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3

The answer to that is because the moment of inertia is not the same for the solid cylinder than for the hollow one. As you write the formula for the moment of inertia, it depends on the distribution of the mass. The further away the mass is from the rotation axis, the more contributes to the moment of inertia (as in distance squared $r^2$). So, since the ...

2

Here, in the above picture $M \ge m$. Note that this is the most general case. We can have $M = m$ and the angle $\theta$ can vary anywhere between $[0;\cfrac{\pi}{2}]$ (Actually, the most general case would have been to take 4 different masses but we will be going out of the bandwidth of your problem, and it would be a pointless discussion.) Now, ...

2

usually linear velocity is the velocity of a point rotating around the axis of rotation given by $$\vec v = \vec{\omega} \times\vec{r}$$ when object has no translational motion but if the object has both translational and rotational motion then $\vec v$ will be measured from Center-of-momentum frame. in the frame where the C.M is translating at velocity ...

1

Suppose we have a particle moving with a velocity $\vec{v}_1$ and a object which has no linear velocity, but is rotating with angular velocity $\vec{\omega}$. Now suppose we look at a point at a displacement $\vec{r}$ from the axis of rotation of this rotating velocity. The velocity of this point is $\vec{v}_2 = \vec{\omega} \times \vec{r}$. Now the ...

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