In the arrangement shown in the figure, a heavy pulley of mass M is suspended from a spring of stiffness constant K and the mass m is suspended from a string tightly wound on the pulley. If the mass m is given a small downward displacement and released, find the time period of small oscillations (Take pulley as Disc) and string does not slips on the pulley.

 

Free body diagrams for pulley of mass M and hanging mass m are shown above.

String does not slip through pulley and pulley gets rotational motion .

If pulley rotates in clockwise direction there is a net force (T₁ - T₂ ) gives the required torque,

where T₁ is the tension force in part of the string between pulley and mass.

T₂ is the tension force in part of the string between puleey and fixed point of string.

 

Let us consider the mass is pulled down slightly and released so that oscillating motion of

small amplitude is set up in the system.

 

If we apply Newton's law to hanging mass m , we get

 

( m g ) - T₁ = - m (2a ) ............................(1)

 

where 2a is acceleration of hanging mass m, if acceleration of translational motion of pulley is a

( for a displacement x of pulley , displacement of hanging mass is 2x due to constant length of string.

Similarly acceleration of hanging mass is twice of acceleration of pulley )

 

If we apply Newton's law to rotational motion of pulley , we get

 

( T₁ - T₂ ) R =  I α

 

LHS of above expression is torque acting on pulley .

I is moenet of inertia and α is angular acceleration of rotational motion.

 

By considering pulley as a disc of mass M and radius R, above expression is written as

 

( T₁ - T₂ ) R =  (1/2) M R² ( a / 2R )

 

Above expression is simplified as ,  ( T₁ - T₂ ) = ( 1/4 ) M a  .........................(2)

 

If we apply Newton's law to translational motion of pulley , we get

 

T₁ + T₂ + ( M g )  - kx =  ( M a )  ............................. (3)

 

By substituting for tension forces T₁ and T₂ using eqn.(1) and (2) ,

eqn.(3) is written for acceleration a of pulley as

 

   .............................(4)

 

First term in RHS is due to oscilatory motion and second term in RHS is due to damping.

 

From the first term of oscillatory motion, we get angular frequency ω as

 

 

where T is period of oscillatory motion

 

Hence period  T of oscillatory motion is given as