### Overview

Source: Ketron Mitchell-Wynne, PhD, Asantha Cooray, PhD, Department of Physics & Astronomy, School of Physical Sciences, University of California, Irvine, CA

This experiment demonstrates the work-energy principle. Energy is one of the most important concepts in science and is not simple to define. This experiment will deal with two different kinds of energy: gravitational potential energy and translational kinetic energy. Gravitational potential energy is defined as the energy an object possesses because of its placement in a gravitational field. Objects that are high above the ground are said to have large gravitational potential energy. An object that is in motion from one location to another has translational kinetic energy. The most crucial aspect of energy is that the sum of all types of energy is conserved. In other words, the total energy of a system before and after any event may be transferred to different kinds of energy, wholly or partly, but the total* ene*rgy will be the same before and after the event. This lab will demonstrate this conservation.

Energy can be defined as "the ability to do work," which relates mechanical energy with work. Flying projectiles that hit stationary objects do work on those stationary objects, such as a cannonball hitting a brick wall and breaking it apart or a hammer driving a nail in to a piece of wood. In all cases, there is a force exerted on a body, which subsequently undergoes displacement. An object in motion has the ability to do work, and thus it has energy. In this case, it is kinetic energy. In this experiment, gravity will be doing work on gliders.

The transfer of the potential energy of gravity to translational kinetic energy will be demonstrated in this experiment by sliding a glider down air tracks at various angles (*i.e., *heights), starting from rest. The potential energy of an object is directly proportional to its height. The net work done on an object is equal to the change in its kinetic energy; here, the glider will start from rest and then gain kinetic energy. This change in kinetic energy will be equal to the work done by gravity and will vary depending upon the starting height of the glider. The work-energy principle will be verified by measuring the starting height and the final velocity of the glider.

### Principles

Potential energy is associated with forces and is stored within an object. It depends upon the position of the object relative to its surroundings. An object raised off the ground has gravitational potential energy because of its position relative to the surface of the earth.This energy represents the ability to do work because, if the object is released, it will fall under the force of gravity and do work upon what ever it lands on. For instance,dropping a rock on a nai lwill do work on the nail by driving it into the ground.

Suppose an object is moving in a straight line at velocity *v _{0}. *To increase the velocity of the object up to

*v*a constant force

_{1},*F*

_{net}would need to be applied to the object. The work

*W*done on an object by a constant force

*F*is defined as the product of the magnitude of the displacement

*d*multiplied by the component of the force parallel to the displacement,

*F*

_{||}*W = F _{||}d.*

**(Equation 1)**

In the case of the moving object, if the force is applied in the direction parallel to the motion of the object, then the net work is simply equal to the net force times the distance traveled:

*W = F*_{net}*d. ***(Equation 2)**

From kinematics, it is known that the final velocity of an object under constant acceleration is:

*v _{1}^{2} = v_{0}^{2} + 2ad. *

**(Equation 3)**

Applying Newton's second law,* F*_{net} = *ma,* and solving for the acceleration in **Equation 3 **gives:

*W _{net} = F_{net }d = mad = md(v_{1}^{2} - v_{0}^{2} )/(2d) = (v_{1}^{2} - v_{0}^{2} )/2. *

**(Equation 4)**

Equivalently:

*W _{net} = ½ m v_{1}^{2}-½ m v_{0}^{2}. *

**(Equation 5)**

If translational kinetic energy is defined as* KE = ½ mv ^{2}, *then this is just the work-energy principle: the net work done on a system is equal to the change in the kinetic energy of the system.

Now consider gravitational potential energy. If an object starting from a height* h* falls from rest under the influence of gravity, the final velocity of the object can be found using **Equation 3**

*v ^{2} = 2gh.*

**(Equation 6)**

After falling from height *h*, the object has kinetic energy equal to *½ mv ^{2 }= ½ m(2gh) = mgh.* This is the amount of work the object can do after falling a vertical distance

*h*and is defined as the gravitational potential energy,

*PE:*

*PE = mgh,*** (Equation 7)**

where *g* is the gravitational acceleration. The higher the object is placed above the ground, the more gravitational potential energy it has. Gravity is acting, or doing work, on the object, so in this scenario, *W _{net} = mgh.* From the work-energy principle, it is known that this gravitational potential energy should then be equal to the change in kinetic energy:

*½ mv ^{2 }= mgh. *

**(Equation 8)**

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### Procedure

- Obtain an air supply, bumpers, two gliders of varying mass, a velocity sensor, an air track, an aluminum block, and a scale (see
**Figure 1**). - Place the lower-mass glider on the scale and record its mass.
- Connect the air supply to the glider track and turn it on.
- Place the aluminum block under the glider stand,close to the air supply. This will be the lowest-height configuration.
- Place the glider at the top of the track and measure the height,
*h*. The measurement should be with respect toits approximate center of mass._{1} - Place the glider at thebottom of the track and measure the lower height,
*h*.The difference_{0}*h*should be the height of the aluminum block, but perform the measurements to verify._{1 }- h_{0 } - Place the glider back on the top of the track, just above the leg and aluminum block, and release it from rest. Record its velocity
*v*at the bottom of the track using the timing gates. Ensure that the velocity is measured with respect to the point where*h*was measured. Do this five times and take the average velocity. Record this velocity in the appropriate box in_{0}**Table 1**. - Place another aluminum block under the glider stand. This will add 3.4 cm to the potential energy calculation. Repeat step 1.7.
- Fill in
**Table 1**. Calculate*KE*and*PE*for each run and compute their differences. - Repeat steps 1.2-1.9 with the heavier glider.

**Figure 1****: Experimental setup.** The components include: (1) air supply, (2) bumper, (3) glider, (4) velocity sensor, (5) air track, and (6) aluminum block.

Energy is one of the most important and yet ambiguous concepts in physics; fortunately, the relationship between energy and work can aid in the understanding of many physics problems.

Energy - particularly *mechanical* energy - is often defined as the ability to do work, that is, to exert a net force on an object causing it to move a certain distance. Mechanical energy can come in the form of position-dependent energy, referred to as potential energy, and motion-dependent energy, called kinetic energy. While the potential and kinetic energy of an object can be converted to one another, the law of energy conservation dictates that the total energy of an isolated system remain constant.

This video will introduce the work-energy principle, discuss the concepts of kinetic and potential energies, and use the law of energy conservation to relate these energies in an experiment involving gliders sliding down a track.

While there are numerous types of energy, mechanical energy most clearly illustrates the idea that energy is the ability to do work. One such example is when a cannonball flies into a brick wall.

In this case, a body, the cannonball, does work on an object, the wall, by exerting a net force and causing the object to move a certain distance. Work is defined as the dot product of the applied force and the distance moved. This applied force must be in the direction of the displacement in order for work to be done, that is, only the component of force parallel to the displacement can do work.

Now, we can relate work to mechanical energy, which is made up of kinetic energy and potential energy. A body in motion from one location to another, such as the cannonball, has *translational* kinetic energy and the ability to do work.

Suppose we accelerate the cannonball from an initial velocity of *v*i to a final velocity of *v*f - a process governed by an equation from kinematics. This event requires a constant net force, driven by Newton's second law, to be applied over a certain distance. By combining the two equations, and noting that translational kinetic energy is defined as *½mv2*, it is clear that the work done on the cannonball, which is Fnet times D, is equal to the difference in the final and initial kinetic energies. This is the work-energy principle.

When it comes to potential energy, a boulder at the edge of a cliff has large gravitational potential energy. Upon release, it has the potential to do work on the ground. This potential work depends on the mass of the boulder, acceleration due to gravity, and height of the fall. And this work is equal to the potential energy before the fall, or Pi.

As per the law of conversation, energy can be converted during an event, but the total energy of the system must remain the same. Therefore, the sum of the initial potential and kinetic energies must equal the sum of the final energies. The boulder's initial velocity and kinetic energy are zero while its final height and potential energy are also zero. Therefore, the initial gravitational potential energy is equal to the final translational kinetic energy. By using our previous equations, a number of relationships can be drawn between the velocity, height, mass and energy.

Now that you've learned the principle of work-energy and law of energy conservation, let's see how these concepts can be applied to an experiment involving mechanical energy.

This experiment consists of a velocity sensor, an air track, a few identical aluminum blocks, a glider, a few weights that can be added to the glider, a scale, air supply and a ruler.

Place the glider on the scale and record its mass. Connect the air supply to the air track and turn it on Measure the height of one of the aluminum blocks and record it in the lab notebook. Place the aluminum block under the foot of the air track that lies closest to the air supply. This will be the lowest height configuration.

Place the glider at its initial position and release it from rest. Using the velocity sensor, record the glider velocity as it passes the final position on the track. Repeat this procedure five times and calculate average velocity.

Place an additional aluminum block under the air track raising the height configuration. Measure the difference between *h*i and *h*f as before and verify this is twice the height of an aluminum block. Repeat the set of velocity measurements for this height configuration.

Place a final aluminum block under the air track, assuming the height difference is now three times the block height and repeat velocity measurements. Next, place some weights to increase the glider's mass, and then repeat the experiment to measure velocities at the three different heights.

Using the equations derived from the work-energy principle, the potential and kinetic energies for each run can be calculated being cognizant of the units for each of the variables. The potential energy differences for the various heights are listed in the PE column of the table. As expected, the potential energy of the system increases with increased height and heavier mass, indicating a greater potential to do work.

The values for the translational kinetic energy are also found in the table in the KE column. Similar to the potential energy, the kinetic energy is greater for the heavier glider and yet the final velocities of the heavier glider are the same as the lighter glider. This is clear from the equation relating the energies where the velocity is only a function of the height. Furthermore, the velocity increases at a rate proportional to the square root of the height as expected.

According to the law of energy conservation, the KE and PE columns in the table should be equal, and they nearly are. The discrepancies in the two sets of values come from errors in the measurements taken which are estimated to be around 10% for this type of experiment.

The applications involving the work-energy principle are ubiquitous and involve all different forms of energy.

Roller coasters are a perfect example of mechanical energy conversion. The massive coaster is initially pulled up to a large height in front of a steep incline. The substantial potential energy gained at the top of the incline is then converted to kinetic energy for the rest of the ride. During the ride the coaster experiences a constant exchange of potential and kinetic energy.

Chemical reactions also exhibit energy conversion with the energy typically being exchanged between chemical potential energy and thermal energy. If the reaction is exothermic, the potential energy is given off as heat to the environment, while the opposite is true for endothermic reactions. Some exothermic reactions can be explosive thereby generating kinetic energy which does work on its surroundings.

You've just watched JoVE's introduction to Energy and Work by Force. You should now understand both the concept and importance of the work-energy principle and how the law of energy conservation can relate potential and kinetic energies. Thanks for watching!

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### Results

Sample calculated values of the initial potential energy at various heights are listed in the *PE* column of **Table 1**, found using **Equation 7**. The final velocities measured from the experiment are also in the table. The translational kinetic energy is calculated using these measured values of the final velocity. According to the work-energy theorem, the *KE* and *PE *columns in the table should be equal, and they nearly are. The discrepancies in the two values simply come from errors in the measurements taken throughout the experiment, where a percent difference of around 10% can be expected from this type of experiment.

Note that as the initial height is increased, the final velocity also increases at a rate that is proportional to the square root of the height increase (c.f. **Equation 6).** The potential energy of the system also increases with increased height. Furthermore, note that the cart with the increased mass (the last three rows in **Table 1**) has both higher potential energy and kinetic energy when compared to the lower-mass cart (first three rows), but the final velocities of this cart are the same as for the lower-mass cart. This makes sense because the final velocity is only a function of the height (**Equation 6). **

**Table 1: Results.**

Cart Mass (kg) | Height (cm) | PE (mJ) |
V (m/s)_{f} |
KE (mJ) |
% difference |

0.23 | 3.4 | 77 | 0.8 | 74 | 4 |

0.23 | 6.8 | 155 | 1.2 | 167 | 8 |

0.33 | 3.4 | 111 | 0.85 | 120 | 8 |

0.33 | 6.8 | 221 | 1.25 | 259 | 17 |

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### Applications and Summary

Applications of the work-energy principle are ubiquitous. Roller coasters are a good example of this energy transfer. They pull you up to a great height and drop you down a steep incline. All the potential energy that you gain at the top of the incline is then converted to kinetic energy for the rest of the ride. The coasters are also massive, which adds to the potential energy. Skydivers use this principle as well. They ride in an airplane that does work on the system to bring them to a height of around 13,000 feet. Their initial velocity in the vertical direction is nearly zero just before they jump out, and they quickly reach terminal velocity (because of air resistance) after jumping. Firing a gun also converts potential energy to kinetic. The gunpowder in the ammunition has a lot of stored chemical potential energy. When it is ignited, it does work on the bullet, which exits the muzzle with a tremendous amount of kinetic energy.

The work-energy principle has been derived in this experiment. Using a glider on an inclined air track, the work done by gravitational force has been experimentally verified to equal the change in the kinetic energy of the system.

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