Title How Does A Moving Cart

Title: How does a moving cart rolling down a slope?
Objectives:
Adapt the traditional vertical and horizontal axes to a coordinate system aligned with a slope.
Determine the acceleration of a cart that rolling down from a frictionless track (our assumption) by calculating theoretically and measuring experimentally.
Compare the experimental and expected values of acceleration.
Show that the acceleration of a cart moving down a slope (from frictionless track) is dependent on the angle of the slope.

Introduction
If you have been on a roller coaster, you experienced a large, downhill acceleration after reaching the top of the first hill. Compare this acceleration to the acceleration you might experience if you coast on a bike down a slightly sloped roadway. The acceleration on the sloped roadway is probably less than the acceleration on the roller coaster. What causes objects to experience different accelerations down different slopes? You have already learned that the force of gravity causes things to roll or slide down a slope. Figure A shows a motion diagram of a box that is sliding down a hill with an acceleration a. From the diagram, you can see that there is simultaneous acceleration in both the positive x- (horizontal) and negative y- (downward) directions Figure A

By turning the coordinate system so that the x-axis is parallel to the slope, it is easy to resolve the downward force due to gravity into a force perpendicular to the board and a force parallel to the slope.

Diagram 2
This parallel force causes, Fg Sin ɵ the block to accelerate down the slope. The free body diagram in Figure B shows the forces that are acting on the block. Notice that the coordinate system is angled with the x-axis in the direction along the slope. The component of Fg, the weight of the box, lying perpendicular to the plane can be shown to be in equilibrium with the normal force of the surface on the box, FN, because there is no acceleration in the y-direction. If is the angle of the slope, then Newton's second law for the y-direction shows that:

The motion diagram in Figure A shows that there is acceleration in the x-direction of
Figure B. So, Newton's second law in the x-direction along the slope shows that:

This equation can be solved for a (acceleration of block down the slope) to provide a way to determine the acceleration of the block.
Sinc Fg=mblock g, you can substitute into the equation (2) above to get (mblockg) Sin = mblock­ a, which reduces to

Equation (3) enable you to calculate the acceleration of an object rolling or sliding down along a slope

From the viewpoint of kinematics, we can also use certain equations to determine acceleration of an object. In our situation, the acceleration a of object is constant. Constant acceleration means a constant change of velocity, e.g the velocity increases uniformly as the object moves. This could mean a constant change of speed, a constant change of direction (such as uniform circular motion), or a combination of both. Object that falls under the gravity is an example of motion with constant acceleration (under certain condition). When this object falls, it is subjected to the acceleration of gravity. Near, the earth the acceleration can be considered as constant and its value is 9.8ms-2.

For an object that moves in a straight line between two positions (let says only in x- direction); x1 at t1 and position x2 at time t2, the average rate of change in its position is given as:

The rate of change of its position is known as the average velocity. Similarly, the rate of change of the velocity is known as the acceleration, and it is given as:

The equations of motion described the position as a function of time x(t), the velocity as a function of time v(t), and the acceleration as a function of time a(t). Hence, the whole motion of the object (if a is constant) can be explained by these equations.
These equations can be represented graphically. The graphical representation of constant acceleration involves many fundamental concepts of kinematics. The slope of a plot of velocity versus time for an object is the acceleration of the object. The ratio of the units along the vertical and horizontal axes of a graph of velocity and time give the units for the object’s acceleration.
In this lab, you have to design an experiment with an aimed to achieve the objective above. You will use two equations, and to archive your objective. At the end of the experiment, you will calculate accelerations based on using both equation, and compare the calculated using both techniques.

Apparatus
Frictionless track
Cart /pascar
Meter Stick

Procedure
Set up the track as shown in Figure 1. Elevate the end of the track until the cart will move downward being exerted by an external force.

Release the cart from rest and use the stopwatch to time how long it takes the cart to reach the end stop. The person who releases the cart should also operate the stopwatch. Repeat this measurement 5 times (with different people doing the timing). Set the distance travelled, d by the cart. Record all the values in Table 1. You need to take measurement of at least 10 angles.

Result:

Table 1
Distance d (m) : 2.000
Data set
Slope angle (θ)
Time(s)

T1
T2
T3
T4
T5
1
3.44 ± 0.03
3.28 ± 0.01
3.47 0.01
3.21 ± 0.01
3.46 ± 0.01
3.44 ± 0.01
2
4.50 ± 0.03
3.22 ± 0.01
2.97 ± 0.01
2.88 ± 0.01
2.91 ± 0.01
2.91± 0.01
3
4.88 ± 0.03
2.91± 0.01
2.84 ± 0.01
2.85 ± 0.01
2.84 ± 0.01
2.87± 0.01
4
6.23 ± 0.03
2.28 ± 0.01
2.22 ± 0.01
2.34 ± 0.01
2.13 ± 0.01
2.25 ± 0.01
5
6.55 ± 0.03
2.47 ± 0.01
2.37 ± 0.01
1.94± 0.01
2.31 ± 0.01
2.47± 0.01
6
8.34 ± 0.03
2.03 ± 0.01
2.03 ± 0.01
2.00 ± 0.01
2.00 ± 0.01
1.94 ± 0.01
7
8.63 ± 0.03
1.97 ± 0.01
1.72± 0.01
2.09 ± 0.01
1.91 ± 0.01
1.97 ± 0.01
8
9.06 ± 0.03
1.93 ± 0.01
1.97 ± 0.01
1.91 ± 0.01
1.93 ± 0.01
1.94 ± 0.01
9
11.40 ± 0.03
1.85 ± 0.01
1.65 ± 0.01
1.84 ± 0.01
1.69 ± 0.01
1.69 ± 0.01
10
14.01 ± 0.03
1.66 ± 0.01
1.53 ± 0.01
1.63 ± 0.01
1.75 ± 0.01
1.56 ± 0.01

Analysis and conclusion
1. Calculate the average time and the square of the average time for each data set and record the values in Table 2. Show calculations in the space below.
Table
Data set
Slope angle (θ)
Average time
(s)
Average time
(s2)
Experimental acceleration(ms-2)
Expected acceleration(ms-2)
1
3.52 ± 0.03
3.37 ± 0.05
11.36 ±0.34

0.60
2
4.50 ± 0.03
2.91 ± 0.05
8.47 ±0.29

0.77
3
4.88 ± 0.03
2.86 ± 0.05
8.18 ± 0.29

0.83
4
6.23 ± 0.03
2.24 ± 0.05
5.02 ± 0.22

1.06
5
6.550 ± 0.03
2.23 ± 0.05
5.00 ± 0.22

1.12
6
8.34 ± 0.03
2.00 ± 0.05
4.00 ± 0.20

1.42
7
8.63 ± 0.03
1.97 ± 0.05
3.88 ± 0.20

1.47
8
9.06 ± 0.03
1.94 ± 0.05
3.76 ± 0.20

1.54
9
11.39 ± 0.03
1.74 ± 0.05
3.03 ± 0.17

1.94
10
14.09 ± 0.03
1.63 ± 0.05
2.66 ± 0.16

2.39

2. Calculate the acceleration for each slope angle using your experimental data and record the values in Table 2. Show calculations in the space below. Use the following equation to calculate experimental acceleration.

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10.

3. Use your experimental results to make an association between the slope of the air track and the acceleration of the glider.

The larger the angle of the slope the faster the acceleration of the glider.

4. Calculate the acceleration that you would expect for each slope using Newton's second law in two dimensions.

Record these values as the expected acceleration for each data set in Table 2. Show calculations in the space below.
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10.

5. Compare the values of experimental acceleration and expected acceleration for each data set. Do your experimental data support the predictions based on Newton's second law?

No, because there some factors occur when do this experiment. The theatrically is calculated based on the no factor of the data.

Extension and Application
1. Imagine that you conduct this same experiment with a cart rolling on a non-frictionless surface. What results would you expect and why? Explain using the free-body diagram.

This would be the same as the second newton law result of acceleration. Because there are no certain factor like friction do manipulated the data.

2. Plot acceleration versus sin θ. Draw the best-fit straight line and calculate its slope.