Question

A string with one end fixed on a rigid wall, passing over a fixed frictionless pulley at a distance of 2 m from the wall, has a point mass M$=$ 2kg attached to it at a distance of 1 m from the wall. A mass m $=$ 0.5 kg attached at the free end is held at rest so that the string is horizontal between the wall and the pulley and vertical beyond the pulley. What will the speed which the mass m will hit the wall when the mass m is released?

Answer 1

Constrained$:V_2\cos \theta =V_1..............\left(1\right)$

Distance traveled by ${}^{\prime }{m}^{\prime }$ $=\frac{2}{\cos \theta }-1$

$\tan \theta =\frac{1}{2}$

$\cos \theta =\frac{2}{\sqrt{5}}$

$=\sqrt{5}-1$

Wall $=\Delta k$

$\begin{array}{c}2g\left(1\right)-0.5g\left(\sqrt{5}-1\right)=\frac{1}{2}\times 2\times =V_2^2+\frac{1}{2}\times 0.5\times V_1^2\\ \Rightarrow 20-5\sqrt{5}+5=V_2^2+\frac{1}{2}\times 0.5\times V_2^2{\cos }^2\theta \\ \Rightarrow V_2^2=\frac{125-25\sqrt{5}}{6}\\ \therefore V_2=\sqrt{\frac{125-25\sqrt{5}}{6}}\end{array}$