Uniform Acceleration - WFU Physics



Uniform Acceleration©98

Experiment 2

Objective: To study the kinematics of uniformly accelerated motion.

DISCUSSION:

According to the laws of falling bodies, an object sliding down a frictionless, inclined surface should do so at a constant acceleration. That is, its speed should increase in direct proportion to the lapse of time it slides from rest. The equation of motion is

(1)

where v is the speed, a the constant acceleration, and t the lapse of time. This equation defines what is meant by constant acceleration. For an object with an initial speed v0, this relation must be modified to add in the initial speed, that is

(2)

The distance s traveled by an object in a time t with an initial speed v0 and under constant acceleration can be shown to be

(3)

As long as there is no friction, the acceleration of the sliding object depends upon the acceleration of gravity, g (whose local value is 9.80 m/s2), and the angle of incline θ of the surface. This angle is measured from the horizontal --- that is, it is the angle between the horizontal and the surface. The theoretical relationship is

(4)

This equation is remarkable in that theoretically the acceleration must be independent of any characteristic of the body --- size, shape, mass --- none affects the acceleration provided there is no friction.

If the object is a sphere which rolls down the surface, Eqs. (2) and (3) continue to apply, but Eq. (4) fails because not all of the parts of a rolling body move at the same speed. For example, the point of the body momentarily in contact with the surface is at rest, and the point of the body farthest from the contact point moves at twice the speed of the center of the sphere. For a solid sphere of uniform density, Eq. (4) becomes

(5)

This acceleration does depend upon certain properties of the body, such as its shape (spherical), and the distribution of matter (uniform density). However, the radius and mass of the sphere do not affect its acceleration.

The reason the sphere does not accelerate down the incline as rapidly as does a sliding object is that there is some friction between the sphere and the surface. The force of friction on the sphere due to the surface is directed up the plane (to keep the point in contact with the plane from sliding), and the pull of gravity down the plane is opposed in part by this friction.

We shall examine these two cases --- sliding and rolling-- in this experiment. The frictionless surface for sliding is the air track --- see Figure 1. Mounted over the air track are two devices, called photogates, from which your computer will record the measured data, corresponding to times for several events. Each photogate will measure two different times, the time when the front part of the body cuts a beam of light across its path (produced by the photogate), and the time when the back part of the body leaves the light beam. These two times will enable you to determine very nearly the instantaneous speed v of the body as it passes through the photogate on the track. We shall label the two times measured by the first photogate t1 and t'1. By measuring the length of the object that passes through the photogates beforehand, labeled x, and taking the time it took for the car to transit the photogate Δt1 = t'1 - t1, we can determine the instantaneous speed at the first photogate v1 by

(6)

In a similar way, the instantaneous speed of the cart as it passes through the second photogate v2 can be determined. Note that the time measurements also allow us to determine the time elapsed in traveling from one photogate to the next, that is

(7)

From these data we may determine the acceleration in one of two ways: either from Eq. (2) or from Eq. (3). To use Eq. (3), we also need to measure the distance s between the two photogates.

Thus, using instantaneous speeds derived from the time (and distance) measurements, we can determine the acceleration, which we label av. By solving Eq. (2), we can write

(8)

This equation is valid only if the acceleration is constant. If we use the first instantaneous speed v1, the travel time between photogates t, and the measured separation between the two gates s, we can again determine the acceleration, which we label as. Solving Eq. (3) for the acceleration gives

(9)

The same time measurements can be made in the case of the rolling sphere, and the same expressions can be used to determine two accelerations from the measured data. In this case, the length of the object x is the diameter of the sphere, assuming you have carefully positioned the height of the photogates to intersect the middle of the metal sphere.

EXERCISES OF THE SLIDING CAR:

Apparatus setup: We will use two photogates and a tilted airtrack to measure the instantaneous acceleration av and the standard acceleration as of a frictionless cart. The two photogates will be set up to give us the time it took the motion tag on the cart to pass through each gate, the time it took to get from one gate to the other and the velocity in each gate.

1. Setup the Signal Interface and photogates as shown in Fig. 2.

a. Plug the power brick into the back of the Interface, then into the strip on your table.

b. Connect the white cable into the back of the Interface then into your computer’s USB port.

c. Plug a photogate into Channel 1. This will be photogate 1.

d. Plug a photogate into Channel 2. This will be photogate 2.

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Figure 2: PASCO Interface setup.

e. Turn on the Interface (switch is on the back).

2. Start DataStudio. It should be located somewhere under your start menu.

3. When prompted, choose ‘Create Experiment’.

4. Set up the software to recognize the hardware:

a. In the software, add a “Photogate” sensor to Channel 1.

b. Repeat for Channel 2.

5. Set up the constants:

a. Find the “Constants” menu for the first photogate. In version 1.9.7r8, this tab is in the menu that is seen when you click on the photogate icon shown connected to the interface.

b. Enter the width, in meters, of the motion (white) tab found on the cart.

c. Repeat for the photogate icon in Channel 2.

6. Set up the display.

a. Notice that there is a list of Data options on the upper left (usually). Also notice that there is a list of Display options on the bottom left (usually).

b. From the list of Displays, click and drag the “Table” icon and drop it on “Velocity in Gate, Ch 1” under the Data Window.

c. Repeat for “Velocity in Gate, Ch 2” and “Time Between Any Gates, Ch 2”. You should now have three tables.

d. Move and resize the tables so you can see everything at once.

Data Acquisition: Two sets of date will be taken, one for each inclination of the air track.

1. Turn on the air supply and test to see that the track is level, or nearly level, by noting whether the cart initially at rest remains at rest. Adjust the leveling support screw at one end of the track, if necessary.

2. Measure the thickness of the spacer(s) (slotted mass(es)) and determine sinθ.

3. Incline the track slightly by placing the spacer(s) under the leveling support screw.

4. Turn on the air supply to the air track.

5. Click on the ‘Start’ button in the DataStudio window to record data.

6. Beginning with the cart at the top of the incline, allow the cart to accelerate through the two photogates, being careful to not give the cart any initial velocity or let it bounce back at the end.

7. Repeat until you have 5 measurements in one run.

8. Click on the ‘Stop’ button to stop recording.

9. Repeat parts 1-6 for several additional angles of inclination, θ.

10. Save your sliding cart data.

Data Analysis and Presentation: Make sure your lab write-up includes the tables with the acceleration data for each run.

1. Compare each of the acceleration means to the corresponding value of atheo = g sinθ that you expected.

2. Give the percent deviations of both av and as from atheo by calculating:

EXERCISES OF THE ROLLING SPHERE:

Repeat the exercises carried out for the sliding car, but using the rolling sphere and its grooved track. The diameter of the ball takes place of the width of the motion tag. Compare the two acceleration measurements from this part to Eq. (5).

1. Make sure you have saved the sliding cart data.

2. Physically move the photogates to the track for the rolling sphere data. Be sure to adjust them so that the middle of the ball cuts the photogate.

3. Set the track to some arbitrary small angle.

4. Measure the height of the track at the mounting rod.

5. Measure the length of the table from the mounting rod to where the track touches the table.

6. Use these two measurements to calculate sinθ and atheo.

7. Under the “Experiment” menu, choose “Delete all Data Runs”.

8. Click on the ‘Start’ button in the DataStudio window to record data.

11. Beginning with the ball at the top of the ramp, allow the ball to accelerate through the two photogates, being careful to not give the ball any initial velocity.

12. Repeat until you have 5 measurements in one run.

13. Click on the ‘Stop’ button to stop recording.

14. Repeat parts 9-13 for several additional angles of inclination, θ.

15. Save your rolling sphere data.

Data Analysis and Presentation: Make sure your lab write-up includes the tables with the acceleration data for each run.

3. Compare each of the acceleration means to the corresponding value of atheo = g sinθ that you expected.

4. Give the percent deviations of both av and as from atheo by calculating:

QUESTIONS:

1. Given that av is only valid when the acceleration is constant, what can you conclude from your data and analysis about your two systems?

2. What is the advantage or disadvantage of using the as calculations to determine the acceleration of the system?

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