Experiment 7
Newton’s Second Law of Motion
Objective:
The objective is to experimentally verify Newton’s Second Law, ΣF = Ma.
Equipment:
A computer with Internet connection, a calculator (The builtin calculator of the computer may be used.), paper, and pencil
Theory:
A brief on Kinematics and Kinetics: If the net force ΣF is known and acceleration a is found from it by using ΣF = Ma, the study is called "kinetics." When force is not used, and instead distance, x, velocity, v, and time, t are used to solve for acceleration (Chapter 2 equations), the study of motion is called kinematics. Equations: { a = (v_{f}  v_{i})/t , x = (1/2)at^{2} + v_{i}t , and v = Δx/Δt }are three kinematics equations none of which involves force. 
Now, Newton's 2nd law simply states that
"A nonzero net force ΣF acting on mass M generates an acceleration a in that mass such that ΣF = Ma."
In verifying this formula or simply verifying the kinetic equation, we may write it as ΣF/M=a. In the course of experiment, you will show that if the ratio of the net force, ΣF, to the total mass being moved M_{total} or M+m is kept constant, the acceleration remains constant as well.
To verify each time that a found from ΣF/M = a is valid, calculate it from the kinematics equation: x = (1/2)at^{2} + v_{i}t as well to show that both a's will turn out the same.
In Fig. 1, assuming no friction, the hanging mass m can cause motion for both m and M. The force of gravity on m is F = mg. This force has to move a total mass of m + M. In the absence of friction, F = mg is the only force causing motion. In this case mg is the ΣF.
Using Newton's 2nd Law: ΣF = (Mass)(Accel.) , we may write: mg = (m + M) a or, a = mg / (m + M).

Fig. 1 
In Fig. 2, assume friction is present for M only because it has to slide on the top of the horizontal table. Since N = w in magnitude for M, we may write or N = Mg.
The force of kinetic friction is F_{k} = μ N , or F_{k} = μ Mg. Here the net force, ΣF, causing motion is mg  μ Mg. The total mass to be moved is m+M. This makes the acceleration a = ( mg  μ Mg ) /( m+M ).
Using Newton's 2nd Law: ΣF = (Mass)(Accel.), we may write: mg  μ Mg = (m + M) a or, a = (mg  μ Mg ) / (m + M).

Fig. 2 
Procedure:
Click on the following link: http://www.walterfendt.de/ph14e/n2law.htm . Note that masses in the applet are in (grams). For ease of calculation, suppose they are given in kilograms (kg) and ignore the grams. This is not going to affect the results because as you see in both of the above figures the grams or kilograms cancel in the numerators and the denominators of the equations for acceleration, anyway.
Part A: No Friction
For the first 9 trials, set the coefficient of friction μ to zero. To change the numbers each time, the applet must be reset. Follow the values for M given in the Table below. Run the applet for the values given in the first row. In the first row, both M and m are given. Calculate the "Measured a" based on these values. Also, record the "Accepted a" from the applet. Find the % error using the %error formula you used in the previous experiments.
For rows 2 through 9, in each row, use the given M and keep running the experiment for different values of m that you choose until you get the same acceleration as you obtained for the first row. The applet calculates the accepted value of acceleration for you, each time.
The applet uses x = (1/2)at^{2} +v_{i}t with v_{i} = 0 and x replaced by s. Solving for a results in a = 2s/t^{2} as used by the applet. This can be observed on the applet and used (or recorded) each time to make that sure the acceleration is kept constant. Note that the accelerations you calculate from
a = ( mg  μ Mg ) /( m+M ) and record under the column "Measured a", must turn out to be the same or very close, each time. In each row, you should find m (experimentally) such that the resulting acceleration is the same as before. Once you find the experimental value of m in a row, use it to calculate the measured value of a as in Column 7. Of course, both ΣF and Total mass (Columns 5 and 6) must be calculated before Column 7.
Part B: Friction Present
Include the given coefficients of kinetic friction in the experiment and calculations. Follow the same procedure as applied to Part A.
Data:
Given and Measured g = 9.81 m/s^{2}.
Trial  M  m  μ_{k}  ΣF
= mg  μMg 
Total mass M + m 
Measured a =
ΣF/(M+m) 
Accepted a
= 2s /t^{2} 
% error 
Part A  (kg)  (kg)  ( N )  (kg)  ( m/s^{2} )  ( m/s^{2} )  
1  100.0  4.00  0  
2  150.0  0  
3  200.0  0  
4  250.0  0  
5  300.0  0  
6  350.0  0  
6  400.0  0  
8  450.0  0  
9  500.0  0  
Part B  
1  100.0  11.73  0.080  
2  150.0  0.080  
3  200.0  0.080  
4  250.0  0.100  
5  300.0  0.100  
6  350.0  0.100  
7  400.0  0.160  
8  450.0  0.160  
9  500.0  0.160 
Comparison of the results:
Provide the percent error formula used.
Conclusion:
State your conclusions of the experiment with reference to the values obtained in columns 5, 6, 7, 8, and 9 of the above Table.
Discussion:
Provide a discussion, if necessary.