PHYS111Q : Labs

8 Conservation of Energy

Introduction

The total energy of a system consists of several kinds of energy. Kinetic energy is the energy of motion. Thus far, we have a definition for translational kinetic energy,

$$K = \frac{1}{2} m v^2$$ (29)

When we study rotational motion, we will also have an expression for rotational kinetic energy. The potential energy of a system is energy available for conversion into other forms of energy via work done by a conservative force. The conservative force we will consider in the experiments described below is gravity. The gravitational potential energy of an object of mass $m$ is expressed

$$U_g = mgy$$ (30)

where $y$ is the vertical position of the object. We define the total mechanical energy of a system as the sum of its kinetic and potential energies,

$$E^\mathrm{mec} = K + U$$ (31)

In addition to mechanical energy, a system can have internal energy. For example, the molecules in the spherical plastic projectiles we launched in Lab 4 vibrate on a tiny scale ( $\approx 10^{-10}$ m). They vibrate more at higher temperature than at lower temperature. When the projectile collides with the lab table, some of its kinetic energy is converted into internal vibrations of the projectile and lab table. This kind of conversion of energy is one-way. The microscopic vibrations of the molecules of the projectile and the table top cannot be harnessed to give the ball back the macroscopic kinetic energy it lost in the collision. We call this one-way kind of internal energy thermal energy. There are also other forms of internal energy, which are not limited to one-way conversions. Your text makes the awkward choice of referring to internal energy as ``nonconservative energy,'' $E^\mathrm{noncons}$.

Experimental observations of both microscopic (elementary particles, atoms, molecules) and macroscopic (soccer balls, solar systems, galaxies) systems support the principle of conservation of energy - that the total energy of an isolated system is constant. The principle of conservation of energy can be expressed quantitatively as

$$W^\mathrm{ext} = \Delta E^\mathrm{mec} + \Delta E^\mathrm{noncons}$$ (32)

where $W^\mathrm{ext}$ is the work done on the system by external forces not accounted for as potential energy. You will apply the principle of conservation of energy to a bouncing racquetball and a swinging pendulum.

Experiments

An Object in Free Fall

1. Measure the mass of the racquetball.

2. You have been provided with the file RBallEnergy.cmbl for this experiment. The bottom two graphs are the usual position vs. time and velocity vs. time graphs. The top graph will show potential, kinetic, and total mechanical energy. (Ignore the top graphs until you get to the analysis stage.)

3. You've done the ``bouncing ball'' experiment many times now, so you know the drill. Collect a few measurements showing several clean bounces.

4. Collect a measurement of the racquetball at rest on the table below the motion detector in order to determine its lowest vertical position $y_o$. (You will need this to calculate the potential energy of the system.)

A Pendulum

1. Measure the mass of the pendulum bob.

2. Hang the pendulum from the stand, and adjust the length to place the bob a centimeter or two above the lab table at its lowest point so that the horizontal motion of the bob can be ``seen'' by the motion detector placed on the lab table.

3. Measure the length $L$ of the pendulum, the distance between the pivot point and the center of the bob.

4. Place the motion detector a little more than 50 cm away from the bob.

5. You have been provided with the file PendulumEnergy.cmbl for this experiment. (Ignore the top graph until you get to the analysis stage.)

6. Start the pendulum swinging by releasing it from rest at a horizontal displacement of no more than 10 cm.

7. Adjust the position and orientation of the motion detector to minimize the fluctuations in distance measurements, and use the Sampling tab under Setup -> Data Collection... to adjust the experiment length so that you collect a measurement long enough to see a substantial (at at least 10%) reduction in the amplitude (maximum displacement from equilibrium) of the swing of the pendulum. (You will be asked to quantify the rate of conversion of the mechanical energy of the pendulum into nonconservative forms, so the more reduction you capture, the better.)

8. Before you move the motion detector (!), measure the equilibrium position $x_o$ of the system by collecting a measurement while the pendulum is stationary at its equilibrium position. (You will need this to calculate the potential energy of the system.)

Analysis

An Object in Free Fall

1. The motion detector is looking down on the ball, so it uses a coordinate system in which ``down'' is positive. It is also mounted at some (measured) distance $y_o$ above the lab table. It is convenient (but not necessary), for the purpose of displaying $K$,$U$, and $E$ on the same graph to use a coordinate system in which the lab table corresponds to $y = 0$. Hence, we will use a modified version of Eq. 30,
   

   $$U_g = -mg(y-y_o)$$ (33)

   
   

2. Put your measured $m$ and $y_o$ values into the calculation of the potential energy in Logger Pro. Click on Data -> Column Options -> Potential Energy to bring up a Calculated Column Options widow. Choose the Column Definition tab. In the ``Equation'' box, you should see

   -0.050*9.8*("Position"-0.70)

   which follows the form of Eq. 33 with $m = 0.050$ kg and $y_o = 0.70$ m. Replace these values with your measured values of $m$ and $y_o$, and click Ok.

3. Put your measured mass into the calculation of the kinetic energy in Logger Pro. Click on Data -> Column Options -> Kinetic energy to bring up a Calculated Column Options widow. Choose the Column Definition tab. In the ``Equation'' box, you should see

   1/2*0.050*"Velocity"*"Velocity"

   which follows the form of Eq. 29 with $m = 0.050$ kg. Replace this value with your measured values of $m$, and click Ok.

4. Devise a plan for using your total mechanical energy data to determine whether or not air friction acting on the racquetball is significant within experimental uncertainties, discuss your plan with your instructor, and then carry it out.

A Pendulum

Figure 6: Free-body diagram of a pendulum.

A diagram of the pendulum is shown in Fig. 6. The mass is subject to both gravity and the tension in the string. The pendulum bob has no displacement parallel to the string, so there is no work or potential energy associated with the tension. We can measure the horizontal position of the pendulum with the motion detector, and we need to find its vertical position. According to the diagram, if we use a coordinate system in which the lowest vertical position of the pendulum is $y = 0$,

$$y = L - \sqrt{L^2 - (x - x_o)^2}$$ (34)

where $x_o$ is the horizontal equilibrium position of the pendulum - its horizontal position when it is directly below the pivot point. With this expression for $y$, the gravitational potential energy (Eq. 30) becomes

$$U = mg \left( L - \sqrt{L^2 - (x - x_o)^2} \right)$$ (35)

1. Put your measured mass, equilibrium position, and length into the calculation of the potential energy in Logger Pro. Click on Data -> Column Options -> Potential Energy to bring up a Calculated Column Options widow. Choose the Column Definition tab. In the ``Equation'' box, you should see

   0.200*9.8*(1.00 - sqrt(1.00^2-(``Position''-0.50)^2))

   which follows the form of Eq. 35 with $m=0.200$ kg, $x_o = 0.50$ m, and $L = 1.00$ m. Note that $L$ appears in the equation twice. Replace these values to your measured values of $m$, $x_o$, and $L$, and click Ok.

2. Put your measured mass into the calculation of the kinetic energy in Logger Pro. Click on Data -> Column Options -> Kinetic energy to bring up a Calculated Column Options widow. Choose the Column Definition tab. In the ``Equation'' box, you should see

   1/2*0.200*"Velocity"*"Velocity"

   which follows the form of Eq. 29 with $m=0.200$ kg. Replace this value with your measured values of $m$, and click Ok.

3. Devise a plan for using your total mechanical energy data to determine the average rate at which the mechanical energy of the pendulum is converted into nonconservative forms of energy, along with its uncertainty, discuss your plan with your instructor, and then carry it out.

Group Assignment

Show your work to your instructor and discuss preliminary answers to the questions below.

Individual Assignment

Hand in a printout of your spreadsheet, if you used one, and answers to the following questions.

1. Describe your strategy for determining whether or not air friction acting on the racquetball is significant within experimental uncertainties, and give your conclusion.

2. Describe your strategy for determining the average rate at which the mechanical energy of the pendulum is converted into nonconservative forms of energy, along with its uncertainty, and give your results.

3. What is the maximum period of time over which we can ignore the nonconservative forces acting on the pendulum within uncertainty? Explain. (This may require some additional analysis of your pendulum data.)

Copyright © 2003-2007, Lewis A. Riley

Updated Tue Nov 30 13:48:34 2004

This work is licensed under a Creative Commons License.