What is Motion of bodies connected by strings2 (1)
What is Motion of bodies connected by strings ?

Motion of bodies connected by Strings



Two bodies: Let us consider the case of two bodies of masses $m_{1}$ and $m_{2}$ connected by a thread and placed on a smooth horizontal surface as shown in fig. A force $F$ is applied on the body of mass $m_{2}$ in forward direction as shown. Our aim is to consider the acceleration of the system and the tension $\mathrm{T}$ in the thread. The forces acting separately on two bodies are also shown in the figure:

What is Motion of bodies connected by strings From fig. $\quad \mathrm{T}=\mathrm{m}_{1} \mathrm{a}$             …..(1)

$\mathrm{F}-\mathrm{T}=\mathrm{m}_{2} \mathrm{a}$



What is Motion of bodies connected by strings2 $F=\left(m_{1}+m_{2}\right) a$

or $ a=\frac{F}{m_{1}+m_{2}} $   …..(3)

The acceleration of the system can be calculated from eq. (3) and tension in the thread by eq. (1).

What is Motion of bodies connected by strings3 Three bodies: In case of three bodies, the situation is shown in fig.

Acceleration $\quad a=\frac{F}{m_{1}+m_{2}+m_{3}}$ …..(4)

What is Motion of bodies connected by strings4 $\mathrm{T}_{1}=\mathrm{m}_{1} \mathrm{a}=\frac{\mathrm{m}_{1} \mathrm{~F}}{\mathrm{~m}_{1}+\mathrm{m}_{2}+\mathrm{m}_{3}}$ ……(5)

$\mathrm{F}-\mathrm{T}_{2}=\mathrm{m}_{3} \mathrm{a}$

or$ \mathrm{T}_{2}=\mathrm{F}-\frac{\mathrm{m}_{3} \mathrm{~F}}{\mathrm{~m}_{1}+\mathrm{m}_{2}+\mathrm{m}_{3}}=\frac{\left(\mathrm{m}_{1}+\mathrm{m}_{2}\right) \mathrm{F}}{\mathrm{m}_{1}+\mathrm{m}_{2}+\mathrm{m}_{3}} $    …..(6)

Ex. Three blocks, are connected by string as shown in fig. below, and are pulled by a force $T_{3}=120 \mathrm{~N}$. If $m_{1}=5 \mathrm{~kg}, m_{2}=10 \mathrm{~kg}$ and $m_{3}=15 \mathrm{~kg}$. Calculate the acceleration of the system and $T_{1}$ and $T_{2}$.

What is Motion of bodies connected by strings5

Sol. (i) Acceleration of the system $\mathrm{a}=\frac{\mathrm{F}}{\mathrm{m}_{1}+\mathrm{m}_{2}+\mathrm{m}_{3}}=\frac{120}{5+10+15}=4 \mathrm{~m} / \mathrm{sec}^{2}$

(ii) $\mathrm{T}_{1}=\mathrm{m}_{1} \mathrm{a}=5 \times 4=20 \mathrm{~N}$           $T_{2}=\left(m_{1}+m_{2}\right) a=(5+10) 4=60 \mathrm{~N}$

Ex. A block of mass $M$ is pulled along a horizontal frictionless surface by a rope of mass $m$ as shown in fig. A horizontal force $F$ is applied to one end of the rope. Find (i) The acceleration of the rope and block (ii) The force that the rope exerts on the block. (iii) Tension in the rope at its mid point.

What is Motion of bodies connected by strings6 Sol. (i) $F=(m+M) a \quad$ or $\quad a=\frac{F}{(m+M)}$

(ii) $\mathrm{T}=\mathrm{Ma}=\frac{\mathrm{M} \cdot \mathrm{F}}{(\mathrm{m}+\mathrm{M})}$

(iii) $\mathrm{T}_{1}=\left(\frac{\mathrm{m}}{2}+\mathrm{M}\right) \mathrm{a}=\left(\frac{\mathrm{m}+2 \mathrm{M}}{2}\right)\left(\frac{\mathrm{F}}{\mathrm{m}+\mathrm{M}}\right)$

or $\quad \mathrm{T}_{1}=\frac{(\mathrm{m}+2 \mathrm{M})}{2(\mathrm{~m}+\mathrm{M})} . \mathrm{F}$



Also Read

JEE Physics Notes

To watch Free Learning Videos on JEE by Kota’s top IITian Faculties Install the eSaral App