Question

The two blocks a and b have a mass of 5 kg and 10 kg, respectively. if the pulley can be treated as a disk of mass 3 kg and radius 0.15 m, determine the acceleration of block
a. neglect the mass of the cord and any slipping on the pulley.

Answer 1

I will name block a as Ma=5 kg, block b as Mb=10 kg and mass of the pulley M=3 kg and radius as R. Since the system will accelerate in the direction of the block b because it has greater mass, I will take that direction as positive. Both blocks and the pulley have the same acceleration because the slipping on the pulley is neglected. First, the equations of motion:

Mb*g-Tg=MbαR and
Ta-Ma*g=MaαR,

where Ta and Tb are the tensions of the cord, g=9.81 m/s^2 and α is the angular accereration. Also a=αR where a is the acceleration of the system. 

Now the equation of rotational dynamics of a solid body:

(Tb-Ta)R=Iα=(1/2)*M*R^2*α, where (1/2)*M*R^2 is the moment of inertia of a disc. 

When we input Tb=Mb*g - Mb*α*R and Ta=Ma*g + Ma*α*R from the first two equations into the third we get: (Mb*g - Mb*α*R - Ma*α*R - Ma*g)*R=(1/2)*M*R^2*α.

We solve for α and get: α=(Mb*g-Ma*g)/((1/2)*MR+Mb*R+Ma*R)=2.97 rad/s^2.

We know that a=α*R and we easily get a=0.4455 m/s^2