TABLE OF CONTENTS
ABSTRACT……………………………………………………………………..2 OBJECTIVE……………………………………………………………………..3 INTRODUCTION……………………………………………………...………..3 THEORETICAL BACKGROUND……………………………………….……..3 EQUIPMENTS…………………………………………………………..……….6 PROCEDURES………………………………………………………….……….6 RESULTS…………………………………………………………………...……7 SAMPLE CALCULATIONS……………………………………………………9 DISCUSSION…………………………………………………………………...10 CONCLUSION………………………………………………………………….11
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ABSTRACT The experiment was designed to be conducted so as to determine the theoretical and experimental angular acceleration of a big and a small rolling disk. The experimental value is calculated by using a simplified kinematics equation for angular motion, ̈ , while the theoretical value is calculated from the summation of moments using mass moment of inertia ̈ . The experiment was conducted using an inclined plane with formula, ∑ angulometer. The disk was places at the starting point of the inclined plane and released allowing it to roll to another point. The time taken for the disk to roll from rest to the next point was recorded. Using the recorded time the experimental value was calculated. The expected result from this experiment is to initially have the experimental value of the rolling disks to be smaller than the theoretical value and as the angle of inclination is increased experimental values should be bigger as the disks start to slip since the static friction can’t support the increased acceleration. The smaller disk should roll faster than large disk on the same inclined angle because lighter mass and lower mass moment of inertia.
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OBJECTIVE To understand the relationships of mass moment of inertia, angular acceleration, and angular velocity of a disk rolling down an inclined plane.
INTRODUCTION For this experiment to have been conducted, two different size and mass of circular disk were used to roll down an inclined plane so as to determine the angular acceleration and the time taken for the disks to finish its movement. From the data collected, the theoretical and experimental values of angular acceleration Ӫ for both the big and small disk used were determined and the percentage error was calculated for comparison reasons.
THEORETICAL BACKGROUND The motion of a wheel which is rolling along the ground, without slipping, can be considered in one of two ways: either as a rotation of the wheel about an axis through its center of gravity C and an additional translational motion of the entire wheel with the same velocity as the center of gravity,
θ mg cos
mg sin
mg O
Inner Radius r
Figure 1: Rolling Disk Dynamic Modeling as shown in Figure 1 above, or as a rotation of the wheel about an instantaneous axis through the point of contact O between the wheel and the ground. A point in the body which is on the instantaneous axis is momentarily at rest. The instantaneous axis itself moves forward as the wheel moves forward, but it always remains parallel to itself and to the axis through the center of gravity. The angular velocity of the wheel about the instantaneous axis is the same as that about the axis through the center of gravity.
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The general motion of a rigid body may be thought of as made up of two parts: one a motion of translation of the center of gravity, with the entire mass of the body acting as though it were concentrated there, and the other a motion of rotation of the body about an axis through the center of gravity. The angular velocity and angular acceleration are calculated by taking the torques and the moment of inertia about an axis through the center of gravity. The linear velocity and the linear acceleration of the center of gravity are then calculated by considering all the forces as though they acted through the center of gravity, and by applying Newton's second law of motion to this case. Considering the case as shown on Figure-1, the disk of radius r and mass m rolls without slipping down an inclined plane of angle β. There are three forces acting on the disk, its weight mg, the normal force mg cos β perpendicular to the plane and passing through the center of the disk, and the forward moving force mg sinβ acting parallel to the plane at the point of contact O and opposing the frictional force, as illustrated in Figure-1. The instantaneous axis of rotation passes through point O and is perpendicular to the plane of the disk. Summing up the moment acting on the objects above at about contact point, O, According to Beer, F.P. (2007), the summation of moment is: ̈
∑ ( ̈
̈
)
(
)
Another equation is used to determine the experimental angular acceleration, According to Hibbeler, R.C. (2010), the kinematics equations of angular motion: ̇ ̇
̇
̈( ̇
) ̈
̇
̈
The third equation of angular motion is used for this experiment. ̇
̈
̈
̈ (
) 4
where
- Moment about point O - Mass moment of inertia (
)
̈ - Angular Acceleration ̇ - Angular Velocity - Angular Displacement - Mass of disk - Gravitational Acceleration (9.81
)
- Inclination angle - Inner radius - Time taken for disk to roll from a point to another point
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EQUIPMENTS The experiment was two disks of different disks and measurements to be conducted on the apparatus shown on Figure-2 The big disk had a mass of 0.623kg and a radius of 0.05m and the small disk had a mass of 0.319kg and a radius of 0.035m As shown in Figure 2, there are two carriers for the inclined plane, a height adjuster, an angulometer, a meter stick, three bubble levels for the alignment, and two disks, the workpieces. The experiment can be set up based on a simple height adjustment system. The angle is indicated by an angulometer, and the distance is measured directly with a meter stick on the track.
Figure 2: Experimental Apparatus
PROCEDURE 1. By observing the bubble levels, the apparatus was set up in a horizontal position 2. With the help of the height adjuster, the first angle ( ) was set up 3. The big disk was placed at the starting rest position and was release, the stopwatch was started simultaneously 4. When the disk hit the end point, the stopwatch was stopped and the time taken was recorded. The experiment was repeated three times to obtain an average time. 5. Step 3 and 4 were repeated using the small disk to obtain an average time for it. 6. Steps 3-5 were again repeated at four more different angles ( )
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RESULTS
Inclination Angle, 1 3 5 7 9
Inclination Angle, ( )
1 3 5 7 9
()
Time taken for disk to roll down in 1m distance, (s) Big Disk Small Disk
25.90 26.03 25.79 25.91 18.41 18.37 18.37 18.38 14.19 14.55 13.94 14.23 9.83 10.05 9.95 9.94 10.80 11.04 10.77 10.87 7.43 7.40 7.47 7.43 8.70 8.61 8.69 8.67 5.96 6.09 6.23 6.09 7.08 7.07 7.06 7.07 4.83 5.01 5.09 4.98 Table-1: Time taken for disk to roll down in 1m distance, (s)
Theoretical Angular ⁄ ) Acceleration, ̈ (
Experimental Angular ⁄ ) Acceleration, ̈ (
Percentage of Error (%)
Big Disk
Big Disk
Big Disk
Small Disk
0.6848 1.3976 0.5958 1.1840 13 2.0537 4.1911 1.9754 4.0484 3.81 3.4200 6.9796 3.3853 7.2457 1.01 4.7822 9.7595 5.3213 10.7851 11.27 6.1385 12.5275 8.0024 16.13 30.6 Table-2: Percentage error of the angular acceleration of the disks
15.28 3.40 3.81 10.51 28.76
Small Disk
Small Disk
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PERCANTGE ERROR OF BOTH BIG AND SMALL DISK 35
30
PERCENTAGE ERROR (%)
25
20
Big Disk Small Disk
15
10
5
0 1
2
3
4
5
ANGLE OF INCLINATION (DEGREE)
Graph-1: Comparison f percentage error between big and small disk
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SAMPLE CALCULATIONS From Table 1, when
From Table 2, when ,
, (
)(
,
(
)
) ( ̈
)(
)(
)
,
( ̈ ̈
) ̈
̈
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DISCUSSION After the time taken for the disks to roll down the plane had been collected, the theoretical and experimental angular acceleration was calculated after which the results were graphed and tabulated. The graph shows a better presentation of the percentage error between both the disks used. After observing the graph, it is clear that a bigger inclination angle will produce a bigger error. This is because as the angle of inclination is increased, the disk tends to slip faster as the static force cannot support the acceleration hence the value of angular acceleration is bigger in the experimental value. We can also see that initially the small disk has a bigger error than the big disk. This is because the small disk tends to slip first as it achieves higher acceleration faster than the big disk because of the lighter mass and smaller mass moment of inertia. The mass moment of inertia signifies the resistance of an object to rotation.
For this experiment, there are quite a few sources of errors since the percentage of error reached slightly above 30% which is very high. The possible reason of error are as follows: 1. Friction at the contact surface – The smaller circle of the disk faces an opposite frictional force when it rolls down the inclined plane. The friction will slow down the roll and increases the time of roll. Thus, it contributes to the error since the theoretical value does not consider frictional effects. 2. Rusty surface/Wear & Tear – The surface of the rolling disk is discovered to be slightly rusty and increases the surface roughness. Thus, the friction on the disk increases. Wear and tear also occurs when the disk is used multiple times which alters the geometric conditions by a bit. 3. Inclined Plane is not accurately aligned – Using bubble level to check the alignment of the plane is not accurate. When the plane is not aligned correctly, the disk may move sideways and the side of the disk will slide against the vertical planes which produces more friction. 4. Mass distribution/Geometry of the Disk – The mass and geometry of the disk may not be well distributed or accurate which may cause the disk to move sideways. 5. Inclined Plane is not stable – The rolling of the disk produces force on the inclined plane. One side of the plane is only supported with a big bolt. If the plane is not stable, it may vibrate and affects the rolling disk to move sideways. 6. Estimation of starting/ending time – The time when the disk starts to roll at and stops when is only estimated. Small error margin may occur while time taken to take reading varies. 7. Parallax error when taking readings – Some readings use scale such as x distance, angulometer and bubble level which may have parallax errors. The error margin is small.
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8. Slipping during rolling – The disk may slip while it is rolling at high speed. When it slips, the time varies and error is produced.
CONCLUSION The experimental angular acceleration of the rolling disk can be calculated from the summation of moments using mass moment of inertia and from a simplified kinematics equation the theoretical angular acceleration was calculated from the results of the experiment where the disk starts from rest. The major errors are caused from the incorrect selection of the starting point of roll and frictional effects from the contact points. From the experiment, we also know that the angular acceleration is constant and angular velocity keeps increasing as the disk rolls down. Hence, the relationship between mass moment of inertia, angular acceleration and angular velocity is successfully determined in this experiment, if the error source is neglected.
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