Question

A pulley system is shown in figure, pulley P1, is fixed to a rigid
support and pulley P2, is capable of moving freely upward or
downward. The pulleys and strings are ideal. Weights of two masses
A and B are 200 N and 300 N respectively. Find the tensions T1, and T2, and also the acceleration of A and B (g = 10 m s^-2 ).
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Answer 1

support and pulley P2, is capable of moving freely upward or

downward. The pulleys and strings are ideal. Weights of two masses

A and B are 200 N and 300 N respectively. Find the tensions T1, and T2, and also the acceleration of A and B (g = 10 m s^-2

Answer 2

let '$a_{1}$' and '$a_{2}$' be the acceleration of the masses $m_{1}$ in downward direction and $m_{2}$ in upward direction respectively

since the displacement of mass $m_{1}$ is twice that of $p_{2}$ and hence mass $m_{2}$ thus the acceleration $a_{1}$ is equal to twice of acceleration of $a_{2}$,

hence $a_{1}$=2$a_{2}$          (1)

equation of motion of mass $m_{1}$ is

$m_{1}$ g-$T_{1}$= $m_{1}$$a_{1}$       (2)

equation of motion of mass $m_{2}$ is

$T_{2}$ - $m_{2}$g=$m_{2}$ $a_{2}$       (3)

also $T_{2}$=2$T_{1}$             (4)

solving the equation we get

$a_{1}$=2$a_{2}$ = $\frac{4m_{1} -2m_{2} }{4m_{1}+m_{2} }$g

and $T_{2}$=2$T_{1}$  =$\frac{6m_{1}m_{2} }{4m_{1}+m_{2} }$g

now substituting the value of mass $m_{1}$ and $m_{2}$

$m_{1}$ = 15kg, $m_{2}$ = 25kg,

we get the Tension in the string,

$T_{2}$ =264.7N  and  $T_{1}$ =132.35N

and acceleration of masses is,

$a_{1}$ =1.176m/$s^{2}$  and $a_{2}$ =0.588m/$s^{2}$.