Lab 4 - Motion on an Incline PDF

Preview

Summary

This is the lab procedures. I learned in Dunwoody Campus. My professor was Dr. Skelton....

Description

Phys 2211L

Motion on an Incline

In a famous set of experiments in the early 17th century, Galileo showed that the distance traveled by a ball rolling down an incline does not increase in proportion to the time of travel. Instead, Galileo found that the distance traveled increases as the square of the time. The images above show Galileo demonstrating his results to a surprised public (left), and a variant of his incline (right) that has small bells placed along its length. Galileo found that if he positioned the bells in a geometric progression (that is, at 1, 4, 9, etc., units of length) and let the ball roll, the bells would ring at equal intervals of time. This was surprising and counter-intuitive to observers of the day. It was from these and other results that the idea of acceleration – the rate at which an object picks up speed – came to be understood as essential to the study of motion. In this lab we will reproduce some of Galileo’s experiments by rolling a dynamics cart down an inclined track and measuring its time of travel. First, we will measure the ratio of distance traveled to time of travel and ask whether the cart picks up speed at a constant rate as it travels. Second, we will explore the effect of increasing the inclination of the track on the acceleration of the cart, and compare our results to the acceleration of an object in free-fall.

Finally we will explore another fundamental question: Do heavier objects accelerate faster or slower than lighter objects?

Some Key Concepts When two quantities are proportional, their ratio is a constant. When proportional quantities are plotted on a graph, their constant ratio is the slope of the graph. When quantities are not proportional the slope of their graph will vary.

A

A

B

B

A is proportional to B

A is not proportional to B

The question of whether the displacement of the cart is proportional to the time of travel, or to the squared time of travel, is best answered by plotting these quantities. To further analyze our results we will use the basic definitions of kinematics: • Displacement:

∆x = x 2 − x1

• Average velocity:

v=

x 2 − x1 ∆ x = t 2 − t1 ∆t

• Average acceleration:

a=

v 2 − v1 ∆ v = t2 − t1 ∆t

Recall also that if acceleration is constant, then vaverage = (vinitial + vfinal)/2.

Forces and Acceleration A fundamental principle in physics is that accelerations are caused by forces. For an object in free-fall, it is the force of gravity (its weight) that causes it to accelerate toward the surface of the Earth. Free-fall acceleration is given the symbol g, and has the value of about 9.80 m/s2 at our latitude. What about an object rolling down an incline? In this case, the force of gravity (the object’s weight, mg) still points vertically downward. L W H

The weight vector and its components

But this force vector can be analyzed into a component parallel to the plane and a component into the plane. It is the component parallel to the plane that accelerates the rolling cart. Galileo realized that the component parallel to the plane must be proportional to the ratio of the height of the end of the track (H) and the length of the track (L). But the ratio H/L is precisely the sine of the inclination angle, θ. Thus, ideally, the acceleration of the cart down the incline should be a = gsinθ In fact, Galileo’s purpose in conducting his experiments was to measure g. The motion of an object in free-fall was too rapid for accurate measurements with the resources he had available. (He used a waterclock, or his own pulse, to measure time.) But the incline reduced the

acceleration considerably, allowing g to be calculated from the above expression. Note that if the inclination angle is increased, the acceleration should also increase, but in proportion to sine θ, not θ itself. At an inclination of 90 degrees, sine θ = 1, and the acceleration should reduce to that of free-fall: g. We will test these ideas in experiment 2.

Rolling Friction A complication in our experiment will be that we cannot avoid some frictional force on the cart, resisting its motion down the incline. We expect this to reduce the acceleration caused by gravity. The experimental acceleration that we will calculate from our measurements will be called anet

(calculated from measurements)

and the theoretical acceleration, for a friction-less system, will be: ag = gsinθ We can compare these two quantities to assess the magnitude of the frictional force on the cart.

Experiment 1: Displacement vs. Time Here we will time the motion of the cart down an incline for various displacements (xfinal - xinitial).

Objectives: • How does the displacement relate to the time of travel? • Is the acceleration constant? If so, what is its value? Procedures: 1. Attach the inclinometer to the side of the dynamics track. There is a slot on the side of the track into which the inclinometer’s nut can be inserted. Adjust the level of the inclinometer so that it reads 0.0 degrees when the track is horizontal. 2. Elevate one end of a dynamics track by resting it on a horizontal rod clamped to an upright rod. The track should be inclined to about 2 degrees. Clamp the low end of the track to the table. 3. Slide the plastic bumper onto the track. Place the dynamics cart on the track and let its plunger rest against the bumper. Adjust the placement of the bumper so that the back end of the cart is at 20.0 cm. This will be xfinal for the first trial. 4. Set the cart so that its back end is at 0.0 cm. This will be xinitial. Note that the scale on the track does not quite go to 0.0 cm. You can make a small mark at 0.0, but do not put any other mark on the tracks. Use a 500-gram mass to hold the cart in place until you are ready to begin timing 5. Time the motion of the cart for displacements of 20.0, 40.0, 60.0, 80.0 and 100.0 cm: a. First practice timing a run until you can consistently time the motion to within 1/10 of a second. The cart should start from rest. Remove the mass to set the cart in motion and start the timer. Stop the timer at the instant the cart hits the bumper. Note: Do not wait for the cart to start moving to start the timer. Release the cart and start the timer at the same time. To avoid reaction time delays, the same person should both release the cart and start the timer. Also, anticipate the moment the cart hits the bumper to stop the timer.

b. For each displacement, time at least three runs and take an average time. c. Record each individual time and the average time for each displacement in a data section in your lab notebook.

Analysis 1. For each displacement, calculate the ratio Δx/t and Δx/t2. Make a chart of your calculations in an analysis section of your notebook. Leave room for more columns. 2. Calculate the average of these ratios. 3. Which ratio is roughly the same over all displacements? (Likely neither is perfectly constant, but which shows significant deviations from the average?) 4. What kinematic quantity is the ratio Δx/t? 5. Using Δx/t, calculate the final velocity for each displacement. (Hint: Recall how final velocity relates to average velocity, if acceleration is constant.) 6. Using the final velocity for each displacement, calculate the average acceleration during each displacement. (Recall the definition of average acceleration.) 7. Looking at your results, what does the ratio Δx/t2 represent? 8. Combine steps 5,6, and 7 above into a single, algebraic expression for the acceleration of the cart in terms of the displacement and the time of travel. You will use this expression in the next part to calculate accelerations form distance and time measurements. 9. Post-Lab: Graph Δx versus time and Δx versus time-squared. Put both graphs on the same sheet of graph paper, using ½ page for each. For your Δx/t2 data, find the deviations in these ratios, and the average deviation. Draw a conclusion about whether this ratio is constant or not.

Experiment 2: Motion on an Incline In this experiment we will measure the acceleration of the cart for increasing inclination angles. We will compare our results with the theoretical value for a friction-less system.

Objectives: • How does the acceleration of the cart depend on the inclination angle? • How significant are resistive forces, such as friction, on the motion? • Can the results for the acceleration be extrapolated to free-fall?

Procedures 1. Time the motion of the cart for five additional inclination angles. • • • •

This time, set up a displacement on the track (xi and xf) of about 80 centimeters. Use inclination angles between 2 degrees and 15 degrees. Do not use angles greater than 15 degrees, since it is difficult to time large angles accurately. Time three trials for each inclination angle. Try a few practice runs until you can get consistent times. Record the time and displacement data to at least 3 significant figures. Read the inclination angle to the nearest 1/10 degree.

2. Calculations and Analysis •

Set up a table for calculations so that you can record the following results: Angle (θ)

Sine θ

Time

anet

ag = gsin θ

anet is the measured acceleration of the cart. ag is the expected (theoretical) acceleration in the absence of frictional forces. • •

For each angle, find the average time of travel and calculate the actual and expected accelerations. Assess your results: Find the percent error of ag compared to the actual acceleration for each angle. That is, what percent of the actual acceleration is ag? Does this difference show a trend?

5. Post-Lab: Graph the data using error bars. •

•

•

•

Calculate the uncertainty in the accelerations (Δa)* from the uncertainty in the measured times. Use the average deviation in the measured times for each angle as the uncertainty in the time (Δt)*. That is, given that the travel time is uncertain to within +/- δ, what is the uncertainty in the experimental acceleration? Plot the experimental acceleration versus the sine of the angle, along with “error bars” for each data point. Error bars are vertical lines above and below each point showing the uncertainty in the value of that point. The length of the vertical lines will be the magnitude of Δa for that point. Considering your points and their error bars, draw a “best fit” line for your graph. Determine the slope of your plot and record the equation of the line in the form: anet = (slope) x sinθ.

•

Now draw a line representing ag = gsinθ on the same graph, from θ=0 to your maximum angle. Assess how the plot for anet compares with ag. What value does anet approach as θ 90 degrees? What about as θ 0 degrees? Do the results make physical sense? If not, why not?

*

In this context, read “Δ” as “the uncertainty in”, or “the variation in”, not as “the change in”. We are considering the range of variation in the calculated acceleration because of the deviations in the travel time for each trial.

Experiment 3: Mass and Gravitational Acceleration You can now address another very important question, which presumably applies to free fall also: How does the mass of an object affect its motion? Do heavier objects roll (or fall) faster or slower? We can use the cart-on-incline set-up to answer this question. This time, you will keep the inclination angle constant but vary the mass on the cart. The results should apply to free fall also. Cart with mass

θ

Procedures 1. First, go on record with a prediction: All other things being equal, will a heavier cart roll faster or slower than a lighter one? 2. Run the same experiment as before, this time keeping the same inclination angle for each trial but varying the mass on the cart. Use 5 different mass loads on the cart – no load, then 200 g, 400 g, etc., up to 800 g. As before, run three trials for each load and use your average times in calculating the accelerations

3. In your analysis, calculate and tabulate the accelerations for each mass load, then calculate the mean of these acceleration over all mass loads. 4. Calculate the standard deviation in the accelerations and compare each individual acceleration value to amean +/- σ. That is, how many data points fall inside or outside the range +/- σ of the mean? 5. Considering your data and analysis, what can you say about the effect of mass on the acceleration of the rolling cart? Note from probability theory: If 68% or more of data points fall within one standard deviation of the mean, no effect other than random error can be inferred....