Question of Physics

Question

Two identical blocks of mass m are connected by a massless rope over a pulley of mass M with radius R and moment of inertia I =2/3MR² about its center of mass. The rope does not slip on the pulley. One of the masses is on an frictionless inclined plane making an angle θ with the horizontal. The system is released at t=0 from rest. Find

a) the acceleration of each block and the angular acceleration of the pulley in

terms of the given quantities and g (the gravitational acceleration), and

b) the period the pulley in terms of the given quantities and g (the gravitational acceleration).

Answer 1

∑F₁ = T - mgsinθ = ma ; ∑F₂ = T - mg = −ma solve for T = mg − ma and substitute into first equation so

mg − ma −mgsinθ = ma So mg − mgsinθ = 2ma ; m cancels dividing by 2 solves for acceleration

a = g − gsinθ ⁄2

The angular acceleration, α, is equal to the linear acceleration, a, divided by the radius R of the pulley so

α = g − gsinθ ⁄ 2R

for part B I’m perplexed since the pulley has angular acceleration so seems the period, which is time for one revolution, would not be constant but decrease each second by 2∏R ⁄ΔV , where ΔV is the change in the linear velocity.