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What would be the period of the free osc...

What would be the period of the free oscillations of the system shown here if mass `M_(1)` is pulled down a little force constant of the spring is `k`, mass of fixed pulley is negligible and movable pulley is smooth

Text Solution

Verified by Experts

The correct Answer is:
`T = 2pisqrt((M_(2) + 4M_(1))/(k))`

(i) Equilibrium position determination
`M_(1)g = T rarr` (from `FBD` of `M_(1)`)
`2T = M_(2)g + kx_(0)` (from `FBD` of `M_(2)`)
`:. 2M_(1)g = M_(2)g + kx_(0)`
`:. kx_(0) = 2M_(1)g - M_(2)g`
`(ii)` Displace block `M_(1)` by small disp, `x` by
At new displacement position
`-Mgx + (1)/(2)M_(1)V^(2) + (1)/(2)M_(2)((v)/(2))^(2) + M_(2)g((x)/(2)) + (1)/(2)K (x_(0) + (x)/ (2))^(2) x = C`
`-M_(1)g'(dx)/(dt)+(1)/(2)M_(1)2v(dv)/(dt)+(M_(2))/(8)2v(dv)/(dt)+(M_(2)g)/(2)(dx)/(dt)+(K)/(2)2(x_(0)+(x)/(2))((1)/(2)(dx)/(dt)) =0`
`rArr - M_(1)g+M_(1)a+(M_(2)a)/(4)+(M_(2)g)/(2)+(K)/(2)(x_(0)+(x)/(2))=0`, (where `a = (dv)/(dt)`)
`rArr - M_(1)g + (M_(2)g)/(2)+M_(1)a+(M_(2)a)/(4)+(Kx_(0))/(2)+(Kx)/(4) = 0` (from equilibrium `- M_(1)g+(M_(2)g)/(2)+(Kx_(0))/(2)=0`)
Hence, `(4M_(1) + M_(2))/(4) a= (-Kx)/(4)`
`:. a = -(K)/(4M_(1) + M_(2))x , omega^(2) = (K)/((4M_(1) + M_(2)))`
`omega = sqrt((K)/((4M_(1) + M_(2)))) , T = (2pi)/(omega)`
`:. T_(2) = 2pisqrt((4M_(1) + M_(2))/(K))`
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Knowledge Check

  • The period of the free oscillations of the system shown here if mass m_(1) is pulled down a little and force constant of the spring is k and masses of the fixed pulleys are negligible, is

    A
    `T = 2pi sqrt((m_(1) + m_(2))/(k))`
    B
    `T = 2pi sqrt((m_(1) + 4m_(2))/(k))`
    C
    `T = 2pi sqrt((4m_(1) + m_(2))/(k))`
    D
    `T = 2pi sqrt((3m_(1) + m_(2))/(k))`

  • Comparing the L-C oscillations with the oscillations of a spring-block system (force constant of spring=k and mass of block=m), the physical quantity mk is similar to

    A
    CL
    B
    V`(1)/(CL)`
    C
    `(C)/(L)`
    D
    `(L)/(C)`

  • Find the time period of oscillation of block of mass m. Spring, and pulley are ideal. Spring constant is k.

    A
    `2pisqrt((m)/(k))`
    B
    `pisqrt((m)/(k))`
    C
    `4pisqrt((m)/(k))`
    D
    `2pisqrt((m)/(2k))`