Question

A glider of mass m_1 slides without friction on a horizontal air track. It is attached to an object of mass m_2 by a massless string. The string between them goes over a pulley, a thin cylindrical shell (with massless connecting spokes) with mass M and radius R. The string turns the pulley without slipping or stretching. Find the acceleration of each body, the angular acceleration of the pulley, and the tension in each part of the string. (Warning!!!: DO NOT assume the tension on different sides of the pulley is the same! In fact, you can prove to yourself it’s not!)

Answer 1

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

let tension in the srtring connecting the glider be T1 and tension in the string connecting the mass m_2 be T2.

then writing force balance equation for m_1:

T1=m_1*a...(1)

writing force balance equation for m_2:

m_2*g-T2=m_2*a

==>T2=m_2*(g-a)..(2)

writing torque balance for pulley:

total torque=moment of inertia*angular acceleration

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

==>T2-T1=M*a

==>m_2*(g-a)-m_1*a=M*a

==>m_2*g=a*(m_1+m_2+M)

==>a=m_2*g/(m_1+m_2+M)....(answer)

part b:

angular acceleration of pulley=a/R=m_2*g/(R*(m_1+m_2+M).)

part c:

tension T1=m_1*a=m_1*m_2*g/(m_1+m_2+M).

tension T2=m_2*(g-a)=m_2*(g-m_2*g/(m_1+m_2+M).)