Newton’s Second Law of Motion
Overview
The purpose of this investigation
is to validate Newton’s Second Law of Motion. In Part A a lab cart will be
accelerated by various net forces while keeping mass constant. In Part B the
lab cart will be accelerated by a constant net force while its mass is varied.
The goal is to determine the relation between acceleration and force and the
relation between acceleration and mass. The force on the lab cart is
controlled and provided by gravity acting on a weight at the end of a string
that passes over a pulley at the end of a lab table.
Force, mass, and acceleration all must be measured in order to
complete this lab. Force data is collected by calculating the weight of the
calibrated masses added to the end of the string. Mass data is collected with
a triple beam balance. Acceleration data is collected by a Go Direct Motion
Detector working in connection with Graphical Analysis on laptop. The
Graphical Analysis app allows the user to analyze the motion data to determine
velocity, acceleration, etc.
In this section of the lab, the cart will be loaded with three masses. Then various combinations of mass will be removed from the cart and placed on the end of a string passing over a pulley. By doing this the amount of net force will be varied while keeping constant the total amount of mass being accelerated. It is important to note that the pull of gravity on the dangling mass causes not only the cart and its contents to accelerate, but also the string itself and the mass or masses attached to the end of the string. Put another way, the weight on the end of the string causes all of the mass to accelerate (and it all accelerates at the same rate).
1. Set up the track with one end hanging slightly over the edge of the table. Attach the pulley to this end. Secure the motion detector to a ring stand and place it at the other end so that the beam of sound from the detector will reflect off the cart as it is moving.
2. Load the cart with the following masses: 20 g, 50g, 100 g. Attach one end of a string to the cart. Pass the other end over the pulley at the end of the track. Give the free end of the string a slight tug so that the cart and string are set into motion. Adjust the feet of the track so that the cart goes slightly downhill toward the pulley and does not accelerate once it is set into motion. (i.e. With just enough tilt in the track the cart will move downhill with constant velocity.) At this point, because the cart is not accelerating it should be true that all forces acting on the cart are balanced and there is no net force acting on it.
3. Important note: The slight slope of the track is a way to counteract friction. Once a proper slant has been determined do not change it. In so doing it follows that any additional weight added to the end of the string will be the net force acting to cause the acceleration. In other words, gravity alone is the force causing the acceleration.
4. Complete the mass data table using the triple beam balance. The values in this table must remain the same for each trial.
5. Start the Graphical Analysis app and connect the motion detector by Bluetooth or USB cable. Under the View Options menu choose Meters so that you see a live readout of the distance measured by the sensor. If you move the cart back and forth you should see the distance vary accordingly – if not, then try adjusting the tilt and height of the motion detector. NOTE: the motion detector cannot measure distances below about 0.3 m. You may wish to find the minimum value that it can measure and note the location of the cart and then consider this as a good starting point for the cart.
6. For the first trial, remove the 20 g mass from the cart and hang it on the end of the string. Leave the 50 g and 100 g masses on the cart.
7. Click on the Collect button to start each trial. You should release the cart and let it accelerate after you hear the steady clicking of the motion detector. Someone must catch the cart! (Before it hits the pulley or runs off the table.)
8. You should now have graphs of position vs. time and velocity vs. time that clearly shows the cart at rest, the cart accelerating, and the cart being caught. If not, you need to repeat the experiment – simply click on the Collect button to repeat. You may need to adjust the direction the motion detector is pointing if it is getting errant reflections (normally it works best when tilted slightly upward).
9. You now need to get the acceleration of the cart. It is best to use the velocity graph. Click and drag to select the region of the graph that you want to analyze (the part where the cart was accelerating). Under the Graph Tools menu choose Apply Curve Fit. Inspect closely the results of the curve fit – if it is unsatisfactory then try repeating the process and adjust the amount of data selected for analysis.
10. If all seems well with the regression, then record the results in the data table, making sure to include units in the spaces provided. The line of best fit should match the data very closely! The correlation coefficient is an indicator of how well the data matches the best fit: the closer r is to 1 the better the match. The computer labels this as Correlation; it is also often called simply r. It should be possible to get values of r of at least 0.990 – if it is less than this, then try again if you feel like you have enough time.
11. For the next trial remove the 20 g mass from the string and put it back on the cart. Remove the 50 g mass from the cart and place it on the end of the string. You now have changed the force pulling the cart without changing the mass being accelerated. Collect, graph, scrutinize and record acceleration data as before.
12. Repeat this process with 70 g, 100 g, 120 g, and 150 g
of mass on the end of the string. Do this by transferring a mass or
masses between the end of the string and the cart – thus keeping constant the
total amount of mass being accelerated.
In this section of the lab the cart will begin with no mass loaded onto it. Then under the influence of the same net force each time, increasing amounts of mass will be loaded onto the cart.
1. Use the same cart and the same track setup. Remove all masses from the cart and the end of the string.
2. Attach a 50 g mass to the end of the string and pass it over the pulley. This same mass will be used to provide the same net force for each trial. Record this value in the mass table.
3. Use the program to collect, graph, scrutinize, and record acceleration data just as explained in part A.
4. Repeat the process with 200 g, 400 g, 600 g, 800 g, and 1000 g of mass placed on top of the cart. Do not change the mass pulling the cart. In this way you are changing the mass being accelerated without changing the amount of force.
1. Complete the bottom table on the data sheet for part A: Determine the net force in Newtons by finding the pull of gravity (i.e. the weight) acting on the mass added to the end of the string for each trial. (Remember – by tilting the track, all other forces were set to balance one another.) Acceleration is simply copied from the regression results.
2. Use these results to construct a force vs. acceleration graph. For this graph only, plot the independent variable (force) on the y-axis. Determine the best fit and equation.
1. Complete the bottom table on the data sheet for part B: Total mass being accelerated includes the cart, the string, and all other masses that accelerated with the cart, including the one on the end of the string. Calculate the reciprocal of this. Acceleration is simply copied from the regression results.
2. Use these results to construct an acceleration vs. mass graph. Determine the equation assuming this to be a hyperbola of the form: y = k/x. Do this by using each datum to solve for a value of k and then take the mean of the k values and use it for the best-fit equation and curve. Show this work on the graph. Use the equation to plot the best fit.
3. Also construct a “curve straightening” graph of acceleration vs. mass –1. On this graph the x-variable is equal to the reciprocal of the total mass and the y-variable is still the acceleration. This is a common practice in science and it represents an alternate way to visualize and analyze data that would otherwise produce a curve. Determine the best fit and equation.
A complete report (50 pts): (5 or 6 pages in this order)
q Completed data/results tables. (8)
q Force vs. Acceleration graph. (10)
q Acceleration vs. Mass graph. (10)
q Acceleration vs. Mass –1 graph. (10)
q On separate paper, answers to the questions using complete sentences. (12)
1. Show the work for the following calculated values that appear in the tables: (a) the net force for the 20 g trial, (b) the total mass for the very last trial in part B, and (c) the reciprocal mass for the very last trial in part B.
2. State whether or not your graphs confirm and/or support the types of relations described in Newton’s 2nd Law and explain how so. Be specific in referring to your results and graphs. Remember to address both aspects of the 2nd Law: how acceleration is related to force and how it is related to mass.
3. Consider the total mass accelerated as shown in the data table for part A. This is the mass that was accelerated by the pull of gravity acting on the hanging weight. This value should be the same as one of the constants, slopes, or y-intercepts found on your graphs. (a) The total mass from Part A should be equal to which constant, slope, or y-intercept? (b) Calculate the absolute value of the difference between these two values. (This is a little like finding the absolute error – however, neither value is a “true” or “accepted” value.)
4. Consider the weight that was pulling the cart in part B. This weight should be the same as one of the constants, slopes, or y-intercepts found on your graphs. (a) The weight of the mass on the end of the string should be equal to which constant, slope, or y-intercept? (b) As in the previous question calculate the absolute value of the difference between the values.
5. Show work and solve: (a) Take the acceleration for the last trial in part A and use this value to calculate the tension in the string that acted on the hanging 150 gram mass. (b) Take the same acceleration for the same trial and use it to calculate the tension in the string that acted on the cart (i.e. the force at the other end of the string). (c) Ideally these two values should be the same – is this true for your values? Repeat the calculations for one or more other trials (but you don’t need to show your work again – same steps). What would cause these values to be different – tension at one end of the string versus the other? Briefly discuss.
6. Discuss error in this lab. (Things to discuss: indications and signs of error – random and/or systematic, the probable and significant cause(s) of the error that is apparent in the results. The goal of discussing error is to explain satisfactorily why the results of your lab are not quite exactly what was expected. Be as specific as possible. You will have unexpected results in almost any lab – but what are the particulars in this one?
Part A – Acceleration vs. Force
Mass of cart and string |
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Combined mass of the three calibrated weights |
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Total mass being accelerated |
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Results of CBR/Logger Pro linear regression of velocity-time graph: |
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Trial |
Slope ( ) |
y-intercept ( ) |
Corr. Coeff. (no units) |
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20 g |
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50 g |
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70 g |
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100 g |
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120 g |
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150 g |
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Trial |
Net Force (N) |
Acceleration (m/s2) |
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20 g |
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50 g |
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70 g |
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100 g |
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120 g |
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150 g |
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Mass of cart and string |
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Mass at end of string |
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Results of CBR/Logger Pro linear regression of velocity-time graph: |
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Mass added to cart(g) |
Slope ( ) |
y-intercept ( ) |
Corr. Coeff. (no units) |
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0 |
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200 |
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400 |
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600 |
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800 |
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1000 |
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Total mass being accelerated: m (kg) |
Reciprocal of total mass being accelerated: 1/m (kg –1) |
Acceleration obtained from regression: a (m/s2) |
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