Determining The Relationship Between Acceleration And Constant Mass

Purpose The purpose of this experiment was to (Using Matlab) determine the relationship between acceleration and applied force when the total system mass is held constant (Constant Mass), and the relationship between acceleration and total mass when the applied force is held constant (Constant Force).

Apparatus
Figure 1

Materials/Apparatus Parts
• Cart and Cart ramp
• Ultra Pulley + Photogate
• C Clamp to mount Smart Pulley/Photogate
• Vernier Lab Pro
• Logger Pro
• Mass Set and 1 Hanger
• Block Masses
• C Clamp

Procedure
Constant Mass
1. We made sure that the apparatus is set up as shown in Figure 1.
2. Filled out data table #1 using several techniques in logger pro.
3. During the first trial we placed a few small weights on the cart
and ran a trial using the hanger with no excess weight on it. (Record Mass of System [Cart + Weights + Hanger])
4. Open up Logger Pro
4.1. Find file called “Modified Atwood,” and open.
4.2. Click Experiment Pull Down
4.3. Select Set Up Sensors and then Show All Interfaces
4.4. In the Dig/Sonic 1 channel click on the photo of the photogate.
4.5. Make sure that Motion Timing is checked.
4.6. Click Set Distance or Length option.
4.7. Make sure Ultra Pulley (10 Spoke) In Groove is selected.
4.8. Set length to 0.015 m.
5. During the first trial we pulled the cart back ~50 cm, pressed the green play button in logger pro (on screen), released the cart, then stopped data collection (stop button).
6. Logger Pro generated a velocity vs time graph. We highlighted the part that had a seemingly straight line, zoomed in and clicked the Linear Fit button to get the slope of the line. We recored this as acceleration for trial 1 using that weight on hanger (0).
7. Performed steps 5 and 6 two more times.
8. During the second trial we moved one of the weights on the cart to the hanger and recorded the new weight of the hanger. We then repeated steps 5 and 6 using this new hanger weight.
9. Performed steps 5 through 8, four more times to get a total of 6 different data sets.

Constant Force
1.
We reassembled as necessary making sure that the apparatus appeared as shown in Figure 1.
2. We used data table 2 to record the many accelerations
3. Put a 100-gram weight on the hanger and left it constant throughout the five data sets.
4. We weighed the cart plus the five 500-gram blocks.
5. We taped down all four of the 500-gram blocks on the cart.
6. During the first trial, the cart had all five 500-gram weights upon it.
7. The cart was pulled ~50 cm back from the stop and released, data was collected using logger pro.
8. We highlighted the part of the graph produced that had the seemingly straight-line slope, zoomed in, and clicked the Linear Fit button to get the slope of the line. This was recorded as acceleration.
9. Steps 7 and 8 were repeated twice more.
10. During the second trial, we removed one of the 500-gram blocks from the cart and subtracted the mass of the block from the mass of the cart plus its former amount of blocks.
11. Steps 7 through 10 were repeated 4 more times, for a total of 5 data sets
Data

Data Table 1: Constant Mass
Total System Mass = 0.630 kg
Hanging Mass (Kg) A1(m/s2) A1(m/s2) A1(m/s2) Mean Acceleration
0.05 0.71 0.72 0.69 0.71
0.06 0.82 0.81 0.78 0.81
0.07 1.0 1.0 1.0
1.0
0.09 1.2 1.1 1.1 1.2
0.1 1.1 1.1 1.1 1.1
0.11 1.2 1.2 1.2 1.2

Data Table 2: Constant Force
Applied Force = 1.47 N
Total Cart Mass (Kg) A1(m/s2) A1(m/s2) A1(m/s2) Mean Acceleration Standard Deviation
2.51 0.50 0.50 0.48 0.49 0.011
2.01 0.56 0.60 0.62 0.59 0.031
1.51 0.80 0.80 0.81 0.80 0.004
1.01 1.2 1.2 1.2 1.2 0.004
0.51 1.9 1.6 2.0 1.8 0.232

Analysis
Force Analysis
Free body diagram of the cart with several weights on it:

Free body diagram for the hanging mass and the masses on it:

The acceleration constraint is: a=a1= -a2
The resulting equation is a= (m2g)/(m1+m2)
This equation means that the acceleration of the cart and hanging mass is dependent on the mass of both and the acceleration due to gravity.
This is derived from Newton’s second law.

Analysis of Constant Mass Experiment
The goal of this subsection is to analyze how the relationship between force and acceleration while the total system mass remained constant. So, on the basis of this graph as applied force increases the acceleration increases.

Log10(a) vs Log10(F) Log10(a)=0.079 + 0.69 (log10(F))
The equation for this graph is a =
1.2F0.69
Standard Deviation Applied Force (N)
0.017 0.49
0.020 0.59
0.018 0.69
0.051 0.88
0.019 0.98
0.060 1.1

Analysis of Constant Force Experiment
The goal of this subsection is to analyze how the relationship between mass and acceleration, while the force acting upon them stayed constant. Therefore, on the basis of the graph produced, as the mass of the system increases the acceleration decreases.

Log10(a) vs Log10(M) Log10(a)=0.039 – 0.83 (log10(M))
The equation for this graph is a = 1.1M(-0.83)
Mean Acceleration Total Cart Mass (Kg)
0.49 2.51
0.59 2.01
0.80 1.51
1.2 1.01
1.8 0.51

Conclusions The over arching purpose or goal of this lab was to apply Newton’s second law. The law suggests that the acceleration of an object is dependent on the mass of the object and also the “net” force acting on the object. As our results validate the acceleration of an object is directly proportional to the net force acting on the object (constant mass). As the net force acting on an object increases the acceleration also increases. Accordingly, acceleration is indirectly proportional to the mass of the object. As the mass of an object increases the acceleration decreases (Net Force acting on object is constant).