Question

Same spring is attached with $2kg,3kg$ and $1kg$ blocks in three different cases as shown in figure. If $x_1,x_2$ and $x_3$ be the constan extensions in the spring in these three cases then find the ratio of their extensions.

Answer 1

clearly from the fig we can write Newton's 2nd law of equation for all systems

for first figure : $2g-2g=2a\to a=0$

this gives $k{x}_1=2g\Rightarrow x_1=\frac{2g}{k}$

For 2nd figure , 3 kg block will move downwards , this gives $3g-T=3a...\left(1\right)andT-2g=2a...\left(2\right)$

adding eq 1 and eq 2 we get $a=\frac{g}{5}$

this gives value of $T=k{x}_2=\frac{12g}{5}\Rightarrow x_2=\frac{12g}{5k}$

From the third figure , we can see that 2 kg block will moves down hence :

$2g-T=2a...\left(1\right)andT-g=a...\left(2\right)$

adding both equations we get : $a=\frac{g}{3}$ and usin this we get value of $T=k{x}_3=\frac{4g}{3}\Rightarrow x_3=\frac{4g}{3k}$

Now taking ratio of $x_1:x_2:x_3=2:\frac{12}{5}:\frac{4}{3}=15:18:10$