into heat. (In exercise physiology, the man is said to be doing isometric exercise, while the woman's is concentric and then eccentric.)
Figure j: Because the force is in the opposite direction compared to the motion, the brake shoe does negative work on the drum, i.e., accepts energy from it in the form of heat.
Work in three dimensions
Figure l: A force can do positive, negative, or zero work, depending on its direction relative to the direction of the motion.
A force perpendicular to the motion does no work.
Suppose work is being done to change an object's kinetic energy. A force in the same direction as its motion will speed it up, and a force in the opposite direction will slow it down. As we have already seen, this is described as doing positive work or doing negative work on the object. All the examples discussed up until now have been of motion in one dimension, but in three dimensions the force can be at any angle θ with respect to the direction of motion.
What if the force is perpendicular to the direction of motion? We have already seen that a force perpendicular to the motion results in circular motion at constant speed. The kinetic energy does not change, and we conclude that no work is done when the force is perpendicular to the motion.
So far we have been reasoning about the case of a single force acting on an object, and changing only its kinetic energy. The result is more generally true, however. For instance, imagine a hockey puck sliding across the ice. The ice makes an upward normal force, but does not transfer energy to or from the puck.
Forces at other angles
Suppose the force is at some other angle with respect to the motion, say θ=45°. Such a force could be broken down into two components, one along the direction of the motion and the other perpendicular to it. The force vector equals the vector sum of its two components, and the principle of vector addition of forces thus tells us that the work done by the total force cannot be any different than the sum of the works that would be done by the two forces by themselves. Since the component perpendicular to the motion does no work, the work done by the force must be
$$W=F_{||} |d|$$
where the vector d is simply a less cumbersome version of the notation \(\Delta{r}\). This result can be rewritten via trigonometry as
$$W=F_{||} |d|cosθ$$
Even though this equation has vectors in it, it depends only on their magnitudes, and the magnitude of a vector is a scalar. Work is therefore still a scalar quantity, which only makes sense if it is defined as the transfer of energy. Ten gallons of gasoline have the ability to do a certain amount of mechanical work, and when you pull in to a full-service gas station you don't have to say “Fill 'er up with 10 gallons of south-going gas.”
Students often wonder why this equation involves a cosine rather than a sine, or ask if it would ever be a sine. In vector addition, the treatment of sines and cosines seemed more equal and democratic, so why is the cosine so special now? The answer is that if we are going to describe, say, a velocity vector, we must give both the component parallel to the x axis and the component perpendicular to the x axis (i.e., the y component). In calculating work, however, the force component perpendicular to the motion is irrelevant --- it changes the direction of motion without increasing or decreasing the energy of the object on which it acts. In this context, it is only the parallel force component that matters, so only the cosine occurs.
Figure m: Work is only done by the component of the force parallel to the motion.
Figure n: Breaking Trail, by Walter E. Bohl. Work is the transfer of energy. According to this definition, is the horse in the picture doing work on the pack?
Example 3: Pushing a broom
◊ If you exert a force of 21 N on a push broom, at an angle 35 degrees below horizontal, and walk for 5.0 m, how much work do you do? What is the physical significance of this quantity of work?
◊ Using the second equation above, the work done equals
$$(21 N)(5.0 m)(cos 35^\circ{}) = 86 J$$
The form of energy being transferred is heat in the floor and the broom's bristles. This comes from the chemical energy stored in your body. (The majority of the calories you burn are dissipated directly as heat inside your body rather than doing any work on the broom. The 86 J is only the amount of energy transferred through the broom's handle.)
Example 4: A violin
As a violinist draws the bow across a string, the bow hairs exert both a normal force and a kinetic frictional force on the string. The normal force is perpendicular to the direction of motion, and does no work. However, the frictional force is in the same direction as the motion of the bow, so it does work: energy is transferred to the string, causing it to vibrate.
One way of playing a violin more loudly is to use longer strokes. Since W=Fd, the greater distance results in more work.
A second way of getting a louder sound is to press the bow more firmly against the strings. This increases the normal force, and although the normal force itself does no work, an increase in the normal force has the side effect of increasing the frictional force, thereby increasing W=Fd.
The violinist moves the bow back and forth, and sound is produced on both the “up-bow” (the stroke toward the player's left) and the “down-bow” (to the right). One may, for example, play a series of notes in alternation between up-bows and down-bows. However, if the notes are of unequal length, the up and down motions tend to be unequal, and if the player is not careful, she can run out of bow in the middle of a note! To keep this from happening, one can move the bow more quickly on the shorter notes, but the resulting increase in d will make the shorter notes louder than they should be. A skilled player compensates by reducing the force.
Up until now, we have not found any physically useful way to define the multiplication of two vectors. It would be possible, for instance, to multiply two vectors component by component to form a third vector, but there are no physical situations where such a multiplication would be useful.
The equation W= |F| |d| cos \(\theta{}\) is an example of a sort of multiplication of vectors that is useful. The result is a scalar, not a vector, and this is therefore often referred to as the scalar product of the vectors F and d. There is a standard shorthand notation for this operation,
$$A\cdot{} B = |A||B|cos\theta{} \; \; \; \; \text{[definition of notation A} \cdot{} \text{B;} \theta{} \; \text{ is the angle between vectors A and B]}$$
and because of this notation, a more common term for this operation is the dot product. In dot product notation, the equation for work is simply
$$W=F\cdot{}d$$
The dot product has the following geometric interpretation: $$A\cdot{}B = |A| \; \text{(component of B parallel to A)}$$ $$=|B| \; \text{(component of A parallel to B)}$$
The dot product has some of the properties possessed by ordinary multiplication of numbers, $$A\cdot{}B=B\cdot{}A$$ $$A\cdot{}(B+C)=A\cdot{}B+A\cdot{}C$$ $$(cA)\cdot{}B = c(A\cdot{}B)$$
but it lacks one other: the ability to undo multiplication by dividing.
If you know the components of two vectors, you can easily calculate their dot product as follows: $$A\cdot{}B = A_x B_x + A_y B_y + A_z B_z$$
(This can be proved by first analyzing the special case where each vector has only an x component, and the similar cases for y and z. We can then use the rule A\(\cdot{}\)(B+C) = A\(\cdot{}\)B + A\(\cdot{}\)C to make a generalization by writing each vector as the sum of its x, y, and z components.
Example 5: Magnitude expressed with a dot product
If we take the dot product of any vector b with itself, we find
$$b\cdot{}b = (b_x \hat{x} + b_y \hat{y} + b_y \hat{y})$$ $$=b_x ^2 + b_y ^2 + b_z ^2$$ $$\text{so its magnitude can be expressed as} \; |b| = \sqrt{b\cdot{}b}$$
We will often write \(b^2\) to mean b\(\cdot\)b, when the context makes it clear what is intended. For example, we could express kinetic energy as 1/2m\(|v|^2\), 1/2mv\(\cdot\)v, or 1/2m\(v^2\). In the third version, nothing but context tells us that v really stands for the magnitude of some vector v.
Example 6: Towing a barge
◊ A mule pulls a barge with a force F=(1100N)\(\hat{x}\) + (400N)\(\hat{y}\), and the total distance it travels is (1000m)\(\hat{x}\). How much work does it do?
◊ The dot product is 1.1x\(10^{6}\)N \(\cdot\) m = 1.1x\(10^{6}\) J.
Varying Force
Up until now we have done no actual calculations of work in cases where the force was not constant. The question of how to treat such cases is mathematically analogous to the issue of how to generalize the equation (distance)=(velocity)(time) to cases where the velocity was not constant. There, we found that the correct generalization was to find the area under the graph of velocity versus time. The equivalent thing can be done with work:
General rule for calculating work
The work done by a force F equals the area under the curve on a graph of \(F_{||}\) versus x. (Some ambiguities are encountered in cases such as kinetic friction.)
The examples provided are ones in which the force is varying, but is always along the same line as the motion, so F is the same as \(F_{||}\).
An important and straightforward example is the calculation of the work done by a spring that obeys Hooke's law, $$F ≈ -k(x-x_o) .$$
The minus sign is because this is the force being exerted by the spring, not the force that would have to act on the spring to keep it at this position. That is, if the position of the cart in figure o is to the right of equilibrium, the spring pulls back to the left, and vice-versa.
Figure o: The spring does work on the cart. The cart is attached to the spring.
Figure p: The area of the shaded triangle gives the work done by the spring as the cart moves from the equilibrium position to position x.
We calculate the work done when the spring is initially at equilibrium and then decelerates the car as the car moves to the right. The work done by the spring on the cart equals the minus area of the shaded triangle, because the triangle hangs below the x axis. The area of a triangle is half its base multiplied by its height, so $$W = -frac{1}{2}kleft(x-x_text{o}right)^2 qquad $$
This is the amount of kinetic energy lost by the cart as the spring decelerates it.
It was straightforward to calculate the work done by the spring in this case because the graph of F versus x was a straight line, giving a triangular area. But if the curve had not been so geometrically simple, it might not have been possible to find a simple equation for the work done, or an equation might have been derivable only using calculus.
Example 7: Energy production in the sun
The sun produces energy through nuclear reactions in which nuclei collide and stick together. The figure depicts one such reaction, in which a single proton (hydrogen nucleus) collides with a carbon nucleus, consisting of six protons and six neutrons. Neutrons and protons attract other neutrons and protons via the strong nuclear force, so as the proton approaches the carbon nucleus it is accelerated. In the language of energy, we say that it loses nuclear potential energy and gains kinetic energy. Together, the seven protons and six neutrons make a nitrogen nucleus. Within the newly put-together nucleus, the neutrons and protons are continually colliding, and the new proton's extra kinetic energy is rapidly shared out among all the neutrons and protons. Soon afterward, the nucleus calms down by releasing some energy in the form of a gamma ray, which helps to heat the sun.
The graph shows the force between the carbon nucleus and the proton as the proton is on its way in, with the distance in units of femtometers (1 fm=10-15 m). Amusingly, the force turns out to be a few newtons: on the same order of magnitude as the forces we encounter ordinarily on the human scale. Keep in mind, however, that a force this big exerted on a single subatomic particle such as a proton will produce a truly fantastic acceleration (on the order of \(10^{27}\) m/\(s^2\)
Why does the force have a peak around x=3 fm, and become smaller once the proton has actually merged with the nucleus? At x=3 fm, the proton is at the edge of the crowd of protons and neutrons. It feels many attractive forces from the left, and none from the right. The forces add up to a large value. However if it later finds itself at the center of the nucleus, x=0, there are forces pulling it from all directions, and these force vectors cancel out.
We can now calculate the energy released in this reaction by using the area under the graph to determine the amount of mechanical work done by the carbon nucleus on the proton. (For simplicity, we assume that the proton came in “aimed” at the center of the nucleus, and we ignore the fact that it has to shove some neutrons and protons out of the way in order to get there.) The area under the curve is about 17 squares, and the work represented by each square is $$(1N)(10^{-15}) = 10^{-15} J$$
so the total energy released is about
$$(10^{-15} J/square) (17 squares) = 1.7 x 10^{-14} J$$
This may not seem like much, but remember that this is only a reaction between the nuclei of two out of the zillions of atoms in the sun. For comparison, a typical chemical reaction between two atoms might transform on the order of 10-19 J of electrical potential energy into heat --- 100,000 times less energy!
As a final note, you may wonder why reactions such as these only occur in the sun. The reason is that there is a repulsive electrical force between nuclei. When two nuclei are close together, the electrical forces are typically about a million times weaker than the nuclear forces, but the nuclear forces fall off much more quickly with distance than the electrical forces, so the electrical force is the dominant one at longer ranges. The sun is a very hot gas, so the random motion of its atoms is extremely rapid, and a collision between two atoms is sometimes violent enough to overcome this initial electrical repulsion.
Applications of calculus
The student who has studied integral calculus will recognize that the graphical rule can be re-expressed as an integral,
$$W= \int_{x_1}^{x_2} Fdx$$
We can then immediately find by the fundamental theorem of calculus that force is the derivative of work with respect to position,
$$F=\frac{dW}{dx}$$ For example, a crane raising a one-ton block on the moon would be transferring potential energy into the block at only one sixth the rate that would be required on Earth, and this corresponds to one sixth the force.
Although the work done by the spring could be calculated without calculus using the area of a triangle, there are many cases where the methods of calculus are needed in order to find an answer in closed form. The most important example is the work done by gravity when the change in height is not small enough to assume a constant force. Newton's law of gravity is
$$F=\frac{GMm}{r^2}$$ which can be integrated to give
$$W =\int_{r_1} ^{r_2} \frac{GMm}{r^2}dr $$ $$=-GMm(\frac{1}{r_2}-\frac{1}{r_1}).$$
Work and Potential Energy
The techniques for calculating work can also be applied to the calculation of potential energy. If a certain force depends only on the distance between the two participating objects, then the energy released by changing the distance between them is defined as the potential energy, and the amount of potential energy lost equals minus the work done by the force,
$$Δ PE = -W $$
The minus sign occurs because positive work indicates that the potential energy is being expended and converted to some other form.
It is sometimes convenient to pick some arbitrary position as a reference position, and derive an equation for once and for all that gives the potential energy relative to this position
$$PE_x=-W_{ref→x} \; \; \text{[potential energy at a point x]}$$
To find the energy transferred into or out of potential energy, one then subtracts two different values of this equation.
These equations might almost make it look as though work and energy were the same thing, but they are not. First, potential energy measures the energy that a system has stored in it, while work measures how much energy is transferred in or out. Second, the techniques for calculating work can be used to find the amount of energy transferred in many situations where there is no potential energy involved, as when we calculate the amount of kinetic energy transformed into heat by a car's brake shoes.
Example 8: A toy gun
◊ A toy gun uses a spring with a spring constant of 10 N/m to shoot a ping-pong ball of mass 5 g. The spring is compressed to 10 cm shorter than its equilibrium length when the gun is loaded. At what speed is the ball released?
◊ The equilibrium point is the natural choice for a reference point. Using the equation found previously for the work, we have
$$PE_x=1/2k(x-x_o)^2$$
The spring loses contact with the ball at the equilibrium point, so the final potential energy is
$$PE_f = 0 $$
The initial potential energy is
$$PE_i =1/2(10 N/m)(0.10 m)^2. $$ $$=0.05 J$$
The loss in potential energy of 0.05 J means an increase in kinetic energy of the same amount. The velocity of the ball is found by solving the equation KE=(1/2)m\(v^2\) for v,
$$v =\sqrt{\frac{2KE}{m}} $$ $$=\sqrt{\frac{(2)(0.05 J)}{0.005 kg}} $$ $$=4 m/s$$
Example 9: Gravitational potential energy
◊ We have already found the equation Δ PE = -FΔ y for the gravitational potential energy when the change in height is not enough to cause a significant change in the gravitational force F. What if the change in height is enough so that this assumption is no longer valid? Use the equation W=GMm(1/\(r_2\)-1/\(r_1\)) to find the potential energy, using r=∞ as a reference point. ◊ The potential energy equals minus the work that would have to be done to bring the object from \(r_1\)=∞ to r= \(r_2\), which is
$$PE=-\frac{GMm}{r}.$$
This is simpler than the equation for the work, which is an example of why it is advantageous to record an equation for potential energy relative to some reference point, rather than an equation for work.
Although the equations derived in the previous two examples may seem arcane and not particularly useful except for toy designers and rocket scientists, their usefulness is actually greater than it appears. The equation for the potential energy of a spring can be adapted to any other case in which an object is compressed, stretched, twisted, or bent. While you are not likely to use the equation for gravitational potential energy for anything practical, it is directly analogous to an equation that is extremely useful in chemistry, which is the equation for the potential energy of an electron at a distance r from the nucleus of its atom. As discussed in more detail later in the course, the electrical force between the electron and the nucleus is proportional to 1/\(r_2\), just like the gravitational force between two masses. Since the equation for the force is of the same form, so is the equation for the potential energy.
Figure r: The twin Voyager space probes were perhaps the greatest scientific successes of the space program. Over a period of decades, they flew by all the planets of the outer solar system, probably accomplishing more of scientific interest than the entire space shuttle program at a tiny fraction of the cost. Both Voyager probes completed their final planetary flybys with speeds greater than the escape velocity at that distance from the sun, and so headed on out of the solar system on hyperbolic orbits, never to return. Radio contact has been lost, and they are now likely to travel interstellar space for billions of years without colliding with anything or being detected by any intelligent species.
When does work equal force times distance?
In the example of the tractor pulling the plow discussed (figure b-d), the work did not equal Fd. To simplify things, Fd is written throughout the article, but more generally everything here would be true for the area under the graph of \(F_{||}\) versus d.
The following two theorems allow most of the ambiguity to be cleared up.
The change in kinetic energy associated with the motion of an object's center of mass is related to the total force acting on it and to the distance traveled by its center of mass according to the equation $$Δ KE_cm=F_{total}d_{cm}$$
This can be proved based on Newton's second law and the equation KE=(1/2)m\(v^2\). Note that despite the traditional name, it does not necessarily tell the amount of work done, since the forces acting on the object could be changing other types of energy besides the KE associated with its center of mass motion.
The second theorem does relate directly to work:
When a contact force acts between two objects and the two surfaces do not slip past each other, the work done equals Fd, where d is the distance traveled by the point of contact.
This one has no generally accepted name, so we refer to it simply as the second theorem.
A great number of physical situations can be analyzed with these two theorems, and often it is advantageous to apply both of them to the same situation.
Example 10: An ice skater pushing off from a wall
The work-kinetic energy theorem tells us how to calculate the skater's kinetic energy if we know the amount of force and the distance her center of mass travels while she is pushing off.
The second theorem tells us that the wall does no work on the skater. This makes sense, since the wall does not have any source of energy.
Example 11: Absorbing an impact without recoiling?
◊ Is it possible to absorb an impact without recoiling? For instance, would a brick wall “give” at all if hit by a ping-pong ball?
◊ There will always be a recoil. In the example proposed, the wall will surely have some energy transferred to it in the form of heat and vibration. The second theorem tells us that we can only have nonzero work if the distance traveled by the point of contact is nonzero.
Example 12: Dragging a refrigerator at constant velocity
Newton's first law tells us that the total force on the refrigerator must be zero: your force is canceling the floor's kinetic frictional force. The work-kinetic energy theorem is therefore true but useless. It tells us that there is zero total force on the refrigerator, and that the refrigerator's kinetic energy doesn't change.
The second theorem tells us that the work you do equals your hand's force on the refrigerator multiplied by the distance traveled. Since we know the floor has no source of energy, the only way for the floor and refrigerator to gain energy is from the work you do. We can thus calculate the total heat dissipated by friction in the refrigerator and the floor.
Note that there is no way to find how much of the heat is dissipated in the floor and how much in the refrigerator.
Example 13: Accelerating a cart
If you push on a cart and accelerate it, there are two forces acting on the cart: your hand's force, and the static frictional force of the ground pushing on the wheels in the opposite direction.
Applying the second theorem to your force tells us how to calculate the work you do.
Applying the second theorem to the floor's force tells us that the floor does no work on the cart. There is no motion at the point of contact, because the atoms in the floor are not moving. (The atoms in the surface of the wheel are also momentarily at rest when they touch the floor.) This makes sense, since the floor does not have any source of energy.
The work-kinetic energy theorem refers to the total force, and because the floor's backward force cancels part of your force, the total force is less than your force. This tells us that only part of your work goes into the kinetic energy associated with the forward motion of the cart's center of mass. The rest goes into rotation of the wheels.
The dot product
Up until now, we have not found any physically useful way to define the multiplication of two vectors. It would be possible, for instance, to multiply two vectors component by component to form a third vector, but there are no physical situations where such a multiplication would be useful.
The equation W=|F||d|cosθ is an example of a sort of multiplication of vectors that is useful. The result is a scalar, not a vector, and this is therefore often referred to as the scalar product of the vectors F and d. There is a standard shorthand notation for this operation,
$$ A \cdot B=|A||B|cosθ, \; \; \text{[definition of the notation A⋅B ; θ is the angle between vectors A and B]}$$
and because of this notation, a more common term for this operation is the dot product. In dot product notation, the equation for work is simply
$$W=F⋅d$$
The dot product has the following geometric interpretation:
$$A\cdot B =|A|\text{(component of B parallel to A )} $$ $$=|B|\text{(component of A parallel to B )}$$ The dot product has some of the properties possessed by ordinary multiplication of numbers,
$$A\cdot B =B\cdot A $$ $$A\cdot (B+C) =A\cdot B+A⋅C $$ $$(cA)\cdot B =c(A\cdot B)$$
but it lacks one other: the ability to undo multiplication by dividing.
If you know the components of two vectors, you can easily calculate their dot product as follows:
$$A\cdot B=A_x B_x +A_y B_y +A_z B_z$$
(This can be proved by first analyzing the special case where each vector has only an x component, and the similar cases for y and z. We can then use the rule A\(\cdot{}\)(B+C)=A\(\cdot{}\)B+A\(\cdot{}\)C to make a generalization by writing each vector as the sum of its x, y, and z components.)
Example 14: Magnitude expressed with a dot product
If we take the dot product of any vector b with itself, we find
$$b \cdot{} b = (b_x \hat{x} + b_y \hat{y} + b_z \hat{z}$$ $$=b_x ^2 + b_y ^2 + b_z ^2$$
so its magnitude can be expressed as |b| = \(\sqrt{b\cdot{}b}\)
We will often write \(b^2\) to mean b\(\cdot{}\)b , when the context makes it clear what is intended. For example, we could express kinetic energy as (1/2)m\(|v|^{2}\) , (1/2)mv\(\cdot{}\)v , or (1/2)m\(v^{2}\) . In the third version, nothing but context tells us that v really stands for the magnitude of some vector v.
Example 15: Towing a barge
◊ A mule pulls a barge with a force F=(1100 N)\(\hat{x}\)+(400 N)\(\hat{y}\) , and the total distance it travels is (1000 m)\(\hat{x}\) . How much work does it do?
◊ The dot product is 1.1×\(10^{6}\) N⋅m=1.1×\(10^{6}\) J .