## H-5 JOULE'S EQUIVALENT
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.
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,
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.
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:
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
ΔW = τθ = F(R+r)Δθ = 2πF(R+r)n
where τ = F(R+r) is the work done against friction, F = F
The tension on one end of the rope is
Some versions of this experiment use a spring balance on each end of the cord.
(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) 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 (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.
(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.
(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.
(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). (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
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
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 Text and drawings © 1996, 2004 by Donald E. Simanek. |