Question

An Atwood's machine consists of two masses, m1 and m2, connected by a string that passes over a pulley. If the pulley has radius and moment of inertia 1/2 MR^2 about its axle. Determine the acceleration if masses m1 and m2. Compare the situation in which moment of inertia is ignored

Answer 1

let tension in string connecting the mass m2 is T2 and tension in string connecting to mass m1 is T1.

let acceleration of the system be a m/s^2.

assume m2> m1

writing force equation for m2:

m2*g-T2=m2*a

==>T2+m2*a=m2*g....(1)

writing force equation for the pulley:

torque=moment of inertia*angular acceleration

==>T2*R-T1*R=0.5*M*R^2*(a/R)=0.5*M*a*R

==>T2-T1=0.5*M*a....(2)

writing force equation for mass m1:

T1-m1*g=m1*a

==>T1-m1*a=m1*g...(3)

from equation 1, T2=m2*g-m2*a

from equation 2, T1=m1*g+m1*a

using these in equation 2:

m2*g-m1*g-m2*a-m1*a=0.5*M*a

==>(m2-m1)*g=(0.5*M+m1+m2)*a

==>a=(m2-m1)*g/(ma+m2+0.5*M)

when moment of inertia is ingored,

acceleration becomes (m2-m1)*g/(m1+m2)

so acceleration in case of pulley with moment of inertia is lesser.