Question

Block $A$ of weight $500\text{N}$ and block $B$ of weight $700\text{N}$ are connected by the rope pulley system, as shown in the figure. The largest weight $C$ that can be suspended without moving block $A$ and $B$ is $W$. The coefficient of friction for all plane surfaces of contact is $0.3$. The pulleys are ideal. The value of $\frac{W}{90}$ is ______.

Answer 1

Drawing FBD of all the blocks, we get,


From FBD of block $A$,

$T=f_1=\mu N_1...\left(i\right)$

$N_1=500...\left(ii\right)$

From FBD of block $B$,

$N_2=N_1+700=500+700=1200...\left(iii\right)\left[\text{From}\right(ii\left)\right]$

$T_1=f_1+f_2+2T...\left(iv\right)$

From FBD of block $C$

$T_1=W...\left(v\right)$

Now, value of,

$f_1=\mu N_1=0.3\times 500=150\text{N}\left[\text{From}\right(i\left)\text{and}\right(ii\left)\right]$

$f_2=\mu N_2=0.3\times 1200=360\text{N}\left[\text{From}\right(iii\left)\right]$

Further,
$\therefore T_1=W=f_1+f_2+2T\left[\text{From}\right(iv\left)\text{and}\right(v\left)\right]$

$\Rightarrow W=f_1+f_2+2f_1\left[\text{From}\right(i\left)\right]$

$\Rightarrow W=3f_1+f_2$

$\Rightarrow W=3\times 150+360=810\text{N}$

$\therefore \frac{W}{90}=\frac{810}{90}=9$