Question

The descending pulley shown in figure (10-E7) 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

A is light pulley and B is the descending pulley having I = 0.20

${\text{kg-m}}^2$ and r = 0.2 m

Mass of the block = 1 kg

According to the question

$T_1=m_{1}a...\left(1\right)$

$(T_2-T_1)r=I\alpha ...\left(2\right)$

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

$T_2-T_1=\frac{Ia}{2r^2}=\frac{5a}{2}$

$\text{and}{T}_1=a$

$(\text{because}\alpha =\frac{a}{2r})$

$\Rightarrow T_2=\frac{7}{2}a$

$\Rightarrow m_{2}g=m_2\frac{a}{2}+\frac{7}{2}a+a\left[\text{using}\right(3\left)\right]$

$\Rightarrow \frac{2I}{r^2}g=\frac{2I}{r^2}\frac{a}{2}+\frac{9}{2}a(\frac{1}{2}m{r}^2=I)$

$\Rightarrow 98=5a+4.5a$

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