# ATWOOD MACHINE: UNIFORM ACCELERATION MOTION

```KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY
COLLEGE OF SCIENCE
FACULTY OF PHYSICAL AND COMPUTATIONAL SCIENCE
DEPARTMENT OF PHYSICS
EXPERIMENT TITLE: THE ATWOOD MACHINE: UNIFORMLY ACCELERATED MOTION
COURSE: EXPERIMENTAL PHYSICS 1
REFERENCE NUMBER: 20605173
EMAIL: [email protected]
1. Abstract
2. Introduction
3. Theory
4. Diagram of set up
5. Method/Procedure
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
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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.
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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.
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/s2). 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.
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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 (m1 and m2). The forces acting on the two
masses are mainly their weight, m1g and m2g. 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 = m1 g – m2g = (m1 -m2) g …………………………………………………………………..…………. (1)
𝐹NET = ma = (m1 +m2) a ……………………………………………………………………………………………… (2)
Combining (1) and (2)
𝑎=
(m1 −m2) g
(m1 +m2)
……………….…………………………………………………… (3)
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 = mR2
So
𝐹 NET = m1g – m2g – mfg………………………………………..………………… (4)
Where mf is the mass added to the mass to determine the length of the string.
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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)
(m1 +m2)
𝑔……………………….…………. (5)
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)
(m1 +m2+mf)
𝑔…………………………………………………………………………………………..…..… (6)
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
1
1
2
2
𝑦 = 𝑎𝑡 2 from the 2nd equation of motion. 𝑦 = ut + 𝑎𝑡2
At rest, the initial velocity is zero, u = 0
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DIAGRAM OF THE SET UP
THE ATWOOD MACHINE
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METHOD/PROCEDURE
PRELIMINARY MEASUREMENT
1. Put masses on the hangers so that both m1 and m2 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 mf. Record this mass, then remove it from the system so that m1 and m2 are
equal again.
3. With m2 on the floor and m1 near the pulley, measure the distance y from the
bottom of m1 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 m1 so that m1= 25g and m2 = 20g.
5. Using the stop watch, measure the time it takes for m1 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 m2 to m1. 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 m1 = 25g and m2 = 20g, repeat steps 5-7 as before.
10. Repeat steps 5-7 four more times, each time adding 5g to both m 1 and m2. (The
difference between m1 and m2, which determines the net force, therefore remains
constant).
ANALYSIS
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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.
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
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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
M1 /g
M2 /g
T1
T2
T3
0.025
0.020
1.91
1.92
1.93
Average Experimental Theoretical 𝐹 NET
Time
Acceleration Acceleration
T
1.92
0.65
1.08
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
2.40
2.83
0.127
0.031
0.014
0.91
0.90
0,86
0.89
3.27
3.70
0.167
0.033
0.012
0.85
0.79
0.77
0.80
4.14
4.57
0.206
3. TOTAL MASS VARYING AND NET FORCE CONSTANT
M1 /g
M2 /g
T1
T2
T3
0.025
0.020
1.91
1.92
1.93
Average Experimental Theoretical 𝐹 NET
Time
Acceleration Acceleration
T
1.92
0.65
1.08
0.O49
0.030
0.025
2.04
2.01
2.05
2.03
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0.54
0.89
0.049
0.035
0.030
2.28
2.29
2.31
2.20
0.45
0.75
0.048
0.040
0.035
2.34
2.40
2.40
2.38
0.39
0.65
0.O49
0.045
0.040
2.80
2.85
2.71
2.78
0.35
0.58
0.O49
GRAPH 1
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Force Vs Acceleration
0,25
y = 0,0453x - 0,0005
0,2
Net Force
0,15
0,1
0,05
0
0
0,5
1
1,5
2
2,5
Acceleration
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3
3,5
4
4,5
5
GRAPH 2
Acceleration Vs 1/Mass
1,2
y = 0,0481x + 0,0115
1
Acceleration
0,8
0,6
0,4
0,2
0
0
5
10
15
1/Mass
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20
25
THEORY AND CALCULATIONS
Percentage difference between theoretical and experimental acceleration.
Table 2.
Percentage difference, 𝑑 =
𝑑=
Theoretical−Experimental
Theoretical+Experimenta
2.828−2.396
5.224
𝑥 100%
𝑥 100%
d = 8.3%
Table 3.
Percentage difference, 𝑑 =
0.784−0.476
1.26
𝑥 100%= 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 =
S=
S=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑛𝑒𝑡 𝑓𝑜𝑟𝑐𝑒
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
0.167−0.088
3.70−1.96
0.079
1.740
S = 0.0453
Hence from the slope, the mass of the system is M = 0.0453Kg
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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
0.89
15.38
0.75
13.33
0.65
11.76
0.58
From graph 2
The slope, S of the graph is given as , S =
S=
S=
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 1/𝑀𝑎𝑠𝑠
1.08−0.65
22.22−13.33
0.43
8.89
S = 0.0481
Hence from the slope, the net force on the system, 𝐹 NET = 0.0481N
RESULTS AND DISCUSSION
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
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theoretical acceleration is 2.828m/s2. It is seen, from the calculations above, that the
experimental acceleration (2.396m/s2) was very close to the theoretical accelerations
(2.828m/s2). 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
0.0453
0.0481
Difference Percentage
error (%)
0.0003
0.67
0.0009
1.84
ERROR ANALYSIS
Error Calculation
Graph 1
Mean deviation, d = actual slope – expected value
d = 0.0453 – 0.045, d =0.0003
Percentage error, e =
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𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒
𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒
× 100
Percentage error, 𝑒
=
0.0003
0.045
𝑥 100%
e = 0.67%
Graph 2.
Mean deviation, d = expected value – actual slope
d = 0.049 – 0.0481, d = 0.0009
percentage error, e =
percentage error, e =
𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒
𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒
0.0009
0.049
× 100%
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
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.
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.
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4. Air resistance was one of the major problems that was encountered. There was
air circulation in the lab which affected the experimental acceleration.
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.
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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.
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.
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3. College Physics Ed. 13 Hugh D. Young.
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