Question

Two masses M and m are connected at the two ends of an inextensible string. The string passes over a smooth frictionless pulley. Calculate the acceleration of the masses and the tension in the string. Given M $>$ m.

Answer 1

For mass m, the forces are given as

$T-mg=ma$ ...... (1)

Similarly, for mass M, the forces are given as

$Mg-T=Ma$ ...... (2)

From equation (1), we have

$T=mg+ma$ ...... (3)

Substituting equation (3) in (2) we get

$Mg-mg-ma=Ma$

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

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

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

$T=\frac{Mm-m^2}{M+m}g+mg$

$T=g(\frac{Mm-m^2}{M+m}+m)$

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