What is the Moment of Inertia of a Cylinder with a Mass Hanging from a Cord?

In summary, the conversation discusses the calculation of the moment of inertia for a cylinder with a radius of 20cm, mounted on an axle and free to rotate. The calculation involves using the torque equation and adding it to the equation for acceleration. The final answer should be 1.4kg.m^2 but the person is struggling to find where they went wrong in their calculation. Suggestions are made to check the average velocity and initial velocity used in the calculation.
  • #1
John O' Meara
330
0
A cylinder of radius 20cm is mounted on an axle coincident with its axis so as to be free to rotate. A cord is wound on it and a 50-g mass is hung from it. If after being released, the mass drops 100cm in 12s, find the moment of inertia of the cylinder? Take down as positive.
Torque=I*(alpha), i.e., b*T=I*(alpha) => T=I*(alpha)/b ...(i)
m*g - T=m*a ...(ii), on adding i and ii, we get
I=m*b(g-a)/(alpha); now alpha=a/b, where a=acceleration, b=radius, T=tension, I=moment of inertia and s=1m.
v_avg=(v+u)/2=.04166 m/s: v^2=u^2+2*a*s therefore a=-u^2/2/s=-8.681*10^-4. Hence I=225.78 kg.m^2.
I also did the calculation from the energy point of view and got the same answer. Yet the answer in the book is 1.4kg.m^2. Can anyone show me where I went off track? Thanks very much.
 
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  • #2
v_avg=(v+u)/2=.04166 m/s: v^2=u^2+2*a*s therefore a=-u^2/2/s=-8.681*10^-4. Hence I=225.78 kg.m^2.

Looks like you used the average velocity for the final velocity in determining 'a.'
 
  • #3
Noein said:
Looks like you used the average velocity for the final velocity in determining 'a.'
Yes. And that calculated average velocity is wrong by a factor of 2, so the v used is off by a factor of 4.
 
  • #4
I used the initial value for the velocity v, and now I get I=56.46kg.m^2; which is still the wrong answer. So I wounder what else is wrong? Any help would be welcome.
 
  • #5
John O' Meara said:
I used the initial value for the velocity v, and now I get I=56.46kg.m^2; which is still the wrong answer. So I wounder what else is wrong? Any help would be welcome.
What initial value would that be? The initial velocity is zero. The final velocity at the end of 12 seconds can be calculated from the average and initial velocities, and the average velocity can be calculated from the distance and the time. Show us the calculation you did using what you believe to be the correct velocity. Then we can find your mistake.
 
  • #6
The torque=I*(alpha); b*T=I*(alpha) therefore T=I*(alpha)/b
m*g - T = m*a therefore I=m*b^2(g/a-1) where (alpha)=a/b; v_avg=1/12 = 8.333*10^-2; therefore v=2*v_avg=16.666*10^-2m/s;
a=16.666*10^-2/12s=1.388*10^-2m/s/s.
I=.04*.05(9.8/1.388*10^-2 - 1) = 1.41kg.m^2. Much thanks for your help OlderDan.
 

1. What is moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the shape, mass, and distribution of mass of the object.

2. How is moment of inertia calculated?

Moment of inertia is calculated by multiplying the mass of each particle in an object by the square of its distance from the axis of rotation, and then summing these products for all particles in the object.

3. What are the units of moment of inertia?

The units of moment of inertia depend on the units of mass and distance used in the calculation. In the SI system, the units are kilogram-meter squared (kg·m²).

4. How does moment of inertia affect an object's rotational motion?

The moment of inertia determines how easy or difficult it is to change an object's rotational motion. Objects with larger moments of inertia require more torque to accelerate or decelerate, while objects with smaller moments of inertia will rotate more easily.

5. Can moment of inertia be negative?

No, moment of inertia can never be negative. It is always a positive value, as it represents an object's resistance to changes in its rotational motion. Negative values would not make physical sense in this context.

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