The system described in figure is composed of a wheel of mass M and radius R , a massless rope which is wrapped at one end around the wheel and is connected at the other end to a block of mass m . The block is placed on a slanted plane with a lift angle α , and the rope is threaded through a massless pulley connected to the higher end of the incline. Initially, the system is held at rest. As the system is released, the block remains at rest, while the wheel unwraps the rope while rolling down. The wheel is made up of five identical rods of length R which are connected at the center of the wheel and distributed evenly over its massless circumference.
Text Solution
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Sol. The mass of each cord is 
. The moment of inertia of each about one of its ends is 
. Therefore, the total moment of inertia of the wheel about its center is :
I = 5 
= 
MR 2 …..(i)
Note: As per the definition of the moment of inertia, I = 
, we can sum the moments of inertial about the given axis of each.
Since the block is at rest, we have v = ω R and a = R 
= R α , where a is the acceleration of the center of mass and α is the angular acceleration of the wheel. The equations of motion are therefore,
…..(ii)
This set of equations a= 
g, α = 
and T = 
Mg.
Using the relation T = mg sin α (m is at rest) , and substituting in the value of T found in the previous section, we obtain:
m = 
…..(iii)
The wheel advance a distance h ; therefore, the potential energy difference is –Mgh. The principle of conservation of energy implies that Δ K = – Δ U, so,
K = Mgh .....(iv)
Where K is the kinetic energy.
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