Question

A mass of $2kg$ hangs freely at the end of a string, which passes over a smooth pulley fixed at the edge of a smooth table. The other end of the string is attached to a mass $M$ on the table. If the mass on the table is doubled the tension in the string increases by one-half. Find the mass $M$.

Answer 1

Consider the two situations as shown

Tension in the first case,

$(M-2)g=(M+2)a$

$a=\left(\frac{m-2}{m+2}\right)g$

$Mg-T=M\left(\frac{m-2}{m+2}\right)g$

$T=Mg-M\frac{(M-2)}{(M+2)}g$

In second case,

$T=2Mg-2M\left(\frac{2M-2}{2m+2}\right)g$

$\frac{T_{S}econd}{T_{F}irst}=\frac{3}{2}$

$\frac{3}{2}=\frac{1-\frac{(M-2)}{(M+2)}}{2-\left(2\frac{2M-2}{2m+2}\right)}$

$M=4Kg$