Question

The descending pullet shown in figure has a radius $20$cm and moment of inertia $0.20$ kg-$m^2$. The fixed pulley is light and the horizontal plane frictionless. Find the acceleration of the block if its mass is $1.0$ kg.

Answer 1

Let the mass of block be $m_1$ and mass of pulley be

Acceleration of the massive pulley will be half of that of the block.

From the free block diagram: we have$[T_1=m_{1}a.........\left(1\right)$

$\left(T_2-T_1\right)r=Ia$

$T_2-T_1=\frac{Ia}{2r^2}=\frac{5a}{2}.........\left(2\right)$

$m_{2}g-m_2\frac{a}{2}=T_1+T_2.......\left(3\right)$

Put the values,we get

$T_1=aand{T}_2=\frac{7}{2}a$

$m_{2}g=m_2\frac{a}{2}+\frac{7}{2}a+a$

On replacing the values in ${m_2}$in$\frac{1}{2}m{r^2} = I$, we have

$\frac{2l}{r^2}g=\frac{2l}{r^2}\frac{a}{2}+\frac{9}{2}a$

Implies that

$98=5a+4.5a$

$a=\frac{98}{9.5}=10.3m{s}^{-2}$