  Question

# A body of mass m hangs from a smooth fixed pulley P1 by an inextensible string fitted with the springs of stiffness constants k1 and k3 The string passes over a smooth light pulley P2 ,  which is connected with another ideal spring of stiffness constant k2. Find the period of small oscillations of the body. A
T=2π(m(2k1+4k2+2k3))12
B
T=2π(m(1k1+4k2+1k3))12
C
T=2π(m(2k1+2k2+2k3))12
D
T=2π(m(1k1+1k2+1k3))12

Solution

## The correct option is D T=2π(m(1k1+4k2+1k3))12Let x1,x2  and x3 be the elongations of the springs k1,k2 and k3 respectively. Using constraint relations we find the total displacement of m as  x=x1+2x2+x3    ........(1) Spring(3) experiences tension force due to mass m From this, we get , k3x3=mg⇒x3=mgk3 Since, a string is connected between the two springs, the tension force remains same .  So, spring (1) experiences same tension force due to mass m ∴x1=mgk1 Tension experienced by spring (2) is given by 2mg=k2x2⇒x2=2mgk2 Using these values in equation (1) we obtain x=mg[1k1+4k2+1k3]    .......(2) Time period of oscillation  T=2π√displacementacceleration=2π√xg Using (2) in the above equation we get, T=2π(m(1k1+4k2+1k3))12 Hence, option (b) is the correct answer.  Suggest corrections   