Experiment 8
Conservation of Energy
Objective:
The objective is to verify the law of conservation of mechanical energy with frictional losses present.
Equipment:
An incline, a wooden box, a pulley with attachments,1meter piece of string, a mass scale, graphing paper, a set of weights, a weight hanger, and a scientific calculator
Theory:
For an inclined plane as shown in Fig. 2, the P.E. at position “2” exceeds the P.E. at position “1” by the amount M_{1}gh where M_{1} is the total mass( box + 300 grams) placed on the incline and h is the change in the elevation of the box.
M_{1}gh is the work done on the box in going from “1” to “2” if no friction is involved ; however, there is friction between the box and the incline and more work must be done to elevate the box from position “1” to position “2.” Mass M_{2} provides the necessary work or energy. If M_{2} is great enough to pull the box up the hill while M_{2} itself moves down under the influence of gravity, and the process occurs at a constant velocity, we may then accept that
Energy loss by M_{2} = Energy gain by M_{1 }+ Energy consumed by friction.
This can mathematically be written as:
M_{2}gS = M_{1}gh + W_{f} (1)
Where h and W_{f} as well as μ, or the coefficient of friction, must be determined.
Calculation of h: To determine h, refer to the right triangle shown in Fig. 3:
Fig. 3
Measurement of μ:
To measure μ, tilt the incline until the box slides down at a constant velocity. Measure that particular angle, θ_{k, } at which constantvelocity sliding occurs and use it to find μ_{k} from the formula (For proof refer to Experiment 5):
μ_{k} = tan θ_{k}
Calculation of W_{f } :
To calculate the work or energy consumed by friction, we need to determine the force of kinetic friction, F_{k}, and then multiply it by S, the displacement. Let's denote the Work done by friction by the upper case w as W_{f}. Lower case w is used to denote the weight of the box or M_{1}g. F_{k} is given by F_{k} = μ_{k} N, where N can be found from the figure on the right. N = w cos θ. We may write: W_{f} = F_{k} S = μ_{k} N S = [μ_{k} wcosθ] S or, W_{f} = μ_{k}M_{1 }g (cos θ) S.

(w), the weight, can be replaced with F_{┴} and F_{║}. F_{┴} is a measure of how strongly the box presses against the incline. Note that F_{┴} must be equal to N for equilibrium in the normal to the surface direction. Since F_{┴ } = wcosθ ; therefore, N = wcosθ as well. 
Finally, Eqn. 1 becomes:
M_{2}g S = M_{1}g S (sin θ ) + μ_{k} M_{1 }g (cos θ) S. ( 2 )
Eqn. 2 is the energy balance equation. If SI units are used, both sides will have units of Joules. Although S can be cancelled out from both sides, but we deliberately keep it in there and let its value be 0.300m, for example. This will result in units of Joules if M_{1} and M_{2} are converted from grams to kg. We will then be comparing the number of joules we get on the left side with the number of joules on the right side of this equation.
Note that since the experiment is to be performed at a constant velocity, the initial and final velocities will be equal and therefore there will not be any change in the K.E. of the system. That is why no mention is made of K.E. in Equation (2).
Procedure:
Side Experiment:
With the box (empty or with some weights in it) on the incline, tilt the incline to θ_{k} where motion at constant velocity can be observed. Repeat this procedure 3 times for the empty box and 3 times with 300 grams of mass in it and each time determine a value for μ_{k } by using the appropriate equation. Find an average value for μ_{k} based on the 6 angles you measure.
μ_{k} = tan θ_{k}
The Main Experiment:
Set up the incline for an angle θ between 20 to 30 degrees. With 300 grams in the box, find M_{2} such that the box slides up the incline at a slow constant velocity. Assuming a displacement of say (S = 0.300m or so), use Equation (2) and calculate the values of each side of the equation, separately, and see how close in value they turn out to be.
Calculate a %difference between the right and left values by using the following equation:
Repeat the above steps for M_{1} = 400 grams and the same θ.
Change θ and repeat the experiment once for M_{1} = 300 grams and once for M_{1} = 400 grams.
Record the results in an appropriate table.
In each case calculate the % difference.
Data:
Given: g = 9.81 m/s^{2}, 20.0° < θ < 30.0° , M_{1} = 300 gr and 400 gr
Measured: Trial 1: Trial 2: Trial 3: Trial 4:
Comparison of the results:
Show the percent difference formula used as well as the calculated values of percent difference for all cases.
Conclusion:
State your conclusions of the experiment.
Discussion:
Provide a discussion if necessary.