A pulley in the form of a uniform disc of mass 2m and radius r is free to rotate in a vertical plane about a fixed horizontal axis through its center. A light inextensible string has one end fastened to a point on the rim of the pulley and is wrapped several times round the rim. The portion of string not wrapped around the pulley is of length 8 r and carries a particle of mass m at its fee end. The particle is held close to the rim of the pulley and level with its centre. If the particle is released from this position, then the impulse of the sudden tension in the string when it becomes taut is J=βmgr ,then the value of β is

Question

A pulley in the form of a uniform disc of mass 2m and radius r is free to rotate in a vertical plane about a fixed horizontal axis through its center. A light inextensible string has one end fastened to a point on the rim of the pulley and is wrapped several times round the rim. The portion of string not wrapped around the pulley is of length 8 r and carries a particle of mass m at its fee end. The particle is held close to the rim of the pulley and level with its centre. If the particle is released from this position, then the impulse of the sudden tension in the string when it becomes taut is J=βmgr ,then the value of  β  is

Infinity Learn · Accepted Answer

Angular momentum is conserved about axis of rotation. ∴ mur=(mrω)r+IϖHere, I=12(2m)r2=mr2 u=4gr ∴m(4gr)r=mr2ω+mr2ω⇒ω=2grConsidering the sudden change in the linear momentum of the particle caused by the impulse tension, we have J=−mrω+(4mgr)Or J=2mgr

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