Physics System of Particles and Rotational Motion

Even if the torque is same, the rotation of a body can be slow or fast depending on the distribution of mass along the surface area. If the mass of a body is more on the axis area, it's speed of rotation will be faster as compared to the body whose mass is accumulated away from the axis due to which the body will resist the angular acceleration.

In simple words, a push or pull applied on a body is known as a force. Force is a concept used in terms of linear motion. On the other hand, the force which causes a body to spin around an axis instead of moving from one point to aother is called torque. Torque is a theory which comes into account while studying rotational motion and is considered as the rotational analogue of force.

Radius of gyration of a solid surface,

${\mathrm{K}}_{\mathrm{S}}=\sqrt[]{\frac{2}{5}\mathrm{R}}$

Radius of gyration of a hollow surface,

${\mathrm{K}}_{\mathrm{H}}=\sqrt[]{\frac{2}{3}}\mathrm{R}$

$\Rightarrow \frac{{\mathrm{K}}_{\mathrm{S}}}{{\mathrm{K}}_{\mathrm{H}}}=\sqrt[]{\frac{3}{5}}=\frac{\sqrt[]{3}}{\sqrt[]{5}}$

μ_min = (I tanθ)/ (I + mR²)
= [ (mR²/2)tanθ] / [mR²/2 + mR²] = (tanθ)/3 = (tan 60°)/3 = √3/3 = 1.732/3 = 0.5773

Since given coefficient of static friction is less than μ_min, so body will perform rolling with slipping and kinetic friction will act
F? = μN = μmg cosθ = (0.4) * mg cos 60° = mg/5

The x and y coordinates of the center of mass are given by:
x? = y? = 4a / 3π

Using conservation of Angular momentum along axis of rotation, we can write
Mr²ω = (Mr² + 2mr²)ω? ⇒ ω? = Mω / (M + 2m)

The number of revolutions can be found using the rotational kinematic equation for angular displacement (θ):
θ = (ω_initial + ω_final)/2 * t
Number of revolutions = θ / 2π
Number of revolutions = [ (ω_final + ω_initial) * t] / (2 * 2π)

Based on the numerical values provided in the document, the calculation is:
Number of revolution = [ (2π * 3360/60 + 0) * t] / (2 * 2π) . with further calculation yielding the result:
Number of revolution = 728