PHYS111Q : Labs

7 Work and Kinetic Energy

Introduction

We have introduced in lecture the concepts of work

$$W = \left< \vec{F} \right> \cdot \Delta \vec{r}$$ (23)

and kinetic energy

$$K = \frac{1}{2} m v^2$$ (24)

which are related by the work-kinetic energy theorem

$$W^\mathrm{net} = \Delta K$$ (25)

This, unlike Newton's second law, is a scalar equation. No information about the directions of forces or velocities can be extracted from energy calculations. In fact, one of the advantages of the energy framework is that we can do calculations in situations in which direction information is difficult or impossible to obtain.

Your present task is to consider a bouncing ball from a work-kinetic energy perspective. We find that in the real world, a bouncing ball never seems to return to its previous maximum height in subsequent bounces. Instead, it performs ever-smaller bounces until it comes to rest. Until now, we have not tried to quantify or even identify the loss(es) which lead to this phenomenon. You will use the work-kinetic energy theorem to study energy loss in this system. Your lab group will be given a racquetball to study. You will consider energy lost both during bounces (possibly to vibrations of the table, internal vibrations and heating of the ball, e.g.) and to work done on the ball by the drag force it is in flight.

Gravity continually does work on a bouncing ball, slowing it down on its way up and speeding it up again on its way down. The net work done by gravity

$$W^\mathrm{grav} = -mg\Delta y$$ (26)

between bounces is always zero, because at each bounce occurs at the same height, so $\Delta y = 0$. Hence, gravity cannot be responsible for any permanent energy loss. It is therefore helpful to separate the net work done on the ball into two parts

$$W^\mathrm{net} = W^\mathrm{grav} + W^\mathrm{loss}$$

$$= -mg\Delta y + W^\mathrm{loss}$$ (27)

where $W^\mathrm{loss}$ is the work done by the force or forces that act to permanently remove energy from the system. By measuring the mass of the ball and its vertical position at two points, you will be able to quantify the work done by gravity, and by measuring the velocity of the ball at the same two points, you can quantify its change in kinetic energy $\Delta K$. Then, you can find $W^\mathrm{loss}$ using the work-kinetic energy theorem (Eq. 27).

The drag force on an object traveling through air always opposes the velocity of the object. It follows that Eq. 23 becomes simply

$$W^\mathrm{drag} = - \left< F^\mathrm{ drag} \right> d$$ (28)

where $d$ is the total distance traveled by the object. Note that $d$ is not the magnitude of the vector displacement of the object. Instead, it is the (positive) length of the actual path taken by the object. It is straightforward to determine $d$ for a projectile moving in one dimension, but the two dimensional case is more difficult. If you find that a significant amount of negative work is done on the ball while it is in flight, you can use Eq. 28 to determine the average drag force acting on the ball.

Experiment

1. Measure the mass of the ball.

2. Connect the motion detector to the LabPro interface, and connect the interface to the USB port of your laptop.

3. Run Logger Pro. For these measurements, we want to have the best time resolution we can manage, so that we can measure velocities as close to the bounces as possible. Use the Data Collection button to set the sampling rate to 50 samples/second.

4. Mount the motion detector so that it is ``looking'' down at the table as it was for your racquetball measurements of Lab 2.

5. Collect five ``good'' measurements of the tallest ($\approx30$ cm) trajectory you can measure. We will be analyzing the work done in the time period from just before the first bounce until just after the second bounce. A ``good'' measurement includes no bad data (spikes, atypical random fluctuation e.g.) in the vicinity of the first and second bounces.

Analysis

You have been provided with a spreadsheet (Work.xls) to aid you in the analysis of your data. Your measurements go into cells highlighted in yellow. Don't forget to enter the mass in cell B2, or all of the energy calculations will be zero even if you've entered all of your speeds and positions. Once all of your data is in the spreadsheet, column H will show $W^\mathrm{loss}$ during the bounces, labeled $W^\mathrm{bounce}$, and column I will show $W^\mathrm{loss}$ while the ball is in free-fall going up and coming down, labeled $W^\mathrm{drag}$. Column J shows the average drag force going up and coming down.

1. Click the Examine button , and you will be able to use your mouse to examine individual data points on each graph.

2. For each measurement,
   1. determine the maximum speeds just before and just after the first two bounces (4 speeds per measurement),

   2. determine the positions corresponding the four speeds collected above, and

   3. determine the position when the ball was at its highest point above the table.

   Record these values in your spreadsheet.

3. Add calculations of the average and the standard deviation of the mean of $W^\mathrm{bounce}$, $W^\mathrm{drag}$, and $\left<F^\mathrm{ drag}\right>$.

Before You Leave Lab

Show your spreadsheet to your instructor. Discuss preliminary answers to the questions below. If you don't have an answer to question 4, make sure you know how to get one before you leave lab.

Group Assignment

Hand in your spreadsheet and answers to the following questions.

1. In your spreadsheet, under what circumstances is the work done by gravity negative and under what circumstances is it positive? Explain why.

2. If you check the calculations in column G of the spreadsheet, you will find that the work done by gravity is calculated using
   
   $$W^\mathrm{grav} = mg \Delta y$$
   
   
   Why isn't the minus sign there (see Eq. 26 above)?

3. What would you expect the signs of $W^\mathrm{bounce}$ and $W^\mathrm{drag}$ to be? Why? Does your spreadsheet support your expectation?

4. Based on your observations, is more energy lost during bounces or to air drag? Explain.

Copyright © 2003-2010, Lewis A. Riley

Updated Fri Aug 27 11:05:11 2010

This work is licensed under a Creative Commons License.