Acceleration of Gravity on an Inclined Plane 

Purpose:

1.  To find the acceleration of gravity by studying the motion of a cart on an incline.
2.  To gain further experience using the computer for data collection and analysis. 

Equipment Needed:

• 1 windows based computer with Logger Pro software
• 1 motion detector
• 1 ballistic cart
• aluminum track
• 1 wood blocks
• 1 meter stick
• 1 small carpenter level  

Introduction:
     In this laboratory you will use the computer to collect position (x) vs time (t) data for a cart accelerating on an inclined track.  By comparing the acceleration of  the cart when moving up and down the track, the effect of friction can be eliminated and the acceleration due to the effect of gravity alone can be found.  Since the force of friction acts with the force of gravity when the cart is going up the track and against the force of gravity when the cart is going down the track, we can average the slightly increased acceleration (when going up) with the slightly decreased acceleration (when going down) to obtain an acceleration that depends only on the force of gravity.  If we call g the acceleration due to gravity when an object is in free fall, then the component of this acceleration along the track is g sinθ where θ is the angle of incline for the track.Because the force of friction acts with the motion of the cart on the way up ramp, and acts against the motion on the way down the ramp, the average of the two accelerations will be taken with the following ratio:

gsin(A)= (a₁+a₂)/2

where g= gravity, A= angle of incline, a₁= acceleration up incline, and a₂= acceleration down incline.

In this lab we will measure acceleration by looking at the slope of the v vs t curve for the cart.

Procedure:
(from lab)

1. Connect labpro to computer and motion detector to DIG/SONIC2 port on labpro. Turn on the computer and load the logger pro software.  Set up the logger pro software by opening the "graphlab" file in the mechanics folder.
2. Set up the Track and slightly increase the incline by putting a wooden block under it. Make sure the track is leveled. Once you have done this, Determine the inclination angle by using the method on FIGURE #1.
3. Place the motion detector at the upper end of the track facing the lower end.  The Ballistic cart should start at the lower end of the track.  Now gently push the cart towards the motion detector.  Make sure the cart does not reach 50 cm close to the motion detector ( it will prevent motion detector from detecting it).
4. Start the data collector a few seconds before you push the cart. As the cart leaves your hand watch the v vs t graph and the x vs t. The v vs t graph should form a parabolic section. If  not repeat the process
5. Once the Graphs are complete. You should now figure out the Acceleration using both graphs. For the Position vs time graph, Use the "CURVE FIT" option in the logger pro program.  For the Velocity vs time graph, use the Linear fit instead. Now use trig to find the y component of the acceleration vector, label acceleration as a1 in x vs t, and a2 for v vs t. 
6. Repeat steps 4 and 5 two more times with the same incline. (the Average of the values should be close to 9.8 m/s^2)
7. Repeat the Experiment with a different incline.

We set up the aluminum track at an incline with the wooden block, and then carefully leveled it out. Then, we measured the change in height of the two sides, to the horizontal length of the inclined ramp. The Calculations of the ramp's angle are as followed: 

Tan^-1((y₂-y₁)/(x₂-x₁))=A

Tan^-1((12.9cm-6.7cm)/(228.2cm))=A

A=1.56^o

The Set-up for our Experiment

Once we set up, three trial runs were completed first for the incline at the angle of 1.56 degrees, and then we conducted three trails for incline set to an angle of 3.6 degrees.

*Question: What type of curve will we expect to see for the x vs. t graphs and v vs. t graphs? 

     We expected that the graph of x vs. t would be a parabolic with the left side having a more drastic change in slope than the right side, since friction will be acting with the motion of the cart on the way up and against it on the way down. We also expected the graph of v vs. t to be linear with different slopes for the motion up and the motion down the ramp, since friction would affect the acceleration of the cart.

Results:

Position vs. Time Graph

Velocity vs. Time Graph w/ Line of Best Fit

A linear fit was applied to the negative velocity portion of the v vs. t graph for a₁, as well as a linear fit to the positive velocity of the v vs. t graph for a₂ in order to get values for a₁ and a₂.

Since the derivative of velocity is acceleration, we used the slopes of our best fit lines in order to obtain out accelerations.

The following is our data for each of the trial runs:

Conclusion:

Once the data had been collected, verification of our numbers compared to the accepted value of acceleration due to gravity was needed. Since the force of friction acts with the force of gravity on the way up the ramp and against it on the way down, the average of the two accelerations will be taken to equate our experimental value of acceleration due to gravity. The formula used to calculate G_exp was:

G_expsin(1.56)=(a₁+a₂)/2

-Trial one of 1.56 degree incline

G_expsin(1.56)=(0.33 m/s²+0.18 m/s²)/2

G_exp=9.4m/s²

In this lab we were able to determine the effect gravity has on objects that are on inclined planes pretty accurately. The percentage differences we calculated in this lab seemed to match well with the accepted 9.8 m/s². Our most inconsistent value was 9.4 m/s² with a 4.1% difference. Potential sources of error in this lab could have included various factors such as the friction between the car and the track, and performing the linear fit to the velocity vs. time graphs in different domains.