Lab VIII: Mechanical Equivalent of Heat

George WongJocelyn Schulz

Instructor: Patrick Cooper

Experiment Date: 6 December 2012

Due Date: 17 December 2012

1 Objective

The objective of this laboratory was to observe the transformation of work into heat; thatis, to examine the relationship between work and energy. Specifically, we were to do work onan object, then measure the resulting change in tempreature, then find a correlation betweenthe two.

2 Theory

In the simplist of cases, work is given to be the dot product of a force vector and displacementvector; that is, dW = ~F ·d~r. It is important to the concept of the relation between work andforce to note that only the force being applied in the direction of the motion contributes towork.

In the case of our cylinder being spun N number of times with a mass of M being hung fromit and a radius of R, this gives the equation for work:

W = 2πMgRN

Heat (which we will be equating to work) is related to temperature by mass and a quantitywe call “specific heat” (for that material). In the case where we wish to measure change inhear, we use the formula:

Q = mc (Tf − Ti)

where m is mass, c is specific heat (for aluminum 0.22 cal/g·◦C, Tf is final temperature, andTi is initial temperature.

For our system, by the first law of thermodynamics, we assume that all work done on thecylinder is transformed into heat. Put another way, we assume the total change in internalenergy is zero. Thus, we can perfectly equate ∆Q = ∆W .

To measure change in temperature, we chose to measure the resistance of the metal cylin-der. As temperature increased, resistance increased; as temperature decreased, resistancedecreased. The exact nature of the relationship (while overall seeming to be indirect) is notknown; therefore, a chart of experimentally-obtained values was used.

There are an accepted 4.182 J / cal (according to the IUNS)

3 Set Up

For this diagram, we used a rotary pully whose resistance could be measured around whichwas strung a nylon cord with a mass hanging at the end of it.

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As the crank was turned, the nylon rope would be pulled slightly upwards (at which point itwould begin to slip and the mass hanging at the end of the string would pull it downwards,thereby applying tension to the pulley. Each complete turn of the pulley, a counting devicewas “ticked”, thereby providing an efficient way to count the number of cranks (and thus thefully “distance” that the was turned.

The nylon string was coated with graphite (to act as a lubricant).

Figure 1: Pully + Mass System diagram, showing sketch of pulley and cord. Note thatthe cord was wrapped around the pulley several times.

4 Procedure

1. The effective resistance of the pulley was measured at room temperature.

2. The cord was strung around the pulley four and a quarter times, then attached to ahanging weight of known mass.

3. Using ice, the pulley was cooled until it had an effective resistance that equated to atemperature of about 7 degrees below room temperature.

4. With the weight hanging, the pulley was cranked (making sure that the cord nevercaught and the mass remained approximately the same height above the ground).

5. Cranking continued until the pulley’s resistance measured was equivalent to a temper-ature of approximately 7 degrees above room temperature.

6. The change in heat was found and equated to work to analyze the relationship, andothers (such as the specific heat of the metal.

5 Data and Calculations

masses of objects

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object mass (g) †

bucket 9674 ± 10cylinder 202.6 ± 0.2

†uncertainty in bucket estimated based on reported value

diameter of cylinder: 4.782 ± 0.001 cm

resisances and corresponding temperatures of pulley

environment resistance (kQ) corresponding temperature (◦ C) ††

room temp. 115.3 ± 0.1 22.0 ± 0.5∼ 8◦ C below 175.2 14.0 ± 0.5∼ 8◦ C above 77.5 31.0 ± 0.5

†† uncertainties in temperature were chosen based on the fact that only per-degree resistanceswere given

counter: 215 cranks

cord wound around cylinder/pulley: 4.25 times

given heat capacity for aluminum: 0.220 cal / (g ◦C)

6 Calculated Values

diameter of pulley: 4.782 cm = 0.04782 ± 0.00001 mradius of pulley: 0.02391 m

mass of cylinder: 202.6 ± 0.2 g = 0.2026 ± 0.0002 kgmass of bucket / weight: 9674 ± 10 g = 9.674 ± 0.010 kg

given heat capacity for aluminum: 0.220 cal / (g ◦C) = 220.0 cal / (kg ◦C)

W = 2πMgRN

W = 2π(9.674kg)9.81ms−2(0.02391 m)215 turns

W = 3064 ≈ 3060 J

Q = mc(Tf − Ti)

Q = 0.202 kg (220.0cal

kg◦C)(31.0 − 14.0 ◦C)

Q = 755.48 ≈ 755 cal

W/Q = 3060/755 = 4.05 J / cal

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7 Error Analysis

7.1 Numerical analysis

W = 2πMgRN

∆W =√

(2πgRN∆M)2 + (2πMgN∆R)2

∆W = 0.003 cal

∆Q =√

(c(Tf − Ti)∆M)2 + (mc(√

(∆Tf )2 + (∆Ti)2))2

∆Q = 30 J

∆W/Q =√

(∆Q/Q)2 + (∆W/W )2(W/Q)

∆W/Q = 0.04 J/cal

7.2 Reasons / Causes

As is noted elsewhere, the system that was used was not by any means a closed system.As such, it cannot truly be assumed that all of the work done on the cylinder contributedto a measurable change in temperature. Some of the work became temperature that wasthen dissipated by the atmosphere (or the atmosphere contributed some temperature to thecylinder that was not accounted for).

There may have been some slippage in the string (or times when the string did not slip andturning the crank actually pulled the weight upwards). Thus, a work that was not accountedfor would have been performed. This would disturb calculations of the values of interest.

The actual temperature of the pulley was not measurable either; it was only possible to knowits temperature to within one degree. Further, a chart relating resistance to temperature wasused, and while it was assumed that this chart was accurate, less of experimental verification,there would be no way to know its accuracy.

Also, the radius of the cylinder was only measurable to within a hundredth of a millimeter.This distance, however, was considered to be relatively quite small, so its effect on error/ uncertainty was ignored. Similarly, the masses of the cylinder and buckets were onlymeasureable to within certain margins; however, the accuracy with which they could bemeasured let it be taken as relatively zero.

8 Questions

1 When you turn the crank, what would be the problem with turning the crank too slowly?

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If the crank were turned too slowly, there would be two problems. First, there might not beenough friction between the pulley and the nylon cord to ensure that the full mass of thehanging weight / bucket contributed to torque–which would make it so there were no reliableequation to find work done on the pulley. Second, much more of the heat that was impartedto the pulley by the cranking motion would be lost to the environment (as compared to thepulley being cranked at a faster speed).

2 What role does heat capacity and heat conductivity of the cord play in the accuracy ofthis experiment?

The greater conductivity of the heat conductivity (and related heat capacity) of the cord,the less the accuracy of the experiment. If heat were transfered to the cord from the heatingcylinder, then the measured temperature change of the cylinder would not be representativeof the true change in heat of the cylinder (as some of the heat would be lost to the cord). Aswe were measuring change in heat / temperature of the cylinder, this could be disastrous.

3 Can you think of advantages and disadvantages of making the temperature interval largeror smaller?

The greater the interval, the more precise our measurements would theoretically be; however,this is misleading, because the greater the interval, the worse off we would be as far as heatlost / interaction between the system and the surrounding environment.

4 Why are the lower and higher temperatures chosen to be symmetric about room tempera-ture?

The lower and higher temperatures were chosen to be symmetric about room temperaturebecause this would effectively minimize the effect that the environment had on the temper-ature change. While the cylinder was chilled, the environment would contribute heat to it;however, once it passed the room temperature mark, the environment would leech energyaway from it. It was hoped that this leeching and contributing would cancel out, therebyrendering the effect of the environment on the cylinder minimal.

5 Any heat flow into or out of the cylinder will contribute to error. How is this errorminimized?

This error is minimized in two ways. First, the entire process is speed up as fast as possibleto minimize the amount of time heat has to flow into and/or out of the cylinder. Second,the temperature change interval was centered around room temperature (the temperature ofthe environment)–see question 4 for more info.

6 What will be the effect on your results if there is moisture on the cylinder when you startturning the crank for data?

If there were moisture on the cylinder, this would affect the data in several ways. First, it

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would likely cause the cylinder to misrepresent the amount of heat change, as the changingtemperature of the water would necessarily account for some of the true heat change. Second,it might decrease friction between the nylon cord and the pulley, which could end up affectingthe amount of work done on the pulley. Third, it might cause the nylon string to stick toitself, in the process causing the turning of the pulley to pull the weight/bucket upwards.This too would affect the work done on the cylinder by the string.

9 Conclusion

All in all, we found an experimental value for joules / calory that agreed rather well withthe accepted value. Our experiment showed that there were approximately 4.05 J/cal, withan uncertainty of 0.04 J/cal. Compared to the accepted value of 4.186 J/cal, this is a 4%deviation.

As noted before, the much of the error came from the fact that the system was not closed.Because it was not closed, there was no way to ensure that heat did not escape or enterthe system in any unaccounted-for ways. Of course, from the final result, it seemed that(assuming the scientifically-accepted value for J/cal is correct) there was very little of thisthat happened regardless. A 4% deviation from accepted value is certainly acceptable for thetype of equipment used.

In addition to the required lab, we also intentionally wet the wheel and attempted to performthe same experiment. First, we noted that there were a greater number of loops (numberof times to string the cord around the cylinder) required in order to keep the bucket off ofthe ground. This is likely due to the fact that there was decreased friction-per-loop, so moreloops were required to maintain the same amount of torque. Also, we began the experiment;however, we noticed that it was taking a significantly greater amount of time to incur thesame change in temperature. After some time of this, we halted the experiment, citing thefact that much heat had already been lost to the environment; therefore, our data wouldlikely not be correct. Later, we determined this greater discrepency was likely due to thefact that the water was absorbing much of the heat imparted to the cylinder.

In the future, if it were possible to measure the J/cal value using different materials, thatmight enhance the experimental experience. Further, using different materials in place of thenylon cord, and different mass weights might also provide for a more in-depth experiment.

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