Expt 3 Determination of "g"

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Experiment 3

Determination of "g"

Objective:
To determine g, the acceleration due to gravity at the Earth's surface.

Introduction

Under everyday conditions, one finds that the presence of friction makes it difficult to test the laws of Newtonian mechanics. The airtrack is a device that eliminates most of the friction between a long track and riders that hover on a cushion of air (similar to a hovercraft).

The airtracks and riders must be handled carefully as any damage occurring to them will increase friction and lead to poor results. Take special care to keep all surfaces clean and to not drop the airtrack.

Figure 1

Theory

For a frictionless inclined plane as shown in Fig. 1, the acceleration of the rider is given by a = F/m = mgsinq/m or
a =	g sinq	(1)
=	gh / L	(2)
From simple kinematics, with an initial velocity of zero one finds that the displacement at time t is given by
s =	at² /2	(3)
=	½ (gh/L) t² by substituting from Equation (2)	(4)
Now by graphing s versus t² with displacement along the y-axis, one obtains a straight line through the origin. The slope of this straight line will be gh/(2L). To see this, compare equation (4) with y = mx + b, where x is t²

Experimental Procedure

1) Level the airtrack with the levelling screw until a rider placed on the track does not tend to move in either direction. Measure L (note, L is not the length of the airtrack but rather the length of the baseline that with h sets the angle q - see figure1). As with all measurements in the lab, be sure to include uncertainties.

Technique A: Distance traveled by rider varied (incline constant)

2) Select a block (one of the thicker blocks) which you will use to incline the air track and measure its thickness h with a vernier caliper. Determine what the vernier caliper reads for zero and record in your data any offset (any offset should be accounted for in your results). Place the block under the air track end with the single foot to raise it by height h.

3) Do some rough measurements to determine the time it takes for the rider to travel various distances. Note the smallest and largest of the displacement and time measurements that the experiment will yield. With this information prepare a data section with a rough graph of displacement versus time. Also prepare a data table to record 5 timing trials for each travel distance selected.

4) Select a displacement s and determine the rider travel time to traverse this distance. The blower must be left on as the rider is released (turning on the blower to start the rider does not work - why?). Record the measurement both in a data table and on the rough graph. Repeat this time measurement for at least 5 trials (to determine the timing variations from trial to trial) and record each in the data table and on the rough graph .

5) Repeat for seven different displacements using the same height h. (be sure to repeat each measurement 5 times.)

Technique B: Angle of incline varied (travel distance constant)

6) Select one fixed distance for the rider to travel. This travel distance is kept constant while the incline angle is changed by using blocks of different thickness h.

7) Do some rough measurements to determine the time it take for the rider to travel the fixed distance for various angles of the incline. Note the smallest and largest of the block heights and time measurements that the experiment will yield. With this information, prepare a data section with a rough graph of block height versus time. Prepare a data table to record 5 timing trials for each block height used.

8) Select one of the blocks to incline the air track and measure its thickness h with a vernier caliper. Determine the time taken by the rider to travel the selected distance with the block under the airtrack foot. Repeat and record this measurement 5 times (with the same travel distance and block height).

9) Repeat for seven different angle of incline (block heights) using the same rider displacement. (Be sure to repeat each measurement 5 times.)

Results

Technique A: Distance traveled by rider varied

For each displacement s, determine the mean travel time and the mean deviation. The mean deviation represents the uncertainty in your timings.
Plot a graph of t² versus s with displacement along the x axis. Include error bars on your graph. Determine the slope of your graph. Give consideration to the origin (Is it a data point? Should it be included in your straight line fit?).
Determine how the slope of your graph slope is related to the experiment’s equations by following the analysis presented in the theory section (notice the theory section considers the case of plotting displacement along the y-axis). From this, determine g.
Compare your result with the accepted value for g of 9.81 m/s² using percent difference.

Technique B: Angle of incline varied

For each height h, determine the mean travel time and the mean deviation. The mean deviation represents the uncertainty in your timings.
Plot a graph of 1/t² versus h with height along the x axis. Include error bars on your graph. Determine the slope of your graph. Give consideration to the origin (Is it a data point? Should it be included in your straight line fit?).
Determine how the slope of your graph is relate to the experiment’s equations by following the analysis presented in the theory section (notice the theory section considers the case of plotting displacement along the y-axis). From this, determine g.
Compare your result with the accepted value for g of 9.81 m/s² using percent difference.

Further

Airtrack with Friction

The forces acting on the rider are show in the figure. F_n is a normal force that the airtrack exerts on the rider. The net force on the rider in the direction perpendicular to the airtrack is zero, hence;
F_n =	mg cos q	(a1)
Frictional force is given by;
f =	m F_n where m is the coefficient of friction.
=	m mg cos q	(a2)
The net force on the rider in the direction parallel to the track is;
ma =	mg sin q - f
=	mg sin q - m mg cos q	(a3)

Now by using a = 2s/t² and sin q = h/L one has;
2s/t² =	(gh/L) - m g cos q	(a4)
Solving for h gives;
h =	(2sL/g)t^-2 + m L cos q	(a5)
This shows that if h is plotted versus t^-2, the result will be a straight line with slope = (2sL/g) and intercept = m L cos q Therefore, friction only affects the intercept and not the slope.

Notes