H-5 JOULE'S EQUIVALENT

1. OBJECT:

To measure the Joule equivalent, sometimes called the "mechanical equivalent of heat". It is the constant which relates the thermal energy given to a body to the work done on it: J = ΔW/ΔH, in joule/calorie.

2. APPARATUS:

Pasco TD8551 apparatus, consisting of an aluminum cylinder with embedded thermistor, which can be rotated by a crank. Accessory equipment includes: a heavy (approx 10 kg) pail, nylon rope or strap, table clamp, C-clamp, ice bath with a large plastic bag, vernier caliper, analytic balance, ohmmeter (10 kΩ to 1000 kΩ), thermometer, electrical leads.

Powdered graphite lubricant, level.

Either: A heavy (~10 kg) pail and a 500 gm spring balance, or a 15 kg spring balance and a 500 gm spring balance. A pulley and weight hanger may be substituted for the smaller spring balance.

3. METHOD AND THEORY.

Whenever mechanical work is done against frictional forces energy is dissipated as heat. You will measure the work and the heat to determine Joule's equivalent, the quotient J = ΔW/ΔH, in the units joule/calorie.

 [1]

ΔW = JΔH

where J is the Joule equivalent.

The apparatus is a modern adaptation of the classic method invented by Callendar. An aluminum cylinder (A) is rotated by turning the crank (E). Friction is applied to the surface of the cylinder by a belt or heavy cord (G) over the cylinder which supports a heavy weight, (C). [Such a belt-and-drum arrangement used to measure work is called a "Prony brake."]

In the original experiment one measured the temperature rise of a measured amount of water in the cylinder. In this modern version there's a thermistor (temperature sensitive resistor) embedded in the aluminum cylinder.

You will measure the thermal energy given to the aluminum cylinder by doing a measured amount of work against friction on that cylinder. The thermal energy is measured by the usual calorimetric technique of determining the rise in temperature. The thermal energy is given by:

 [2]

ΔH = msΔT

where m is the mass of the material, s is its specific heat capacity and DT is the change in its temperature.

As you apply a torque to turn the crank the friction of the stationary belt exerts a counter torque on the cylinder. The work done against friction may be calculated from the belt tension and the angle the cylinder turns.

 [3]

ΔW = τθ = F(R+r)Δθ = 2πF(R+r)n

where τ = F(R+r) is the work done against friction, F = F2 - F1 is the difference in the tension at the ends of the belt, r is the radius of the rope, and Δθ = 2πn is the total angle the cylinder turns through in n revolutions.

Lifting a heavy weight. A heavy weight (C), perhaps a one- gallon paint can filled with sand and lead, is attached to one end of the cord. The cord wraps around the aluminum cylinder (A) about three times. The other end of the cord runs over a pulley (B) to a weight hanger (D) with a much smaller weight, say 100 to 200 gm. The crank (E) is turned just fast enough to hold the can a few centimeters above the floor.

Do not lift the can more than a few centimeters above the floor. If you were to let go of the crank its backlash could give you a painful blow to your hand. Do not let go of the crank while turning it. When finished, relieve the tension in the cord by turning the crank backward slowly.

The tension on one end of the rope is much smaller than on the other; small enough to neglect! Pasco recommends holding the low tension end of the rope in your hand, and not measuring that force at all. The idea of holding it in your hand is a very practical way to control the tension, but you can achieve this and measure the force simply by holding that spring balance.

Pulling against a spring balance. A spring balance (S) capable of reading forces of 10 to 20 kg equivalent is attached to one end of the cord. The other end passes over a pulley and a weight hanger is attached to that end. The crank is turned at such a rate as to keep the spring balance reading constant.

Some versions of this experiment use a spring balance on each end of the cord.

4. PROCEDURE.

(1) If you are using the weighted paint can, weigh the can, including its contents.

(2) Connect the ohmmeter to the two electrical terminals on the apparatus. If you are using a multimeter (multi-function meter) be sure that it is set on the "ohms" or "resistance" function.

(3 Check that the meter is functioning properly. Record the thermistor's resistance at room temperature. Record the room temperature on an ordinary thermometer, as a check. There's a calibration table for these thermistors in the appendix.

(4) Practice Run: Assemble the apparatus as described. The cord should have about three non-overlapping wraps around the cylinder. Be sure to position the apparatus so that the rope around the cylinder will not rub against the electrical connecting wires, or against anything else. Turn the crank a few times to get the feel of how fast you must turn it to achieve stable operation, and to adjust the lengths of cord and position of the spring balances. A speed of about 1 rev/sec or a bit more should do it. Certainly you won't want to exceed 2 rev/sec. Do a practice run, counting turns, and timing them, until the ohmmeter indicates a temperature rise of a few degrees.

Pasco recommends applying graphite lubricant to the cylinder. You may wish to use the lubrication if you find the friction to be variable, making it difficult to maintain the can at a reasonably constant height.

(5) Carefully remove the aluminum cylinder by first unscrewing the retaining screw (F) (which has a black plastic knob). Remove the cylinder, measure its diameter, and weigh it. Measure the diameter of the rope, or thickness of the strap, whichever you are using. [You need this to determine the actual radius of application of the tension force, which is the cylinder radius plus the rope radius.]

(6) Do not reassemble the apparatus yet, but note that the two notches in the plastic engage and lock into place with two metal rods extending out from the main shaft. Also notice the two metal slip rings which make electrical contact with two springy metal contacts. These are necessary to provide the electrical connection to the thermistor which is inside the aluminum cylinder.

(7) As usual, in a calorimetric experiment, you will start with a temperature about the same amount below room temperature as the final temperature will be above room temperature. So you must cool the aluminum cylinder, while keeping it dry. Put the aluminum cylinder in a water-tight bag and immerse the bag in an ice-water bath to bring it down to, or near, 0°C.

(8) Remove the cold cylinder, avoiding warming it, and quickly reassemble it in the apparatus. (9) Now do the experiment. Turn the crank while recording the resistance. The total number of crank revolutions is recorded by an automatic counter. Stop when the thermistor indicates the desired final temperature.

5. ANALYSIS.

(1) Using Eq. 1, 2 and 3, derive an expression for J in terms of the measured quantities.

(2) Use this equation to determine the Joule constant.

(3) Derive the error equation for the formula for J, and find the experimental error.

6. ALTERNATE METHOD AND ANALYSIS

(9') In the trial run you found out about how many crank turns are required to raise the cylinder temperature 1°C. Use this fact to decide how often to take resistance readings (i.e., how many crank turns between readings). Make out a data sheet in advance, and coordinate your efforts with your partners so you can take data of resistance, numbwer of turns, and time. When finished, you'll be able to make a plot of temperature vs. time. It would also be a good idea to take resistance vs. time readings before you begin to crank and after you have finished, to establish the warming and cooling curves and allow you to make any necessary correction for thermal lag in the thermistor. Your analysis will be as follows:

(1) For each resistance in your data table, calculate the corresponding temperature, using the table in the appendix.

(2) Calculate the work done as a function of the number of revolutions, using Eq. 2.

(3) Plot temperature vs. work done. You might expect this to be a linear relation. Is it? Fit a straight line to this data. Determine the Joule equivalent from the slope of this line.

7. QUESTIONS.

(1) We asserted that the work against friction is equal to the work you do in turning the crank. You calculated the work as T(R+r), where T is the rope tension, R is the radius of the cylinder and r is the radius of the rope. Yet the work done by friction is fR, where f is the frictional force at the surface of the cylinder. Why does one use different values of the radius in these two methods of expressing the work? That is, why must you add the rope radius to the cylinder radius when multiplying by the tension, rather than simply using the cylinder radius?

(2) What percent error in the computed value of the mechanical equivalent of heat will be caused by an error of 0.2°C in the thermometer which measures the initial and final temperatures of the water. Assume a temperature rise of 20°C?

(3) [S&S] In this experiment a better balance of the heat gained from the surroundings and the heat lost to the surroundings is obtained when the difference between the room and final temperature of the aluminum cylinder is 10 to 15 percent less than the difference between the initial temperature and room temperature. Discuss the reasons for this.

(4) [S&S] The British Thermal Unit, (BTU), still used by some engineers, is defined to be the amount of heat required to raise one pound of water one degree Fahrenheit. Convert your value of the mechanical equivalent of heat from joules per kilocalorie to foot pounds per BTU. Use the conversion relations, 1 Kg weighs 2.20 pounds and 2.54 centimeters equal one inch.

(a) Express the conversion factor in the form: (fp/BTU)/(J/kC).
(b) Express the Joule equivalent in the form: J = ( )fp/BTU.

(5) [S&S] What percent error in the computed value of the mechanical equivalent of heat will be caused by a 3% error in the measurement of the friction force?

(6) James Prescott Joule (1818-1889) did some of his early measurements of the relation between thermal energy and mechanical energy while on his honeymoon in the Swiss Alps. He measured the difference in water temperature between the top and bottom of a waterfall, and accounted for the difference as due to the change in mechanical potential energy of the water falling the height of the waterfall. To see how large this effect might be, calculate how much the water is warmed by passing over Niagara Falls and falling 50 m. Is it reasonable that Joule could have measured such a small temperature difference accurately enough to draw a correct conclusion?

(7*) We asserted that the work done against friction is equal to the work you do in turning the crank. You calculated the work as TR, where T is the rope tension and R = Rcylinder + Rrope. Yet the work done by friction is fRcylinder, where f is the frictional force at the surface of the cylinder. Why does one use different values of the radius in these two methods of expressing the work? That is, why must you add the rope radius to the cylinder radius when multiplying by the tension, rather than simply using the cylinder radius?

APPENDIX 1: TABLE OF RESISTANCE VS. TEMPERATURE FOR THE THERMISTOR

```R in      T(C)     R in      T(C)
Kilohms            Kilohms

351.020     0      95.447     26
332.640     1      91.126     27
315.320     2      87.022     28
298.990     3      83.124     29
283.600     4      79.422     30
269.080     5      75.903     31
255.380     6      72.560     32
242.460     7      69.380     33
230.260     8      66.356     34
218.730     9      63.480     35
207.850    10      60.743     36
197.560    11      58.138     37
187.840    12      55.658     38
178.650    13      53.297     39
169.950    14      51.048     40
161.730    15      48.905     41
153.950    16      46.863     42
146.580    17      44.917     43
139.610    18      43.062     44
133.000    19      41.292     45
126.740    20      39.605     46
120.810    21      37.995     47
115.190    22      36.458     48
109.850    23      34.991     49
104.800    24      33.591     50
100.000    25      32.253     51
```
APPENDIX 2: A HISTORICAL NOTE

Until the 18th century heat was thought to be a material substance, called "caloric", which could flow from one body to another. Chemists and physicists attempted to weigh caloric, with inconclusive and confusing results.

Physicist Count Rumford (Benjamin Thompson) (1753-1814) is credited with casting the first serious doubt about the caloric model, based on his experiments measuring heat generated while boring cannon barrels. Even with a very dull boring tool, causing a great amount of heat, he could detect no increase in the mass of the cannon and metal shavings. He suspected that heat was not a substance, but had something to do with motion. We now know that thermal energy in a body is the total of the kinetic energies of its constituent particles.

It wasn't until the middle of the 19th century that scientists finally and conclusively showed that thermal energy and mechanical energy were directly related, and determined that relation. Previously the two had been measured by independent units, and were defined independently.

In 1850 Joule carried out laboratory experiments to measure the mechanical equivalent of heat. In one experiment (out of many) Joule used falling weights to drive a paddle wheel inside a thermally insulated water-filled container. His results, in modern units, gave a value of 1 calorie = 4.186 Joule, within 1% of the currently accepted value of 4.185 J. This relates the previously defined units of thermal energy to those of mechanical energy.

Today the kilocalorie is defined to be 4,185 J, replacing the older definition which defined it from calorimetery and declaring the specific heat capacity of water to be 1.

Text and drawings © 1996, 2004 by Donald E. Simanek.