Figure shown, two block of mass $m$ and $M$ connected by a string passing over a pulley. The horizontal table over which the mass $m$ slides is smooth. The pulley has a radius $r$ and moment of inertia $I$ about its axis and it can freely rotate about this axis. find the acceleration of the mass $M$ assuming that the string does not slip on the pulley.

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Solution

Mg - $T_{1}$ = Ma ........ ( 1 )

$T_{2}$ = ma ............. ( 2 )

( $T_{1} - T_{2}) = I \frac{a}{r^{2}}$ .......( 3 )

[ because a = $r α$ ] .... $[T . r = I (\frac{a}{r})]$

if we add the equation 1 and 2 we will get

Mg + ( $T_{2} - T_{1}$ ) = Ma + ma ... ( 4 )

Mg - $I \frac{a}{r^{2}}$ = Ma + ma

( M + m + $\frac{I}{r^{2}}) a$ = Mg

a = $\frac{M g}{(M + m + \frac{I}{r^{2}}})$

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Mg - T1 = Ma ........ ( 1 )T2 = ma ............. ( 2 )( T1−T2)=Iar2 .......( 3 )[ because a = rα] .... [T.r=I(ar)]if we add the equation 1 and 2 we will get Mg + ( T2−T1) = Ma + ma ... ( 4 )Mg - Iar2 = Ma + ma ( M + m + Ir2)a = Mg a = Mg(M+m+Ir2)