GOAL

To calculate the acceleration of an object rolling down an inclined plane.

INTRODUCTION

Acceleration is the change in the velocity of an object. Velocity is a vector quantity with both direction and magnitude. Acceleration is also a vector quantity with both direction and magnitude. If the speed of an object is changed, that object has accelerated either positively or negatively depending on whether it increased or decreased in speed. Another way to accelerate an object is to change its direction of movement. This means that a car going around a corner is undergoing acceleration because its velocity in terms of direction is changing even if the car's speed, as seen on the speedometer, is constant.

As discussed above, an object falling under the influence of gravity accelerates. From your studies, you can recall the key kinematic equations for the uniformly accelerated motion of an object starting from rest, where v = velocity, a = acceleration, and d = distance.

Using these equations it is then possible to solve for the unknown variables.

In this lab experiment, we will measure the time it takes for a marble to roll down an inclined plane. From the experimental data, we will then estimate the value of gravity (g).

PROCEDURE

Set up a ramp as shown in Figure 1 that will be our inclined plane. Depending on the distance of your ramp, mark intervals of 40 to 60 cm. For example, in Figure 1, the marks are at 50 cm. Ideally, the marks should be as separated as possible in order to obtain a better reading. The height of the ramp is also going to play a role in the accuracy of your measurements. The steeper the slope is, the ball will run down faster and will make measurements less accurate. Before starting with the measurements, you may want to run some trials in different conditions.

Figure 2: Experimental setup

Figure 2 shows the setup used by a student who did this lab some time ago. In this case, the marks are spaced 40 cm. This experiment works best with gentle angles of elevation. You may get better result if you use a smaller slope than Figure 2 shows. The recommended angles should be around 5 degrees, 10 degrees, and 15 degrees.You also need to record the angle of the inclined plane making a plumb line with the protractor as shown in Figure 3.The angle of elevation from this protractor is around 12⁰.

Figure 3: Measuring the angle of an inclined plane

Measuring Time

Our procedure will consist of dropping the marble at the top of the inclined plane and measuring with the stopwatch the time it takes to reach each one of the markings, starting with the closest to the point where you release the marble. Using the example shown in Figure 1, we will first measure the time it takes for the ball to go from 0 cm to 50 cm. As we learned in Laboratory Experiment #1, it is always good practice to repeat the measurements several times (in our case 5 times) to reduce errors.

When we have taken and recorded the 5 trials for this first measurement, we will proceed by repeating the experiment but in this case measuring the time that it takes for the ball to reach the second mark (in the example of Figure 1, it will be from 0 to 100 cm).

Equations used for this experiment

The known variable in this experiment is the distance between the marks. The measured variable in this experiment is the time it takes for the ball to travel a specified distance.

For a body undergoing accelerated movement, the equations that we will use are:

INITIAL PARAMETERS

QUESTION 1:

What is the distance between two marks in your inclined plane?

QUESTION 2:

What is the angle of your inclined plane with respect the horizontal? (0 would mean the inclined plane totally horizontal, so the ball would not move; 90 would mean the board totally vertical)

Write down these two values in the table we will use to record all our measurements. It will also be used at the end of the lab.

EXPERIMENTAL RESULTS

When we are finished with this experiment, we will have all the data in our table complete. We will, however, complete the table step by step.

Measurements from release point to 1^st marking

At this point, we will measure the time that it takes from the ball to travel from the release point to the first marking. Insert this distance in the appropriate column in the table below. Note that the distance should be the same for all the 5 trials in this first step. For example, if the distance between markings is 50 cm, d should be equal to 50 cm and so on.

Because we are here only concerned with the time between the release point and the 1^st marking, we can ignore the shaded section of the table.

Repeat these measurements 5 times, recording the time it took for the ball to reach the first marking.

Using the equations fromStep 3, calculate the velocity and the acceleration. Calculate and record the Average and Standard Deviation of time, velocity and acceleration.

Measurements from release point to 2^nd marking

Repeat the previous step, now measuring the time that it takes for the ball to travel from the release point to the second marking. Complete the unshaded part of the table below. Keep in mind that d₂ should be the double of d₁.

Once again, measure the time and calculate velocity and acceleration. Calculate also the Average and Standard Deviation of your measurements.

Measurements from release point to 3rd marking

Repeat the same procedure, now measuring the time it takes for the ball to reach the 3^rd marking. Complete the time below, taking into account that the values for the first and second markings are the same as you calculated in Steps 1 and 2.

If your inclined plane allows it, you may want to repeat the process for a 4^th marking.

Completion of the table with time, velocity and acceleration

At this point, you should have all the data in the table completed.

ANALYSIS OF RESULTS

QUESTION 3

Newton's first law says a body at rest will remain at rest unless acted upon by an outside force, and a body in motion will continue in motion at the same speed and in the same direction unless acted upon by an outside force. What forces were acting on the marble as it traveled down the ramp?

QUESTION 4

Did the measured acceleration was about the same for the three (or four) sections of the experiment (Release point to 1^st marking, to 2^nd marking, etc) ?

QUESTION 5

Do you expect this acceleration to be constant or different for the three (or four) sections of this experiment? Explain your reasoning.

QUESTION 6

By looking at the Standard Deviation results for the calculated acceleration, which section of this experiment is the more precise? Explain your reasoning.

QUESTION 7

What was the average value of acceleration for the most precise section of this experiment?

QUESTION 8

Intuitively, we can understand that the velocity (and therefore the acceleration) of the ball will increase as we increase the angle of the inclined plane. We can make the assumption that the acceleration of the ball is equal to:

Expected acceleration = (5/7)gsin() where g=9.8 m/s²

The angle of the inclined plane is the value that you measured in Question 2 and transcribed into the table. Using the measured value of the angle of the inclined plane, calculate the expected value of the acceleration.

QUESTION 9

Calculate the relative error between the measured value of acceleration (from Question 7) and the expected value of acceleration (from Question 8).

QUESTION 10

What do you think are the elements that may contribute to increasing this error? How would you solve them?

LABORATORY REPORT

make a laboratory report using Word or another word processing software that contains at least these elements:

• Introduction: what is the purpose of this laboratory experiment?
• Description of how you performed the different parts of this exercise. At the very least, this part should contain the answers to questions 1-10 above. You should also include procedures, etc. Adding pictures to your lab report showing your work as needed always increases the value of the report.
• Conclusion: What area(s) you had difficulties with in the lab; what you learned in this experiment; how it applies to your coursework and any other comments.