Bed of nails – Watch as Dr. Fazzini hits me with a sledgehammer as I lay on a bed of nails during STEM-CON. I don’t get injured because the kinetic energy from the sledgehammer is used up breaking the brick. This is the full length version of the clip included in the lecture video.
PhET Energy Skate Park Simulation
Take a look at how energy is conserved in a skate park. You can see what the effects of gravity and friction are on speed and energy.
Hello there! Welcome to lecture 7: work and energy!
Work and energy are important concepts that allow us to understand and quantify the ability of objects to do something useful for us. That utility might be moving a piston in an engine, turning a wheel about an axis, or spinning a turbine to generate electricity. Work and energy are so intimately related that they have somewhat circular definitions that describe how intertwined they are. Energy is defined as the capability of an object to do useful work. And work is defined as the transfer of energy by applying a force to that object over a distance.
Work and energy as concepts can allow us to understand simple machines such as pulleys and levers, and also enable us to analyze the motion of projectiles, planets, and roller coasters, among other things.
Each of the following concepts will be discussed in this video: forms of energy, conservation of energy, mechanical energy, work, the work-energy theorem, power, and simple machines.
Forms of energy
As mentioned in the introduction, energy is defined as the capability of an object to do useful work. Before discussing forms of energy, let’s discuss the unit that we use to quantify it: the Joule. A Joule is a unit that contains other units hidden inside of it. A Joule is a Newton times a meter. And as we know, a Newton is a kilogram times a meter divided by a second-squared. A Joule can be broken down as kilogram meter-squared per second-squared. All three of these units mean the same thing: Joule, Newton-meter, and kilogram meter-squared per second-squared. As we’ll see in lecture eight, the unit of Newton-meter is typically not used when discussing work and energy, as it’s used as the unit for something called torque. Therefore, we will primarily use Joule as the unit for work and energy.
Energy, and work, are scalar quantities. That means that there is no direction associated with either of the two concepts, only a magnitude or strength.
There are many different forms of energy. In general, energy can be potential, or stored energy, which is energy that exists based on the arrangement or configuration of something. Or energy can be kinetic, which is energy due to motion. Each form of energy has its own equation used to quantify that energy. Let’s discuss different types of stored energy first.
Some forms of potential energy include gravitational, electric, chemical, nuclear, and elastic. Gravitational potential energy, or GPE, will be discussed in more detail in just a moment.
Electric potential energy comes from placing charged particles some distance apart from each other. We’ll talk about this type of energy in more detail in lecture 22. For now it’s enough to say that it’s the type of energy that can be stored in elements used in electric circuits.
Chemical energy is a type of stored energy that’s based on the configuration of atoms or molecules. For example, gasoline contains stored energy that we can use when we ignite it. Chemical energy is also how we as humans obtain energy from the food we eat during the digestive process.
Elastic potential energy
Elastic potential energy is stored in the configuration of some type of deformable object. We can see this in a spring. This video shows a mass bouncing up and down on a spring. When the mass is pulled downward, the deformation of the spring causes it to start moving once I let go of the mass. The up and down motion, known as an oscillation, shows us the conversion of energy from elastic potential energy, to kinetic energy, to gravitational potential energy, over and over again as the mass bounces up and down.
Gravitational potential energy
Let’s spend a little time talking about gravitational potential energy, which is stored energy that comes about due to the gravitational attraction between the Earth and other objects of mass. Gravitational potential energy has a symbol of GPE. The units we use, because it’s a form of energy, are Joules. Let’s consider where GPE comes from. Let’s say I have an object sitting on a table. That object has gravitational potential energy. I could push it off the table and it would move downward, gaining kinetic energy. How can I cause the GPE to increase? I could increase the gravitational potential energy by increasing the mass of the object. The more massive the object is, the more energy it has. GPE is therefore directly proportional to mass. I could also increase the GPE by raising the height of the table. The larger the distance away from our reference, the larger the GPE. GPE is therefore directly proportional to height. (Exactly what the height is measured in relation to – sea level, ground level, floor level, or something else – is up to you to decide.) Finally, I could increase gravitational potential energy if I could somehow move my object of mass to a planet with a larger gravitational acceleration. The object would have more GPE if I had it on a planet like Jupiter or Saturn, and less GPE if I had it on a planet like Mars.
Tying this all together, we can look at the equation for gravitational potential energy. GPE equals m times g times h. M is the mass of the object, g is the gravitational acceleration, and h is the height above the reference level.
Kinetic energy is energy that objects have by virtue of their motion. We discussed kinetic energy briefly in lecture 6 when we talked about momentum and collisions. To recap, kinetic energy has a symbol of KE, units of Joules, and an equation of one-half mass times velocity squared. Remember that kinetic energy, like all forms of energy, is a scalar quantity. This is an important concept. Energy has no direction associated with it. When calculating kinetic energy, be very careful to use parentheses in your calculator when squaring the velocity. When taking the square of any number, positive or negative, the result will be positive. However, calculators are not so smart and may give you a negative answer if you do calculations without parentheses. It’s not possible to have negative kinetic energy. It would imply that it is possible to have less energy of motion than an object that isn’t moving, which doesn’t make sense at all!
Heat, or thermal energy, is energy that moves between objects that have different temperatures. While we’ll spend quite a bit of time discussing heat in lectures 15 through 18, it’s important enough to bring up in some detail right now. The so-called useful types of energy are generally potential energy and kinetic energy. Heat can be utilized to do useful work for us, but is also where a lot of useful energy is wasted and cannot be used again. For example, when two objects rub together, the force of friction generates heat. We consider that heat to be wasted energy, as it cannot be utilized as useful work.
Conservation of energy
Energy can be converted from one form to another. However, it cannot be created or destroyed. If I could somehow measure all of the energy in the entire universe today, and then measure it again tomorrow, those two numbers should be the same. This principle doesn’t just apply to the entire universe; it also applies to any closed system. Generators, motors, and engines are examples of things that perform energy conversions. If you start making observations, you’ll probably notice energy conversion at work all around you.
In this example, I have two steel balls that I get moving very quickly. I then strike the two balls together, with a sheet of paper in between them. The motion of the steel balls demonstrates that before they come together, they have kinetic energy. When they come together, they stop moving, so they no longer have any kinetic energy. Upon colliding, they make a loud clacking noise, showing that there is sound energy created as a result of the collision. But also, they actually burn a hole through the piece of paper where they come together, showing very clearly that heat was generated in the process. KE was converted to sound and heat energy. If I could measure each type of energy, I should see that the heat and sound energy is equal to the original KE of the system.
In a rocket launch, chemical energy is converted to heat, sound, kinetic energy, and gravitational potential energy. Pretty impressive!
A rather smashing demonstration of conservation of energy comes from the bed of nails. In this video, I am lying in between two large pieces of wood that are studded with nails. Each nail is individually very sharp. The reason the nails don’t hurt me is because there are enough nails to distribute my weight over a relatively large area. That concept, known as pressure, will be explored in more detail in lecture 13. A cinder block is then placed over the upper piece of wood, and my colleague, Dr. Fazzini, professor of physics at the College of DuPage, swings a sledgehammer to break the brick. Dr. Fazzini raises the sledgehammer high above his head, giving it gravitational potential energy. As it swings downward, it gains kinetic energy. That kinetic energy is transferred to the cinder block and causes the block to break apart into many pieces. The energy from motion goes into rearranging the molecules in the block. Some of the energy goes to heat, and some of it goes to sound. The force from the blow is distributed along the upper layer of nails, which doesn’t cause too much discomfort due to the large number of nails. I’d say this demo was a big hit!
Mechanical energy provides us with a way to describe the amount of gravitational potential energy and kinetic energy of an object. In the absence of friction and air drag, this quantity will be conserved in a system. Conservation of mechanical energy can therefore be a very useful way to calculate properties of an object.
In this demo, a set of tracks is used to demonstrate conservation of mechanical energy. The tracks all have equal starting heights, and equal ending heights. That means that all of the GPE that the balls have at the start will be converted to equal amounts of KE at the end. Because each of the balls is identical, they’ll have the same velocity when they reach the end of the track. How far the ball goes as it launches off the end of the track is related to the velocity. When I release the balls from the tracks, while they might take different amounts of time to reach the end, they all hit the ground the same distance away, indicating that they all had the same amount of mechanical energy throughout their motion.
The amount of time it takes for the balls to reach the end has to do with how long each ball travels at each speed. If you go fast for a longer period of time, then you’ll reach the end first. This is why the ball moving down the ramp with the earliest down-slope reaches the end of the track the fastest. But regardless, the conservation of mechanical energy still holds up.
A roller coaster provides a great example of conservation of mechanical energy. This animation shows a roller coaster cart as it travels from one end of the ride to the other. It starts out at the tallest height. At this moment, the cart has all GPE and no KE. As it moves downward, the GPE converts to KE. At the lowest point in the track, the KE of the cart, and therefore also the speed, is at its highest value. As long as the level of the track remains lower than the starting hill, the energy the roller coaster has at the start will keep it moving throughout the ride.
If the roller coaster were designed with a taller hill after the start, there would not be enough energy to get the roller coaster up that hill. In an actual ride, there would need to be a drive chain that provides that energy from chemical or electrical sources to get the cart up to the top of that tall hill.
I find the concept of mechanical energy to make the analysis of accelerated motion questions much simpler than using the equations we learned in earlier lectures.
Because we’re ignoring friction and air drag, we can use the conservation of mechanical energy to do all of these calculations. This ballistic cart launches a ball with a mass of 9.7 grams with an initial velocity of 3.6 meters per second. Because we aren’t using kilograms to measure the mass of the ball, we must be careful with the units used in our energy calculations.
We will use conservation of mechanical energy to determine the height that the ball is launched to. At the start of the ball’s motion, we’ll say that the ball has a height of zero meters. The ball’s starting height will therefore be established as our reference point. At this height, the ball has zero GPE. Because it is moving, it does have KE. That KE is equal to one-half times 9.7 grams times 3.6 meters per second, squared. The mechanical energy at this point is therefore equal to 62.86 gram meters-squared per second-squared.
Because mechanical energy is conserved throughout the ball’s motion, it should have 62.86 gram meters-squared per second-squared of energy at the top of its motion. The ball is not moving at this instant, meaning that it has zero KE at this time. All of its mechanical energy is now in the form of GPE. And that GPE is equal to 62.86 gram meters-squared per second-squared, which equals to mass times gravitational acceleration times height. We can solve for height, and find that it is equal to 62.86 gram meters-squared per second-squared, divided by 9.7 grams times 9.8 meters per second-squared. The calculated height of the ball is therefore 0.66 meters.
This is extremely accurate, as I measured the height that the ballistic cart launched the projectile to be 0.66 meters. We can use conservation of mechanical energy to calculate the speed of the ball at any height above the reference point.
Here’s a good example of a system that contains some dissipative forces like air drag and friction. In this case, I start with a pendulum that’s right up against my nose, as I stand against a wall. The pendulum is not moving, so all of its mechanical energy is in the form of GPE. I can release the pendulum and the large mass will swing away from me, gaining KE as it gets to the bottom of its motion. It gains GPE again on the upswing, and this conversion occurs again as it swings back to my face.
By the time the heavy mass reaches my face again, the ball has lost some of its mechanical energy, which means that it no longer is able to get to the same height anymore. This means that as long as I don’t move my face away from the wall, I can be very confident that the ball will not hit me in the face when it returns. The heavy mass experiences air drag as it travels through the air, and the pin holding it to the ceiling experiences friction as it rubs against the ceiling supports.
In much the same way, a roller coaster, which we discussed earlier, must start out with a sufficient amount of mechanical energy to give the ride enough energy to continue throughout its entire motion without any more input work. If there’s only one drive chain brining the roller coaster cart up to the top of the tallest hill, that GPE must be enough to not only get the roller coaster to the end of the track, but also be able to account for the energy converted to sound and heat due to friction and air drag.
We need to be careful when discussing the concept of work in a physics context. You may use the word work to talk about something you do to make money. Or maybe you think about going to the gym or otherwise physically exerting yourself. Be careful! Work, when we’re using it to refer to a physics phenomenon, has a very specific meaning that cannot be interchanged with our loose definition of “work” that we use in the English language.
Work is energy that is transferred to an object by exerting a force on that object over a distance. Work has a symbol of the uppercase letter W. The units are Joules. The equation for work is that W = F parallel times d. D is the displacement of the object. F parallel is the component of force that is parallel to the displacement of the object.
Let’s discuss that concept of parallel component of force. When a force acts on an object, Newton’s second law tells us that the object will experience an acceleration (which could cause the object to speed up, slow down, or change direction). It’s possible that the displacement, or motion of the object, and the force, may not be pointed along exactly the same line. I think one of the best examples of this is the motion of a wheeled suitcase.
A wheeled suitcase can be rolled along a floor, which is much easier than older styles of luggage that you had to carry. This way, most of the weight of the suitcase can be supported by the floor. The wheels allow the force we apply to the handle to cause the suitcase to move in the same direction as our body. We apply the force on the handle at an angle, but the suitcase does not move along the same line as our force.
Let’s visualize the motion of the suitcase as a vector. That’s the displacement. Then, we can visualize the force we apply to the handle as a vector. We can break that vector into components: a component that’s parallel to the displacement, and a component that’s perpendicular to the displacement.
Only the parallel component of force is doing work. And the work is equal to the parallel component of force multiplied by the displacement of the suitcase. The perpendicular component of the force is keeping the suitcase upright, but it isn’t doing any actual work, from a physics perspective.
This animation shows another example of this. Say the box is moving in the direction indicated by the displacement vector. The force is applied in the direction indicated by the force vector. The force vector can be split into parallel and perpendicular components. If the force has 5 Newtons of parallel component, and that force is used to move the box a distance of 2 meters, then we can multiply those two values together to calculate that 10 Joules of work was done on the box.
While work is a scalar quantity, indicating that it has no direction associated with it, it is still possible to do positive or negative work. Positive work means that the parallel force points in the same direction as the displacement, and negative work means that the parallel force points in the opposite direction as the displacement.
Let’s go back to our rolling suitcase example. Let’s say the traveler exerts 100 Newtons of parallel force on the suitcase as they pull the suitcase along the ground a distance of 2 meters. The work exerted on the suitcase by the traveler is 200 Joules, and that work is positive.
Now let’s say there’s 60 Newtons of friction opposing the motion of the suitcase. Friction therefore does 120 Joules of negative work on the suitcase. In that case, the net work that is done on the suitcase is 200 Joules minus 120 Joules, which is 80 Joules.
What if I apply a force perpendicular to the motion of an object? This video shows me holding a heavy bowling ball as I walk from one side of the camera to the other. Let’s look at the displacement vector, which is a horizontal line from left to right. The force that I’m applying on the bowling ball, however, points up. The force and the displacement are perpendicular. While it’s hard for me to carry the bowling ball, and I feel like I’m exerting myself, from a physics perspective, I am doing zero work on the bowling ball.
As just described, when a force is applied to an object parallel to its displacement, work is done on that object. Newton’s second law tells us that force equals mass times acceleration, implying that the object will speed up, slow down, or change direction. If the work done on the object is positive, it will speed up, gaining kinetic energy. If the work done on the object is negative, it will slow down, losing kinetic energy.
By how much? The work-energy theorem can be used to quantify this. The work-energy theorem states that work is equal to the change in kinetic energy of an object. Work is equal to Delta KE, or KE final minus KE initial. If ten Joules of positive work is done on an object, then the kinetic energy of that object will increase by ten Joules.
In this example, a cart is released from the top of a ramp. It rolls down the ramp, gaining speed, until it reaches the bottom. We can consider the motion of the cart and use it to discuss the work-energy theorem. A few moments after releasing the cart, the motion detector measures the velocity of the cart as 0.621 meters per second, at which point it is 0.295 meters away from the motion detector. Just before it reaches the bottom of the ramp, the motion detector measures the velocity of the cart as 1.002 meters per second, at which point it is 0.667 meters away from the motion detector.
Let’s first measure the initial and final kinetic energy of the cart. The cart has a mass of 0.349 kilograms. Initially, the cart has a KE of one-half times 0.349 kilograms times 0.621 meters per second, squared. This KE is 0.067 Joules. At the end, the cart has a KE of one-half times 0.349 kilograms times 1.002 meters per second, squared. This KE is 0.175 Joules. The change in kinetic energy, Delta KE, is equal to final KE minus initial KE, or 0.108 Joules.
Work was done on the cart by gravity. It moved a distance of 0.372 meters from start to finish. The force done by gravity must be carefully calculated by splitting the weight of the cart into components. We know that gravity points down, but we are only interested in the component of force acting parallel to the track. This value, which I used trigonometry to calculate, is 0.298 Newtons. Multiply 0.372 meters and 0.298 Newtons, we see that the work done by gravity is 0.111 Joules.
Note that work and energy are in very close agreement with each other! The slight difference is very likely due to dissipative forces such as friction and air drag, that are not taken into account in these calculations. Some of the work done by gravity throughout this experiment wasn’t converted into KE, but was converted into heat and sound instead.
Mechanical work is useful to us, especially when harnessed in a device such as a motor. Mechanical work is done in the engine of a car to move us around from one place to another. But work by itself isn’t always enough information. How do we describe the engine of a car? Usually we talk about how much horsepower it has. What is horsepower, and what does it measure?
Horsepower is one unit that’s used to quantify the concept of power. While we will not use horsepower in this class, it is used a lot in the United States. It was originally used to relate the capability of motors to horses.
Power has a symbol of the uppercase letter P. The SI unit that we will use are Watts, which is equal to Joules per second. The equation for power is equal to work divided by time. Power therefore tells us how quickly work is done.
A golf cart and a sports car will both be able to move us from point a to point b. But a golf cart will speed up much slower than the sports car. The golf cart is therefore less powerful.
Let’s say a golf cart with a mass of 500 kg speeds up from rest to 9 meters per second in one minute. We can use the work-energy theorem to calculate the work done by the motor, and then divide by time to calculate the power. Work is equal to the change in kinetic energy. Because the golf cart was initially at rest, the initial KE is zero. The final KE is one half times 500 kg times (9 meters per second) squared, or 20,250 Joules. Because want to know what the power is, we divide by the amount of time it takes to speed up to 9 meters per second. One minute is sixty seconds. 20,250 Joules divided by 60 seconds is a power of 337.5 Watts.
Compare that golf cart to a sports car with a mass of 1600 kg that can speed up from rest to 27 meters per second in three seconds. Let’s use the work-energy theorem to calculate its power. The work is equal to one half times 1600 kilograms times (27 meters per second) squared, or 583,200 Joules. Divide by time to calculate power, which is 194,400 Watts. In case we needed mathematical evidence, a sports car is way more powerful than a golf cart!
A simple machine is used to change either the direction or magnitude (or both) of an applied force. There are six types of simple machines: levers, wheel and axles, pulleys, inclined planes, wedges, and screws.
Perhaps you’ve heard the quote: “give me a lever long enough and a fulcrum on which to place it, and I shall move the world,” which was said by Archimedes. While literally moving the world would be impractical, not to mention inconvenient, levers can be used to lift a heavy object, using less force than we get out of the system.
In this demo, I was able to lift a one kilogram mass, which had a weight of 9.8 Newtons. The lever allowed me to apply a force of merely 2.6 Newtons to lift the object. While it might seem like we are getting something for nothing, that’s not the case, which we’ll discuss in just a few moments.
Some simple machines can multiply an input force. This force multiplication is called mechanical advantage. Mechanical advantage has a symbol of MA. The equation for mechanical advantage is MA equals output force divided by input force. Because we divide two forces by each other, mechanical advantage is a unitless quantity. The mechanical advantage of the lever arm used to lift the mass is 9.8 Newtons divided by 2.6 Newtons, giving a mechanical advantage of 3.8.
What is the tradeoff that we have to deal with when using a simple machine? Hopefully your understanding of physics at this point makes you feel profoundly uncomfortable with the fact that something could come out of nowhere. In fact, when I used the lever, note that the mass only lifted a short distance above the table. Meanwhile, I had to push down on the lever a much longer distance in order to get the mass to lift upward. So what I gain in force multiplication, I lose in distance. Remember that work is equal to force times distance. The work is conserved in the simple machine. That means, in the absence of friction or other undesired events (such as bending of the lever), I should get exactly as much work out as I put in.
Let’s do the math for lifting the mass. I applied a force of 2.6 Newtons a distance of 9 centimeters. My work input was therefore 23.4 Newton-centimeters. On the other end, a weight of 9.8 Newtons was lifted a distance of 2 centimeters. The work output was 19.6 Newton-centimeters. Efficiency quantifies how well work input is converted to work output. The symbol for efficiency is the lowercase letter e. The equation for efficiency is work output divided by work input. Efficiency is unitless, but we usually multiply our result by 100 to express it as a percentage. If I put in 23.4 Newton-centimeters of work and got out 19.6 Newton-centimeters of work, then the lever I used had an efficiency of 83.8 percent.
The efficiency is somewhat limited in this simple machine because of the bending of the meter stick I used as the lever. If I had chosen to use a stiffer material for my lever, I could expect the mass to lift higher off of the table, giving me more work output for the same amount of work input.
A wheel and axle is also a simple machine. A linear force can be used to cause rotation, and the force can be multiplied if the wheel and axle system is set up to have a mechanical advantage. Wheelchairs, bicycles, and cars are all examples of objects that use the wheel and axle simple machine.
Pulleys are capable of both changing the direction of a force and also multiplying a force. Pulleys are used in many different applications. From redirecting forces from an input yoke and rudder pedals to flight controls on an airplane; to cable machines at the gym; to elevators; pulleys are all around us. The mechanical advantage of a pulley system is related to the number of pulleys or load strings that are used. A single fixed pulley doesn’t create any mechanical advantage, but allows us to pull down on an object to lift it, rather than pulling up, which may be more difficult for us to do, especially if we have bad knees or a bad back! The more pulleys in the system, the less input force we need to apply to lift an object. However, the caveat is that we must apply the force over a longer distance. The concepts of mechanical advantage and efficiency still apply. You will explore pulleys in quantitative detail in your lab about work, energy, and simple machines.
Inclined planes are also known as ramps. You’ve probably seen them at building entrances, providing an accessible way to enter a building that would otherwise require steps. Not only do ramps provide accessibility, but they also reduce the amount of force that’s required to move along the ramp. The mechanical advantage of the inclined plane is related to the steepness of the slope of the ramp. As mentioned, the lower the force, the longer the distance that the force must be applied in order to obtain the desired amount of work output. You will look at inclined planes in a quantitative way when you do the work, energy, and simple machines lab.
A wedge is a simple machine found in things such as axes, knives, and scissors. A force applied to the blunt edge of the wedge is transmitted to the pointy edge, which can be used to split something apart. A screw takes rotational motion and converts it to linear motion. You may have used screws when putting together furniture or building something.
Thanks for taking the time to learn about work and energy! Until next time, stay well.