Acceleration of Gravity on an Inclined Plane Lab

The purpose of this lab is to find the acceleration of gravity by observing the motion of a cart on an inclined plane. In this lab we will be using:

• Logger Pro Software
• motion detector
• ballistic cart
• aluminum track
• woodblocks
• meterstick
• small carpenter level

Intro:
We will be using Logger Pro to collect the position vs. time data for the ballistic cart as it accelerates along the aluminum track. We are not going to include the effect of friction because friction will act with the force of gravity as the cart moves up the track and against gravity while it travels down the track. With this, we can use the average acceleration of the cart moving up and down the track with:

(a1+a2) / 2

to determine the average acceleration of gravity as the friction will slightly increase acceleration of gravity while the cart moves up the track and decrease slightly as the cart moves down the track. The average acceleration of the cart is equal to the acceleration due to gravity, g, times the sine of the angle of the track.

We can use this picture to find the acceleration of an object on an incline plane. We have θ because that is equal to the angle of the plane with the table. We also have the acceleration of the freefall and the acceleration of the object parallel with the inclined plane. If we look at them as vector components we see that we can add | a_parallel |+| a_{perpendicular} | = a_freefall

We see that g is the same as the a_freefall and can come up with the equation:

sinθ = opposite / hypotenuse

sinθ = | a_parallel | / g

Knowing this we can substitute in the average acceleration of the cart going up the track and going down the track:

| a_parallel | = (a1+a2) / 2

and we can arrange the previous equation:

sinθ = [(a1+a2)/2] / g

g sinθ = (a1+a2) / 2

and in order to find g, the acceleration due to gravity, we divide both sides by sinθ:

g = [(a1+a2)/2] / sinθ

where a1 and a2 are the accelerations of the cart, θ is the angle of the track and the table, and g is acceleration due to gravity.

Procedure:

First, we hooked up the Logger Pro software and opened up the graphlab file which is where we recorded all of our data. We then set up the aluminum track with one side having 1 wood block underneath its feet which raised up the track to give it an incline. Using the bubble level, we made sure the width of the track was level to make sure there wouldn't be any other effects of friction that could taint our results. After that we set the motion detector at the top of the track having it face the bottom part of track where the cart will be coming from that way we could measure its position, velocity, and acceleration.



Once having the lab set up we then solved to find the angle that θ would be by doing some simple trigonometry. We measured out 50 cm on the length of track and used the level to make marks on the table vertically below where the 0 cm were on the track and where 50 cm were on the track. We found that length to be 49.95 cm. Now, in order to find the measure of angle θ, we use the equation:

cosθ = adjacent side of the angle / hypotenuse

which, when using our measurments, comes to be:

cosθ = (49.95cm/50cm)

(arccos(cosθ)) = arccos(49.95/50)

θ = 2.56°

Before starting we opened up two graphs in Logger Pro, one of the position vs. time and one of the velocity vs. time to compare the two as they are occuring. We did a few practice trials to get ourselves comfortable trying the system out and making sure everything was going to work smooth for the lab. Once doing that, we did 3 trials of pushing the cart up the track and letting it come back down with the effects of gravity.

Results:

When doing this experiment and collecting data graphs, we expect to see the Position vs. Time graph to be a positive parabola because as time goes from 0 to

We did two linear fits on the velocity vs. time graph to obtain our a1 and a2. When finding a1, we did a linear fit when the velocity was moving at a constant rate less than 0 and for a2, we did a linear fit when the velocity was moving at a constant rate greater than 0. Those are going to make up our total average velocity that we used in the equation.

Trial 2:
We kept the set up the same but then we raised the incline to become steeper by adding another wood block underneath it. We measured out 50 cm on the track again and also used the level to make marks on the table vertically below the 0 cm and 50 cm mark to measure out the the length of the base of the "triangle" that the track and table make and that came to be 49.75 cm.

cosθ = (50cm/49.75cm)

arccos(cosθ) = arccos(50cm/49.75cm)

θ = 5.73°

Results:

Again, we took a linear fit of the negative velocity (a1) and the positive velocity (a2) that way we could take an average of the two for one average acceleration.

Conclusion:

In the first set of trials when we set the angle of the incline, θ, at 2.56°, our g-experimental value was 7.09 m/s^2. Our accepted value of g, g-accepted, is 9.80 m/s^2. With these numbers we can calculate a percent error that occurred  with the equation:

% error = |[(measured-actual)/actual]| X 100

=|[(7.09m/s^2 - 9.80m/s^2) / 9.8m/s^2]| X 100

=27.6%



Our second set of trials which ran on the elevated track were set at an angle θ = 5.73°. The average g-experimental value was 7.24 m/s^2. We can calculate the percent error:

|[(7.24m/s^2 - 9.8m/s^2) / 9.80m/s^2]| X 100

=26.1%

When we look at the position vs. time graphs of the cart in motion, we expect to see a positive parabola because as it starts traveling up the track it moves closer to the motion detector at a decreasing velocity which gives it a curve and not a line because the velocity is changing at a rate and not a constant. Once it gets to its highest point on the track, which is the closest point to the motion detector, the acceleration due to gravity takes over the cart and cart's position slowly starts to increase at a rate away from the motion detector which gives it an increasing curve. The graph would never have a negative position either because the cart approaches the motion detector which would have 0 position and once gravity takes over the carts velocity, the cart's position increases.

The velocity vs. time graph would be different though. It would be a line going in the positive direction that starts negative, crosses the x-axis which means it has velocity of 0 m/s at one point, and then becomes positive. This happens because the carts position is decreasing until it reaches its highest point as time increases which gives the cart negative velocity. The graph for velocity increases though since g is moving in the direction opposite direction until velocity reaches 0 m/s, which is the carts closest position to the motion detector. Then the velocity becomes positive because it starts increasing a changing rate as it travels in the same direction as acceleration and moves away from the motion detector increasing its position.

In this lab we observed the acceleration due to gravity of a cart as it travels up and inclined aluminum track as the velocity decreased and reached 0 m/s once the cart made it to its closest point to the motion detector and increased as the position increased. I learned how to calculate the acceleration due to gravity on an incline plane and showed the proof on how to find your experimental g. I also learned how to read and interpret the graphs such as the describing why the object has a parabolic position graph while it moves on the inclined plane and why the velocity graph is linear since the acceleration is constant. We had a large percent error when working with this lab but one major thing that could have contributed to the large error was that we could have taken a larger measurement when we were trying to find our angle of θ. We should have taken a larger measurerment of the track to find our angle because we could have gotten a more accurate measurement because there would have been a larger difference in our hypotenuse and the base leg of the triangle. I also think we had a lower percent error when we raised the angle of θ to be a bit higher also because the acceleration due to gravity could have more effect on the object since the plane is steeper.