The purpose of this lab is to measure the conversion factor between mechanical energy and heat energy.
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Historically, the relationship between heat flow into a material and its resulting temperature change was deduced prior to mankind’s understanding of heat as a form of energy. A unit of heat (the calorie) was invented to quantify heat flow. A calorie is defined as the amount of heat necessary to raise the temperature of one gram of water by one degree Celsius. The equivalence of heat energy and mechanical energy can be deduced by measuring for example the amount of heat created when an object undergoes a known amount of work due to a frictional force. We will use this technique to measure the proportionality constant between the heat unit calorie and the energy unit Joule.
A simple schematic diagram of the apparatus is shown above. The inner brass cup is partly filled with water. The outer brass cup is connected to a crank handle and turned about the axis of rotation shown. The inner cup is stationary. Thus, with the inner cup lowered into the outer cup, there is friction. The work done by the frictional force is converted to heat. You will measure the mechanical work done (in Joules) and the heat generated (in calories) and thus determine the conversion factor between the 2 units.
If you apply a constant torque to a disk with radius R by applying a constant force F tangentially to the disk, the torque is given by . The work W done by turning the disk through an angle is given by . If N turns were made this angle is (N ) radians. Thus the work done in N turns is
In your experiment you don’t turn the disk holding the inner brass cup, but you turn the outer cup with a crank and hold the disk stationary with a string exerting a force, which is measured by a spring balance attached to the string. The frictional force between the two cups does not set the inner cup and disk in motion, but is balanced by the force in the string. Thus the formula (9.1) above is valid for the way you execute the experiment.
The amount of heat input to the system can be analyzed by measuring the change in the temperature of the system. In general, the amount of heat absorbed or released by a single material, which does not undergo a phase change , can be calculated by using the equation
In this lab, we will use the Greek letter to denote ‘change’, not which will be used to denote errors (as we have done for the other labs).
When you begin the experiment (generating heat by friction turning the crank) you will start with water at a temperature a few degrees C below room temperature and end the experiment at a temperature which should be about the same amount above room temperature. This is done so that the heat gain from the environment (the room) when the water temperature is below is roughly canceled by the heat loss when the water temperature is above .
mb = mass of inner brass cup and outer brass cup combined
ms = mass of stirring rod
mth = mass of thermometer
Calculate the work W done by turning the outer brass cup using equation 9.1. Calculate its error according to expression 1.3 and 1.7 in Lab1 from the errors of the effective diameter d of the disk and the force F. Calculate the temperature rise of the system and its absolute error according to expression (1.6) from Lab 1.
In this experiment, the generated heat is absorbed not just by one single material, it is absorbed by 4 different elements: the water, the brass cups, the stirring rod, and the thermometer. In order to calculate the total heat generated in this system, you use equation 8.2 for each of the elements. Assume that the stirring rod is made of aluminum and the thermometer is made of glass. The specific heats c of these materials are provided in the table below.
|Material||Specific Heat c [cal g-1oC-1]|
|aluminum (stir rod)||0.215|
Calculate the heat absorbed by the water, Qw , using equation 9.2. Repeat the calculations for the brass cups, stirring rod, and the thermometer. Record these values on your worksheet. Calculate the total heat Q by taking the sum of the heat absorbed by each of the four elements. You should see that the main contributions to the overall absorbed heat are from the brass cups and the water. The relative errors in the masses of both the brass cups and the water are fairly small, so we can consider only the error in our change of temperature () when calculating the error in Q. Therefore, we can find the absolute error in Q simply by mutliplying it by the relative error in .
Calculate your experimental proportionality constant between the heat unit of the calorie and the energy unit of the Joule by calculating . Calculate the error for this ratio according to equations (1.3) and (1.7) from Lab1. Compare your measured value of with the accepted value of 4.187 J/cal and check whether your measurement is consistent with it within expermental error.