Effective Moment of Inertia of A Rotating Pop Can

PURPOSE:

To study the effect of the relative motion between the liquid and the wall of an unopened spinning pop can on its moment of inertia when under a low and high internal pressure.

KEY EQUIPMENT:

• A low friction rotating platform
• Smart Pulley
• Pasco 500 interface system
• Computer

DATA:

Radius for torque r = 5.65 cm
Mass for torque m = 15.2 g
Mass of pop can M = 386.8 g
Radius of pop can R = 3.3 cm
a0 (without pop can) m/s2	a1 (pop can at low pressure) m/s2	a2 (pop can at high pressure) m/s2	Angular Acceleration	Angular Acceleration	Angular Acceleration
a0 (m/s2)	a1 (m/s2)	a2 (m/s2)	a0 (rad/s2)	a1 (rad/s2)	a2 (rad/s2)
1.72	1.48	1.35	30.48	26.14	23.93
1.77	1.49	1.36	31.26	26.43	24.12
1.71	1.53	1.39	30.22	27.15	24.62
1.72	1.53	1.43	30.35	27.06	25.38
1.72	1.54	1.39	30.41	27.26	24.65
1.71	1.47	1.28	30.28	26.07	22.73
1.74	1.48	1.42	30.87	26.16	25.15
1.70	1.46	1.40	30.12	25.91	24.69

Torque: t	Moment of Inertia: I0	Moment of Inertia: I1	Moment of Inertia: I2	Low Pressure: Ilow	High Pressure: Ihigh)
0.008416	0.000276	0.000322	0.000352	4.58E-05	7.56E-05
0.008416	0.000269	0.000318	0.000349	4.9E-05	7.96E-05
0.008416	0.000279	0.00031	0.000342	3.14E-05	6.33E-05
0.008416	0.000277	0.000311	0.000332	3.37E-05	5.43E-05
0.008416	0.000277	0.000309	0.000341	3.2E-05	6.46E-05
0.008416	0.000278	0.000323	0.00037	4.49E-05	9.24E-05
0.008416	0.000273	0.000322	0.000335	4.91E-05	6.2E-05
0.008416	0.000279	0.000325	0.000341	4.54E-05	6.15E-05
	Average Moment of Inertia			4.14E-05	6.92E-05
	Percentage of Solid Moment of Inertia			19.7%	32.8%

Theoretical Moment of Inertia

For a solid pop can:
I_theory = M R² = 21.1 x 10^-5 kg m²

I_experimental = 23.7 x 10^-5 kg m²

Moment of Inertia (low pressure): I_L = 4.14 x 10^-5 kg m²

Moment of Inertia (high pressure) I_H = 6.92 x 10^-5 kg m²

DISCUSSION / ANALYSIS:

If a pop can (unopened) is spun about its axis of symmetry , there is relative motion between the container and the liquid inside. The liquid inside rotates with smaller angular speed and also there is an angular speed gradient from the wall to the center of the container. Hence the effective moment of inertia of the pop can is less than if it were solid.

Now if the pop can is shaken up so that a lot of CO₂ is released, tremendous pressure is built up inside and the liquid is pushed to the wall with greater force increasing the frictional force. Thus when the pop can is spun under high internal pressure, the liquid follows the container more closely and hence effective moment of inertia of the container increases (though still less than that of the solid can).

This effect can be verified by doing an experiment with the set up shown in he diagram above.

A known torque is applied to a platform which can rotate about a vertical axis. Data can be collected by a lab interface such as smart pulley. Linear acceleration and hence the angular acceleration is then obtained. The moment of inertia of the platform (without pop can on it) is thus given by I₀ = t/a₀.

Now a pop can is placed on the platform coaxial with it. The liquid is at low pressure. Above procedure is repeated to determine the combined moment of inertia I₁ of the platform + pop can at low pressure.

Now the pop can is shaken till no splashing sound of the liquid inside is heard an the procedure above is quickly repeated to determine the combined moment of inertia I₂ of the platform + pop can at high pressure.

Thus I_L = I₁- I₀ is the effective moment of inertia of the pop can at low pressure and I_H = I₂- I₀ is the effective moment of inertia of the pop can at high pressure.

REFLECTIONS:

This lab is appropriate for AP Physics C students and probably with honors H.S. Physics students, too.

Various possible modifications/extensions of the lab are:

1. Oscillating the whole system: Take the string going over the smart pulley small enough so that the string around the platform unwinds completely before the mass m hits the floor. The string will rewind due to the inertia of the system and oscillations will take place. Obtain a graph of either displacement or velocity versus time for the cases of low and high pressure. Compare the two graphs and derive your conclusions from it.

2. Try pop cans of different brands of soft drinks and see if there is any remarkable difference in the results.



Above is the graph of one such trial. The results are inconclusive though.

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Created by Hasan Fakhruddin July 1997 for Project PHYSLab.