Physics

Physics 161

Newton’s Laws II

Introduction

In this laboratory you will explore a few aspects of Newton’s Second Law using Capstone and the Dynamics Track. In Part I, you will use hanging weights to accelerate a cart away from a motion sensor and use the data you record to measure the acceleration of gravity and test Newton’s Second Law. In Part II, you will observe the magnitude of the frictional force on an object at rest as well as the force of kinetic friction once that object has been brought to a constant velocity. This will be done indirectly by observing the force needed to accelerate an object at rest to a constant velocity and comparing it to the force needed to maintain a constant velocity.

Reference

Young and Freedman, University Physics, 12th Edition: Chapter 4, section 4.3-4.4, Chapter 5 sections 5.1-5.3

Theory

Part I: Newton’s Second Law is written mathematically as. From this equation we see that two things affect the acceleration of a cart: the applied force and its mass. For example, a more massive cart will require a greater force in order to achieve the same acceleration as a less massive cart.In this experiment, the force used to accelerate the cart is applied by hanging weights. When the cart is pulled by the hanging weights, the total system (both the cart and the hanging weights system) is accelerated. As a consequence, Newton’s Second Law can be written as (ignoring the friction force):

(1)If we define mtotal = mcart + mhang , this can be rewritten as:

(2)By plotting mtotal*a vs. mhang, one can find g as the slope of this plot

Note: To decrease the acceleration of the cart, you will add 500g to the cart, and mcart in these equations will be equal to the measured mass of the cart + added weight. We shall keep that combined mass constant. You will measure the mass of the cart (with the attached force sensor, which is just providing the hook for pulling) using the scale.

Part II: We consider two classes of frictional forces, those for objects at rest (static friction) and those for moving objects (kinetic friction). The force of kinetic friction on a moving object acts in the direction opposite to the velocity of the object. The magnitude of the force of kinetic friction is given by

(3)

where is the coefficient of kinetic friction and is the normal force. In this experiment you will study friction on objects moving horizontally and the normal force is equal to the weight of the object mg. Thus,

,

which can be solved for as:

(4)In the static case, things are slightly more complicated. Similar to Equation 3, we can write:

. (5)

This however gives the maximum force that static friction can generate (in resisting another force trying to move the object) before the object starts to slip and move. In general, the force generated is less than this, i.e. . We can solve Equation 5 for the static coefficient of friction;

. (6)

Procedure

Part I:

1. Make sure the track is level. You can check by making sure that the cart does not roll to one end or the other spontaneously.

2. Measure the mass of the cart plus added weight, convert it to kilograms and record it in Excel in the column mcart. This mass will be constant in our experiment.

3. Connect the motion sensor to the PASCO 850 Universal Interface. In Capstone under Hardware Setup digitally plug in the Motion sensor II from the Sensors menu. Change the Motion sensor’s trigger rate to 50 Hz. Drag a Graph icon into the workspace area and set the y-axis to Velocity.

4. Set the cart on the track at the end closest to the motion sensor. Attach a string to the end of the cart and place it over the pulley. The other end of the string should be connected to the hanging mass of 150 g.

5. One student should hold the cart in place while the other student presses RECORD to collect data in Capstone . You may want to delay releasing the cart for a second in order to make sure you get all the data.

6. Release the cart, being sure to catch the cart before it hits the end of the track, then hit Stop in Capstone.

7. Highlight the points on your graph that follow a straight line (the slope should be positive because the cart is accelerating away from the motion sensor). Do NOT include the end points of this line segment in the data (those values may be affected by forces your hands apply in releasing and stopping the cart.)

8. Using the Linear Fit tool, measure the slope of the velocity graph and record this value (the acceleration) in the table you made in Excel. Your table should have the following information:

where mtotal == mcart + mhang

9. Now remove 10 grams of mass from the hanging mass. Perform steps 6-8. If you can, include all the velocity data on the same graph. Remember to record the Total Mass in kilograms!

10. Repeat step 9 until the hanging mass is 100 grams. Once you have recorded all of your mass and acceleration values, calculate the total mass times acceleration of the cart during each run and complete the above table in Excel.

11. Using your data in the table, plot Total mass *a vs. hanging mass . Place a trendline in this plot and find the acceleration of gravity.

12. To find the uncertainty in g, rewrite equation (1) as

Add a column to your Excel table from step 8 to calculate g from the above equation. Then find the average g and the standard deviation, σg.

Part II:

1. Connect the force sensor to the PASCO 850 Universal Interface and drag the Student Force Sensor icon to the input. Create and plot a graph of Force vs. time.

2. Measure the mass of the friction cart with the cork bottom and add 500 g to the cart and connect the cart to the force sensor.

3. Press Tare on the sensor and then RECORD in Capstone and then gradually begin pulling the cart at as constant a velocity as possible. Since static friction is greater than kinetic friction, the force should drop once the cart has started to move.

4. Drag the friction cart a short distance along the dynamics track while maintaining a very low constant velocity.

5. Press Stop. If you were successful in maintaining constant velocity the force should remain constant. If the force increases or decreases at points on the graph it may be because the cart’s velocity was not constant for that period of time. Your graph should look as much as possible like the one in Figure 4.3, but your data will have negative values. It may take several tries to produce good data. Record data until it reaches a constant value.

Figure 4.3

6. Select a portion of the graph with relatively stable data and determine the average force over that time. (Highlight the data of interest then click on the icon and record Mean value.) Record the mean into your spreadsheet in Excel. Do NOT erase the data from the graph; you will want to compare it with subsequent trials. Capstone saves all data unless you close the session or turn off the PASCO 850 Universal Interface box. If you accidentally delete a graph, do not despair for there is hope in the form of a handy undo feature given by the  icon.

7. Select a portion of the graph that includes the peak (most negative value) of the static friction portion of the data. Click on the icon and record the Min value. This is your value for the maximum force of static friction.8. Repeat the above steps 4 more times. Make a table in Excel (as shown below) and record the values of static and dynamic friction forces in that table. Find the average and standard deviation of the friction forces.

9. Repeat steps 3-5 with added masses of 1 kg, 1.5 kg and 2 kg. Observe the trend between kinetic friction (which directly opposes the applied force that keeps the cart at constant velocity) and the normal force (equal to the weight of the cart because it is on a horizontal surface.)

0.5kg 0.5kg 1.0kg 1.0kg 1.5kg 1.5kg 2.0kg 2.0kg
Trial Ffr-static

(N)

Ffr-kinetic

(N)

Ffr-static

(N)

Ffr-kinetic

(N)

Ffr-static

(N)

Ffr-kinetic

(N)

Ffr-static

(N)

Ffr-kinetic

(N)

1
2
3
4
5
Average
Standard

Deviation

10. In Excel, calculate the normal force making sure to include the mass of the friction cart.

11. Create a graph of (use the average values) vs. for both kinetic and static frictions. Are the graphs linear? Add appropriate trend lines. The slope of the graph should be the coefficient of static/kinetic friction between cork and Aluminum. Calculate both and . See Equations (3) through (6) for reference.

For Your Lab Report

How does the value of g obtained from the linear fit compare with the value obtained from average in step 12? Within how many multiples of σg is the experimental value to the accepted value? For part II, show a sample calculation for and with no error propagation.**

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