A uniformly thick wheel with moment of inertia I and radius R is free to rotate about its centre of mass (see fig). A massless string is wrapped over its rim and two blocks of masses m1 and m2 (m1 m2) are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when m1 descents by a distance h is :

Question

A uniformly thick wheel with moment of inertia I and radius R is free to rotate about its centre of mass (see fig). A massless string is wrapped over its rim and two blocks of masses m1 and m2 (m1 > m2) are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when m1 descents by a distance h is : (A) [(2(m1+m2)gh/(m1+m2)R2+I)](1/2) (B) [(m1+m2/(m1+m2)R2+I)](1/2)gh (C) [((m1-m2)/(m1+m2)R2+I)](1/2)gh (D) [(2(m1-m2)gh/(m1+m2)R2+I)](1/2)

Tardigrade Admin · Accepted Answer

by using work energy theorem Wg = Δ KE (m1-m2)gh= (1/2)(m1+m2)V2 + (1/2)Iω2 (m1-m2)gh= (1/2)(m1+m2)(ω R)2 + (1/2)Iω 2 (m1-m2)gh= (ω2/2)[(m1+m2) R2+I] ω = √(2(m1-m2)gh/(m1+m2)R2 +1)

$[\frac{2 ( m _{1} + m _{2} ) g h}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}}$

$[\frac{m _{1} + m _{2}}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}} g h$

$[\frac{( m _{1} - m _{2} )}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}} g h$

$[\frac{2 ( m _{1} - m _{2} ) g h}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}}$

[(m1​+m2​)R2+I2(m1​+m2​)gh​]21​

[(m1​+m2​)R2+Im1​+m2​​]21​gh

[(m1​+m2​)R2+I(m1​−m2​)​]21​gh

[(m1​+m2​)R2+I2(m1​−m2​)gh​]21​

by using work energy theorem Wg=ΔKE (m1​−m2​)gh=21​(m1​+m2​)V2+21​Iω2 (m1​−m2​)gh=21​(m1​+m2​)(ωR)2+21​Iω2 (m1​−m2​)gh=2ω2​[(m1​+m2​)R2+I] ω=(m1​+m2​)R2+12(m1​−m2​)gh​​

$[\frac{2 ( m _{1} + m _{2} ) g h}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}}$

$[\frac{m _{1} + m _{2}}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}} g h$

$[\frac{( m _{1} - m _{2} )}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}} g h$

$[\frac{2 ( m _{1} - m _{2} ) g h}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}}$