Question

The $10.2kg$ block in FIGURE $P7.37$is held in place by a force applied to a rope passing over two massless, frictionless pulleys. Find the tensions $T_{1}to{T}_5$and the magnitude of force $\overrightarrow{F}$.

Short answer

Tension in the rope,$T_1=100N,T_2=T_3=T_5=F=50Nand{T}_4=150N$.

Step-by-step solution

Given information

Mass of the block, $m=10.2kg$

Given the mass and pulleys system

Explanation

Consider the equilibrium block

$T_1=mg$

localid="1649647716658" $$\begin{gathered} T_1=\left(10.2kg\right)\times \left(9.81m/s^2\right) \\ =100N \end{gathered}$$

Consider the equilibrium of a small pulley

$T_1=T_2+T_3$

Since T₂ and T₃ are attached to the same pulley, therefore the tension $T_2=T_3$

localid="1649647731493" $$\begin{gathered} 100N=2T_2 \\ {T}_2=T_3 \\ =50N \end{gathered}$$

Similarly, T₂ and T₅ are attached to the same pulley, therefore the tension $T_2=T_5$

$T_5=50N$

Since the tension in the ropeT₅would be equal to the forceF

$$\begin{gathered} F=T_5 \\ =50N \end{gathered}$$

Consider the equilibrium of a large pulley

localid="1649647749005" $$\begin{gathered} T_4=T_2+T_3+T_5 \\ {T}_4=50N+50N+50N \\ =150N \end{gathered}$$

Tension in the rope,$T_1=100N,T_2=T_3=T_5=F=50Nand{T}_4=150N$.