DEPARTMENT OF BIOMEDICAL ENGINEERING FACULTY OF ENGINEERING UNIVERSITY MALAYA
Lab 2 : Inertia of Rigid Body
OBJECTIVE The main objective of this experiment is to identify the relationship between the moment of inertia to mass and its radius. Besides that, this experiment also investigates about how the shape of the object will affected its moment of inertia.
INTRODUCTION Moment of inertia, which also known as mass moment of inertia, rotational inertia, polar moment of inertia, or the angular mass. The algebraic representative for 2 moment of inertia is I, and it having a unit of kg m . Moment of inertia(I) is actually use to measure resistance of an object to changes its rotation motion. As to calculate nd nd the moment of inertia(I), Newton ’s 2 Law are being used. Newton’s 2 Law says how a net forces cause accelerations. This law applied to rotational motion as well. Besides that, the moment of inertia of an object is being affects by few quantities such as the mass, radius(distance) and also time. The moment of inertia also will be affected by the shape of the object. Principles of measurement As to calculate the moment of inertia, a set of apparatus to measure inertia in rotational motion(TM 610) is needed. the picture below shows the apparatus.
Figure 1 : Apparatus for measure moment of inertia(TM 610). Adapted from Lab Sheet. As what the Figure 1 had shown, the drive weights will being pulled by the gravitational force, it will bring the system to an accelerating circular motion. Time travel of the driver weights from a fixed height to the ground are being measure using a stopwatch. The time is needed to calculate the inertia of motion of the object. The equations to calculate the inertia of motion are as follows, 1. Moment of inertia Icorrected = Iexperiment - Io
Note: Icorrected – moment of inertia calculated from experiment data but had been corrected with inherent moment of inertia. Iexperiment – moment of inertia calculated from the experimental data. Io – Inherent moment of inertia for rotational axle and pipe, is a constant with -3 2 value of 3.3 x 10 kg m 2. Moment of inertia calculated from experimental data The formula used to calculate the experimental moment of inertia is, Iexperiment = (
mgr 2h
)t 2
3. Equations for calculating the moment of inertia depending on the type of cylinder. 1
2
Solid cylinder : Ix = md 8
1
2
2
Hollow cylinder : Ix = m(D + d ) 8
Figure 2 : Solid cylinder and hollow cylinder. Adapted from Lab Sheet.
EQUIPMENT 1. Apparatus for investigating inertia in rotational motion (TM 610) 2. Digital Stop Watch (can replace with hand phone having the function of stop watch) 3. A steel meter ruler 4. A set of weights (100g, 200g and 400g)
PROCEDURE
Figure 3 : Mounting the pin-point masses A. Pin Point Mass : Moment of inertia as a function of mass 1. The set of apparatus for investigating inertia of rotational motion (TM 610) are being set up on a table. 2. The thin walled pipe is inserted to the apparatus TM 610 by placing it in the center of the rotational axle. 3. The two 100g of weights are fastened to both side of the pipe. 4. The effective radius R is fixed at 20cm. 5. A 100g of drive mass are being use for all experiments in this section. 6. The drive mass is adjust to the height of 40cm from the ground for each experiments and the lapsed time(t) is being calculate by using the stop watch. At the moment of the drive mass is start dropping, the stop watch is started and is stopped when the drive mass had reach the ground. 7. The experiments are repeated three times using the same heights in order to limit error in time measurement. After three trials, the experiments are carried on by replacing the two 100g weights with 200g and 400g. 8. The average time obtained is recorded into the result ’s table. The I experiment and Icorrected are being calculated and graphs are being plot as to identify the relationship of rotational inertia to the mass. B. Pin Point Mass : The moment of inertia as a function of radius 1. As in Part A, the pipe is being fixed to the center of the rotational axle. 2. Two 400g of weights are being fixed on the pipe by placing each of the weights on each sides. 3. The effective radius (R) is set sequentially at 0.055m, 0.095m, 0.155m and 0.245m.
4. Besides that, the drive mass for this part experiments are fixed to 100g as well. 5. The drive mass is adjust to the height of 40cm from the ground for each experiments and the lapsed time(t) is being calculate by using the stop watch. At the moment of the drive mass is start dropping, the stop watch is started and is stopped when the drive mass had reach the ground. 6. The experiments are repeated three times using the same heights in order to limit error in time measurement. After three trials, the experiments are carried on by adjusting the radius R as mention ion step 3. 7. The average time obtained is recorded into the result ’s table. The I experiment and Icorrected are being calculated and graphs are being plot as to identify the relationship of rotational inertia to the radius. C. Comparative Investigation : Solid cylinder-hollow cylinder 1. The hollow cylinder is fastened to the rotational axle and the pipe is being removed. 2. For this part of experiments, a drive mass with 100g are being used. 3. The drive mass is adjust to the height of 40cm from the ground for each experiments and the lapsed time(t) is being calculate by using the stop watch. At the moment of the drive mass is start dropping, the stop watch is started and is stopped when the drive mass had reach the ground. 4. The experiments are repeated three times using the same heights in order to limit error in time measurement. After that, the hollow cylinder is replaced with solid cylinder and the experiment is repeated. 5. The average time obtained is recorded into the result ’s table. The I experiment and Icorrected are being calculated and graphs are being plot as to identify the differences between the hollow cylinder and solid cylinder.
Figure 3 : Hollow cylinder(left) and solid cylinder(right)
RESULTS A. Pin Point Mass : Moment of inertia as a function of mass 2 Weight (kg) Time (t) Avg Time (t) Iexperiment (kgm ) 0.100 4.9 5.0 0.0613 5.2 4.9 0.200 6.5 6.5 0.518 6.6 6.4 0.400 8.2 8.37 0.859 8.2 8.7 B. Pin Point Mass : The moment of inertia as a function of radius 2 Radius (mm) Time (t) Avg Time (t) Iexperiment (kgm ) 55mm 3.6 3.57 0.156 3.5 3.6 95mm 4.8 4.73 0.274 4.8 4.6 155mm 6.8 6.87 0.579 6.9 6.9 245mm 10.3 10.33 1.309 10.4 10.3 C. Comparative Investigation : Solid cylinder-hollow cylinder Cylinder Time (t) Avg Time (t) Iexperiment Icorrected 2 2 (kgm ) (kgm ) Hollow 2.6 2.67 0.0874 0.0841 2.7 2.7 Solid 1.9 1.93 0.0457 0.0424 2.0 1.0 Pin Point Mass : Moment of inertia as a function of mass
2
Icorrected (kgm )
0.0580
0.515
0.856
2
Icorrected (kgm )
0.153
0.271
0.576
1.306
I(from equation) 2 (kg m )
0.00323
0.0018
Iexperimental Vs Mass 1 0.9
) 2 0.8 m g 0.7 k ( a i t r 0.6 e n I 0.5 f o 0.4 t n e 0.3 m o 0.2 M
0.1 0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.3
0.35
0.4
0.45
Mass(kg)
Iexperimental Vs Mass 1 0.9
) 2 0.8 m g 0.7 k ( a i t r 0.6 e n I 0.5 f o 0.4 t n e 0.3 m o 0.2 M
0.1 0 0
0.05
0.1
0.15
0.2
0.25
Mass(kg)
Pin Point Mass : The moment of inertia as a function of radius
Iexperimental Vs Radius 1.4 1.2 ) 2 m g 1 k ( a i t r 0.8 e n I f o 0.6 t n e m0.4 o M 0.2 0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.25
0.3
Effective Radius(m)
Iexperimental Vs Radius 1.4 1.2 ) 2 m g 1 k ( a i t r 0.8 e n I f o 0.6 t n e m0.4 o M 0.2 0 0
0.05
0.1
0.15
0.2
Effective Radius(m)
DISCUSSION Explanation 1. Based on the results and the graphs obtained, we can conclude that the moment of inertia of the rigid body is directly proportional to the weights pinned on the thinpipe. From the graph, we can see that as the weights is slowly increase, the time travel by the driver weights are increased as well. This further explain that the moment of inertia of the rigid body increase as well. This will happen because as the weights increase, the thin pipe will become more heavy and harder to rotate. Therefore, more time will be
needed for the drive weights to trave until it reached the ground. Thus, as results the moment of inertia of the rigid body will increase. 2. Based on the results and the graphs obtained, we can conclude that the moment of inertia of the rigid body is directly proportional to the effective radius. From the graph, we can see that as the effective radius is slowly increase, the time travel by the driver weights are increased as well. This further explain that the moment of inertia of the rigid body increase as well. This will happen because as the effective radius increase, the thin pipe will become harder to rotate. Thus, the time needed for the drive weights to travel to the ground will be longer. This will, as result increases the moment of inertia of the rigid body. 3. As for hollow and solid cylinder, the results shows two readings with huge differences. The moment of inertia of a hollow cylinder is almost twice as much as the moment of inertia of solid cylinder. The reading for hollow 2 cylinder is 0.0841 kgm and the moment of inertia of solid cylinder is 0.0424 2 kgm . However, these two readings are far too differences with the values calculate using the general equations provided. The I from equation for hollow 2 cylinder is 0.00323 kgm and moment of inertia of solid cylinder is 0.0018 2 kgm . One of the reason why the experimental value of the moment of inertia is so large differences with the value calculated from the general equation is because the general equation is assuming that all the particles of the cylinder are being distributed evenly. The results had tell us that the particles of the cylinder are actually not distributed evenly. Therefore, the results will be having differences when being compared. Precautions 1. Systematic errors such as zero errors should be avoid when measuring the height of the drive weights from the ground. This is to ensure that correct readings can be obtained. 2. Random errors such as reaction time can be limited by repeat the experiments few times. This can enable us to find out the average travel time of the driver weights.
CONCLUSION As conclusion, we can say that the moment of inertia of a rigid body is actually being affected by two variables, mass and the effective radius. Both of this variable are directly proportional to the moment o inertia of the rigid body. In other words, the moment of inertia are increases as the mass increase or as the effective radius increase. Besides that, the hollow cylinder has the larger moment of inertia when compared with the solid cylinder. The value of moment of inertia of the hollow cylinder are twice as much as the moment of inertia of the solid cylinder. From 2 general equation, the moment of inertia of a hollow body is 0.00323 kg m and the 2 moment of inertia of a solid cylinder is 0.0018 kg m .
REFERENCES 1. R. C. Hibbeler. (2004). Planar Kinetics of a Rigid Body: Force and Acceleration. th
Engineering Mechanics - Dynamics (10
edition). (pp. 377-378). United State of
America: Pearson. 2. http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html
3. http://www.newton.dep.anl.gov/askasci/phy00/phy00124.htm
