Angela Leong Feng Ping M13608
PC6402 Practical: Inertia Wheel – Lab Report Introduction For a rigid body when r
p = 0, or when the center of mass is taken as the origin – origin – both both of which apply to the
inertia wheel – wheel – the the torque is given by 2
------- (1) -2
where I is the moment of inertia of the wheel in kg∙m kg ∙m , and α is its angular acceleration in rad∙s . The moment of inertia is a measure of the resistance of an object to changes in its rotational motion. The moment of inertia of a uniform solid disk is
------- (2)
where M is the mass of the disk in kg and R is the radius of rotation (radius of the disk) in meters. The relationship between linear acceleration and angular acceleration of a point is given by
------- (3)
where r is the distance between the point and the origin of rotation, which is equal to the radius of the circle the point traces as it rotates. In this experiment, a mass is tied to a string wound around the pulley of the inertia wheel, and when released, the mass accelerates to the floor, unwinding the string which causes the inertia wheel to rotate. In this case, the radius r in equation (3) is the outer radius of the pulley, given by r2.
Experimental Procedure The inner and outer diameters of the pulley were measured with a pair of vernier calipers, and the radius of the disk measured with a meter ruler. The disk was mounted to a metal frame at approximately eye level. A string long enough to reach the floor was fixed to the edge of the pulley, and the other end of the string was attached to a 50 gram (g) weight holder, which acts as the trip platform. The string was wound onto the pulley by rotating the disk and pulley anticlockwise, until the base of the weight holder was 1 meter above the floor (measured with the meter ruler), with the string taut. Markings were made on the string to facilitate winding the weight holder back to the same position during repeat trials.
A series of weights (5, 10, 20 and 50 g) was provided. A 20 g weight was added to the weight holder so that the mass attached to the string was 70 g. The trip platform was released from 1 meter above the floor and the stop watch was simultaneously started. The stop watch was stopped when the trip platform reached the floor, and the time t taken for the weight holder to travel to the floor was recorded. This measurement was repeated four times to obtain a total of five measurements. The procedures in the second paragraph were repeated for ten different combinations of masses, each with an increment of 20 g (to list: 70 g, 90 g, 110 g, 130 g, 150 g, 170 g, 190 g, 210 g, 230 g, and 250 g).
Results & Discussions Mass of disk, M = 1295.2 g = 1.2952 kg (given) Mass of pulley, M p = 99.2 g = 0.0992 kg (given)
Outer diameter of pulley = 0.0697 m 0.0001 m Inner diameter of pulley = 0.0604 m 0.0001 m (measured with vernier calipers)
Distance from trip platform to floor, h = 1.000 m Radius of disk, R = 0.118 m 0.001 m (measured with meter ruler)
0.001 m
= 0.0349 m 0.0001 m Inner radius of pulley, r = = 0.0302 m 0.0001 m Outer radius of pulley, r 2 = 1
(measured with vernier calipers) Theoretical moment of inertia of the inertia wheel,
( ) ( )( ) ( )( ) ( ) ( ) ( ) ) () √ ( ) ( Uncertainty in Uncertainty in Uncertainty in Uncertainty in
Table 1: Experimental data Mass (g) 70 90 110 130 150 170 190 210 230 250
t 1 4.78 4.34 3.91 3.56 3.28 3.19 3.00 2.84 2.72 2.56
Time taken for weight holder to travel to the floor, t (s) t 2 t 3 t 4 t 5 4.97 4.81 5.03 5.03 4.31 4.31 4.31 4.38 3.90 3.87 3.91 3.88 3.63 3.66 3.63 3.65 3.31 3.41 3.40 3.38 3.15 3.16 3.16 3.15 3.00 2.91 3.00 2.97 2.84 2.88 2.81 2.81 2.72 2.69 2.78 2.75 2.69 2.62 2.62 2.63
Mean t 4.92 4.33 3.89 3.63 3.36 3.16 2.98 2.84 2.73 2.62
Table 2: Extrapolation of experimental data Mass (g) 70 90 110 130
Mean t (s) 4.92 4.33 3.89 3.63
-2
a (m∙s ) 0.0825 0.107 0.132 0.152
-2
α (rad∙s
2.37 3.06 3.78 4.36
)
T (Newtons) 0.680 0.872 1.06 1.25
τ (N∙m)
0.0237 0.0304 0.0371 0.0437
150 3.36 0.178 5.10 1.44 0.0503 170 3.16 0.200 5.74 1.63 0.0569 190 2.98 0.226 6.48 1.82 0.0634 210 2.84 0.249 7.14 2.01 0.0699 230 2.73 0.268 7.69 2.19 0.0764 250 2.62 0.290 8.33 2.34 0.0829 (t = mean time for the repeated trials, a = acceleration, α = angular acceleration, T = tension, τ = tor que) Table 3: Error Analysis Mass (g) Mean t (s) (by standard deviation) 0.108 0.0276 0.0162 0.0350 0.0516 0.0147 0.0350 0.0258 0.0306 0.0413
70 90 110 130 150 170 190 210 230 250
-2
a (m∙s ) (by error propagation) 0.00366 0.00136 0.00111 0.00294 0.00547 0.00187 0.00531 0.00453 0.00601 0.00914
Error -2 α (rad∙s ) (by error propagation) 0.105 0.0391 0.0318 0.0843 0.157 0.0537 0.153 0.130 0.172 0.262
T (Newtons) (by error propagation) 0.0301 0.0111 0.00894 0.0242 0.0444 0.0433 0.0428 0.0365 0.0492 0.0748
τ (N∙m)
(by error propagation) 0.00105 0.000388 0.000312 0.000845 0.00155 0.000532 0.00149 0.00127 0.00171 0.00261
From the hand-drawn graph, y-intercept, c = 0.0005 N∙m Gradient,
) (( )
Error propagation in subtraction
:
Error propagation in subtraction
:
Propagated error in
) ( ) √ (
Since the equation of a straight line is given by y = mx + c, where m is the gradient and c is the y-intercept, the graph of torque τ against angular acceleration
α,
plotted from experimental data, is a straight line with an
equation τ
2
= (0.00985 kg∙m )α + 0.0005 N∙m
Comparing with the formula (0.0005
where τ is in N∙m and α is in radians , the value of the y-intercept is so small it can be approximated to be zero
0), in which case the experimental moment of inertia of the inertia w heel is
I = 0.00985 kg∙m
2
The percentage error is
My experimental value of the moment of inertia of the wheel is larger than the theoretical value by 8.00%, a small value, implying that my results are accurate to a large extent. This 8.00% is also very close to the calculated uncertainty of
, thus the error is expected as it falls very close within the
uncertainty range of the results. The graph supports the high level of accuracy of the results, with the data points lying on or very close to the straight line, and small error bars. The discrepancy can be attributed to experimental factors. Firstly, there are limitations with the measuring devices and thus uncertainties associated with the measured values. This leads to propagated errors. Because this experiment involves much extrapolation of the raw data, these small and seemingly negligible uncertainties may propagate to significant errors, especially in the derivation of the torque and angular acceleration. Secondly, the mass tends to swing side-to-side and rotate (in the horizontal plane) where the string is attached to the weight holder, and the string is slightly elastic, adding additional torsional forces and energy due to the elasticity of the string and the oscillations of the mass, which are not taken into account in the calculations. Thirdly, there is some friction between the pulley and the axle where the pulley is connected to the stand, which opposes the rotation of the inertia wheel. Lastly, there are experimental errors like the reaction time in starting the stopwatch at exactly the point when the trip platform is released, and stopping the stop watch at the exact point the weight holder reaches the floor. As the time taken for the weight holder to travel to the floor is rather short, the reaction times can lead to significant errors in the time measured. Further, it is difficult to release the trip platform from exactly 1.000 m above the floor for every trial. The results can be improved by changing to a less elastic string, oiling the axle to reduce the friction between the pulley and the axle, and conducting the experiment in pairs to facilitate winding the trip platform back to the same height after every trial. The frictional torque acting on the inertial wheel T f is due to the friction between the pulley and the axle where it is connected to the stand. This frictional torque acts anticlockwise as the wheel rotates clockwise, reducing the torque on the wheel due to the tension T in the string. Thus the net torque is given by τ
= Tr 2 – τ f = Iα
Rearranging,
Tr 2 = Iα + τ f
Comparing with the equation of the gr aph,
Tr2 = (0.00985 kg∙m ) α + 0.0005 N∙m
2
The best experimental value of the friction torque acting on the inertia wheel is τf = 0.0005 N∙m
Conclusion Despite limitations with equipment and experimental errors, the results show a very strong linear relationship between the torque and angular acceleration of the inertia wheel, that corroborates strongly with the theoretical formula τ = Iα. The moment of inertia for the inertia wheel used in this experiment is experimentally 2
determined to be 0.00985 kg∙m , with a percentage error of 8.00%.
