- #1
etotheipi
For instance, in the case of a simple pendulum, it is quite acceptable to write down ##-mg\sin{\theta} = ma_{tangential}##, and go from there.
However, if we introduce a rotating body which is not a particle, we may still calculate its torque from its centre of mass, however we can no longer resolve tangentially to solve for the tangential acceleration (presumably of the centre of mass). Instead, it is required to calculate the body's moment of inertia and proceed with ##\tau = I\alpha##.
I'm not sure about how I could go about proving this mathematically; I wonder if it has something to do with the added rotation of the non point-like body during the motion (since, from an energy perspective, we will end up with some ##\frac{1}{2} I \omega^{2}##, which will "reduce" the velocity of the centre of mass since we now have less ##\frac{1}{2} m v^{2}##?). If so, how might I start to prove this with a force, as opposed to energy, approach? Thank you!
However, if we introduce a rotating body which is not a particle, we may still calculate its torque from its centre of mass, however we can no longer resolve tangentially to solve for the tangential acceleration (presumably of the centre of mass). Instead, it is required to calculate the body's moment of inertia and proceed with ##\tau = I\alpha##.
I'm not sure about how I could go about proving this mathematically; I wonder if it has something to do with the added rotation of the non point-like body during the motion (since, from an energy perspective, we will end up with some ##\frac{1}{2} I \omega^{2}##, which will "reduce" the velocity of the centre of mass since we now have less ##\frac{1}{2} m v^{2}##?). If so, how might I start to prove this with a force, as opposed to energy, approach? Thank you!