KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY
COLLEGE OF SCIENCE
FACULTY OF PHYSICAL AND COMPUTATIONAL SCIENCE
DEPARTMENT OF PHYSICS
TH

TABLE OF CONTENTS
1.
Abstract……………………………………………………………………………………………………………… 1
2.
Introduction………………………………………………………………………………………………………. 2
3.
Theory……………………………………………………………………………………………………………….. 3
4.
Diagram of set up………………………………………………………………………………………………. 3
5.
Method/Procedure……………………………………………………………………………………………. 4
6.
Observation table (Data)
7.
Graph 1
8.
Graph 2
9.
Theory and Calculations
10.
Results and Discussion
11.
Error Analysis
12.
Precautions
13.
Conclusions
14.
References
1 | P a g e

ABSTRACT
The purpose of this experiment is to verify the predictions of Newton’s law. In this experiment, an Atwood Machine was used in the experiment to verify the mechanical laws of motion with constant acceleration. An Atwood Machine consists of two objects of different masses hanging vertically over a pulley of negligible mass. The weights were attached to hooks to stay connected to the string around the pulley.
When the system was released, the system accelerated in the direction of the larger mass.
It was assumed that the tension is the same in each part of the string. This leads to calculation of the time intervals for the system to be in motion and the experimental acceleration of the system. There were three trials, to record the time it takes the system to be in motion. The data recorded was then applied into kinematic equations to calculate the experimental acceleration of the system.
There were some likely sources of error such as frictional forces in the pulley and the weight of the string which were not considered in the experiment.
2 | P a g e

INTRODUCTION
Force is a vector quantity which is measured in Newtons. There are different types of forces. These include, force of gravity, applied force, frictional force, and normal force.
Normal force is perpendicular to the surface and is exerted by the surface.
The force of gravity is equal to an object’s mass multiplied by gravity (9.8
m/s 2 ) . The sum of the forces, or net force, is equal to an object’s mass multiplied by acceleration. Tension is the force applied by a rope, string, or cable. Tension is the same throughout a string.
The Atwood machine consists of a pulley, which connects two masses. When these masses are unequal, the system will accelerate in the direction of the larger mass. The masses are connected by a light string. An Atwood machine is used in experiments to verify the mechanical laws of motion with constant acceleration. The purpose of this experiment is to verify the predictions of Newton’s Law. Newton’s Law predicts that the acceleration should be proportional to the net force and inversely proportional to the total mass of the system.
3 | P a g e

THEORY
According to Newton’s Second Law of Motion, it states that, the acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force and inversely proportional to the mass of the object (
1
2
). The forces acting on the two masses are mainly their weight,
1 g
2 g. These forces act downwards. The resultant forces on these masses (as far as the motion of the masses and the string are concerned) are however in opposite direction.
Thus 𝐹
NET
= m
1
g – m
2 g = (m
1
-m
2
) g …………………………………………………………………..…………. (1)
𝐹 NET = ma = (m
1
+m
2
) a ……………………………………………………………………………………………… (2)
Combining (1) and (2) 𝑎 =
(m1 −m2) g
……………….…………………………………………………… (3)
(m1 +m2)
The analysis so far has been idealized for clarity. In this experiment, there is an existence of frictional force, f associated with the pulley, opposing the motion and the string. The pulley also has moment of inertial I, which can be represented as I = m R 2
So 𝐹
NET
= m
1 g – m
2 g – m f g ………………………………………..………………… (4)
Where mf is the mass added to the mass to determine the length of the string.
The total mass of the system is m1 and m2 remains constant, so from Newtons second law of motion predicts theoretically that: 𝑎 =
(m1 −m2−mf) 𝑔 ……………………….…………. (5)
(m1 +m2)
It was also recognized the moment of inertial and the rotational inertial of the pulley, then the acceleration of the system becomes: 𝑎 =
(m1 −m2−mf) 𝑔 …………………………………………………………………………………………..…..… (6)
(m1 +m2+mf)
In this lab, the acceleration of the masses will be identified both experimentally and theoretically. The system will begin at rest. Therefore, having recorded the distance traveled, y and the time it took to do so, t the experimental acceleration will be calculated using kinematic equations 𝑦 =
1
2 𝑎𝑡 2 from the 2 nd equation of motion. 𝑦 = ut +
1
2
𝑎𝑡 2
At rest, the initial velocity is zero, u = 0
4 | P a g e

DIAGRAM OF THE SET UP
THE ATWOOD MACHINE
5 | P a g e

METHOD/PROCEDURE
PRELIMINARY MEASUREMENT
1.
Put masses on the hangers so that both m
1 and m
2 equal 20g. this should include the mass of the hangers. The system should now be in equilibrium.
2.
Add small masses to (in steps of 1g) to overcome friction, until a slight tap causes the system to move at constant speed. This should be judge by eye. The added mass is m f.
Record this mass, then remove it from the system so that m
1
and m
2 are equal again.
3.
With m
2 on the floor and m
1 near the pulley, measure the distance y from the bottom of m
1 to the floor. This is the distance that will use in calculating the acceleration. Record it.
TOTAL CONSTAN MASS AND VARYING NET FORCE
4.
Add 5g to m
1 so that m
1
= 25g and m
2
= 20g.
5.
Using the stop watch, measure the time it takes for m
1
to fall the distance y. Repeat twice and find the average of the time taken.
6.
Calculate the net force on the system using equation (2),
7.
Calculate the experimental equation using equation (5) and the theoretical acceleration using equation (3). Find the percent difference between the two.
8.
Repeat step 5-7 four more times, each time transferring 2g from m
2
to m
1.
The sum of the masses will remain the same, but the net force will change from one trial to the next.
TOTAL MASS VARYING AND NET FORCE CONSTANT
9.
Start with m
1
= 25g and m
2
= 20g, repeat steps 5-7 as before.
10.
Repeat steps 5-7 four more times, each time adding 5g to both m
1
and m
2
. (The difference between m
1
and m
2
, which determines the net force, therefore remains constant).
ANALYSIS
11.
Construct a graph of the net force versus the acceleration, using the measurements in which the total mass is constant. Draw a best fit straight line, measure its slope. Based on the theoretical equation, state the significant of the slope. Calculate the percent difference between the actual slope and its expected value.
6 | P a g e

12.
Construct a graph of the acceleration versus the reciprocal of the total mass, using the measurement in which the net force is constant. Draw a best fit straight line, measure its slope. Based on the theoretical equation, state the significant of the slope. Calculate the percent difference between the actual slope and its expected value
7 | P a g e

OBSERVATION TABLE (DATA)
1.
PRELIMINARY MEASUREMENT
M1 /g M2 /g M f /g Y /cm
20.00 20.00 2.00 130.00
2.
TOTAL CONSTANT MASS AND VARYING NET FORCE
M
1
/g M
2
/g T
1
T
2
T
3
Average
Time
T
0.025 0.020 1.91 1.92 1.93 1.92
Experimental
Acceleration
0.65
Theoretical
Acceleration
1.08
𝐹
NET
0.O49
0.027 0.018 1.29 1.39 1.35 1.34 1.52 1.96 0.088
0.029 0.016 1.08 1.10 1.09 1.09
0.031 0.014 0.91 0.90 0,86 0.89
0.033 0.012 0.85 0.79 0.77 0.80
2.40
3.27
4.14
2.83
3.70
4.57
0.127
0.167
0.206
3.
TOTAL MASS VARYING AND NET FORCE CONSTANT
M
1
/g M
2
/g T
1
T
2
T
3
Average
Time
T
0.025 0.020 1.91 1.92 1.93 1.92
Experimental
Acceleration
0.65
Theoretical
Acceleration
1.08
𝐹
NET
0.O49
0.030 0.025 2.04 2.01 2.05 2.03 0.54 0.89 0.049
0.035 0.030 2.28 2.29 2.31 2.20
0.040 0.035 2.34 2.40 2.40 2.38
0.045 0.040 2.80 2.85 2.71 2.78
0.45
0.39
0.35
0.75
0.65
0.58
0.048
0.O49
0.O49
8 | P a g e

GRAPH 1
0,25
0,2
0,15 y = 0,0453x - 0,0005
0,1
0,05
0
0 0,5 1 1,5 2 2,5
Acceleration
3 3,5 4 4,5 5
9 | P a g e

GRAPH 2
1,2
1
0,8
0,6
0,4
0,2
0
0 5 10
1/Mass
15 y = 0,0481x + 0,0115
20 25
10 | P a g e

THEORY AND CALCULATIONS
Percentage difference between theoretical and experimental acceleration.
Table 2.
Percentage difference, 𝑑 =
Theoretical−Experimental
𝑥 100%
Theoretical+Experimenta 𝑑 =
2.828−2.396
𝑥 100% d = 8.3%
5.224
Table 3.
Percentage difference, 𝑑 =
0.784−0.476
1.26
𝑥 100% d = 24.4%
Calculation of the Slope
Graph 1
The slope of the graph (1) indicate the relationship between the net or the total force and the theoretical acceleration of the system. The slope value of graph 1 is the total mass use in the system.
Newton’s second law of motion, the significant of the slope indicates the mass of the system From.
The slope, S of the graph is given as, S = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑛𝑒𝑡 𝑓𝑜𝑟𝑐𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
S =
0.167−0.088
3.70−1.96
S =
0.079
1.740
S = 0.0453
Hence from the slope, the mass of the system is M = 0.0453Kg
11 | P a g e

Graph 2
The slope of graph 2 indicate the relationship between the mass and the acceleration of the system. The slope value of graph 2 is the total net force on the system.
The significant of the slope indicate the net force on the system.
Plotted value
1
(m1 + m2)
22.22
Theoretical
Acceleration
1.08
18.18
15.38
13.33
0.89
0.75
0.65
11.76 0.58
From graph 2
The slope, S of the graph is given as , S =
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 1/𝑀𝑎𝑠𝑠
1.08−0.65
S =
22.22−13.33
S =
0.43
8.89
S = 0.0481
Hence from the slope, the net force on the system, 𝐹
NET
= 0.0481N
12 | P a g e

In this experiment, we measured the acceleration of the masses in an Atwood machine, and compared these results to the theoretical values calculated using the equation from
Newton’s Law. The average experimental acceleration is 2.396m/s 2 whereas the theoretical acceleration is 2.828m/s 2 . It is seen, from the calculations above, that the experimental acceleration (2.396m/s 2 ) was very close to the theoretical accelerations
(2.828m/s 2 ). By comparing the two values, the data further corroborates Sir Isaac
Newton’s Second Law of Motion. Acceleration is directly proportional to the net force acting on the system. This was also shown by the inversely proportional relationship between the acceleration of the masses and the total sum of those masses. When the total masses remain constant, acceleration increases with the increase in net force acting on the system.
In comparing the difference between the expected value of the mass and the slope value given by the calculation, there is relatively minimal difference between them. This can be corrected when the errors encountered are taking into consideration.
The table shows the results from the calculations.
Mass/Kg
Force/N
Expected value
0.045
0.049
Slope value
Difference Percentage error (%)
0.0453 0.0003
0.0481 0.0009
0.67
1.84
13 | P a g e

ERROR ANALYSIS
Error Calculation
Graph 1
Mean deviation, d = actual slope – expected value d = 0.0453 – 0.045, d =0.0003
Percentage error, e = 𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒
100
Percentage error, 𝑒 =
0.0003
𝑥 100%
0.045
e = 0.67%
Graph 2.
Mean deviation, d = expected value – actual slope d = 0.049 – 0.0481, d = 0.0009 𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒 percentage error, e = 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒
100% percentage error, e =
0.0009
0.049
𝑥100%
e = 1.84%
During the lab, there were some errors that affected the values of the experimental acceleration of the system. During which the system is set in motion. These errors are addressed as follows.
1.
Friction in the pulley.
The pulley contributed some amount of frictional force in the system during when the system is set in motion. This is by when the string used moves around the pulley. The neglection of the frictional force by the pulley affects the values of the experiment.
14 | P a g e

2.
The fact that the mass of the string was ignored.
The string used during the lab has some amount of weight that can be considered during the lab. But due to the neglection of the mass affects the values of the experimental acceleration of the system and thus can be corrected.
3.
The masses of the weight might not have been exact. Considering the weight of the hooks attached to the masses was not identified. The weight could have been verified to eliminate some amount of the error that appear in the calculation.
4.
Air resistance was one of the major problems that was encountered. There was air circulation in the lab which affected the experimental acceleration.
15 | P a g e

PRECAUTIONS
1.
It was ensured that parallax error in reading from the meter rule be avoided, that is when measuring the distance y for the weights to be in motion.
2.
It was ensured that the doors and windows in the lab were closed to reduce air resistance.
3.
It was ensured to reduce the friction in the pulley.
4.
It was ensured that the materials used during the lab were clean and dry.
16 | P a g e

CONCLUSION
The purpose of this lab was to measure the net force on a system using Atwood machine. The net force of the system was calculated when varying the masses and keeping the constant to calculate for the theoretical acceleration. The discrepancies between the expected value and the slope values were relatively minimal for both comparison of the mass and the net force.
The theoretical acceleration is the most accurate compare to the experimental value. This is because the theoretical acceleration values were calculated using the formula from Newton’s second law of motion. (F=ma) to calculate for the acceleration of the system. While the experimental acceleration involves the consideration of the moment of inertial of the pulley.
The additional source of error may have been the string mass, but it is unlikely that the mass would change the results. The frictional force encountered by the pulley constitute some amount of error.
17 | P a g e

REFERENCES
1.
"Atwood's Machine." Atwood's Machine. Ed. Hyper A. Physics. Hyper Physics,
Mar.-Apr. 2007. Web. 27 Oct. 2014.
2.
"Newton's Laws." Newton's Laws. Ed. Physics T. Classroom. The Physics
Classroom, Feb.-Mar. 2001. Web. 25 Oct. 2014.
3.
College Physics Ed. 13 Hugh D. Young.
18 | P a g e