A uniform round object of mass `M`, radius `R` and moment of inertia about its centre of mass `I_(cm)` has a light, thin string wrapped several times around its circumference. The free end of string is attaced to the celling and the object is released from rest. Find the acceleration of centre of the object and tension n the string. [ Take `(I_(cm))/(MR^(2))=k`]

Explanation

Equation for translatory motion
`mg-T=ma`…….`(1)`
Equation for rotatory motion
`TR=I_(cm)alpha`……..`(2)`
Condition for no slipping
`a=Ralpha`…….`(3)`
Solving the equations
`Mg-(I_(cm)a)/(R^(2))=Ma`
`a=(Mg)/(M[1+(I_(cm))/(MR^(2))])=(g)/(1+k)`
`T=(I_(cm)a)/(R^(2))=((k)/(1+k))Mg`