Question

A wire of linear density $9.8\times {10}^{-3}\text{kg/m}$ passes over a frictionless light pulley fixed on the top of a frictionless inclined plane which makes an angle ${30}^0$ with the horizontal. Masses $m$ and $M$ are tied at the two ends of the wire such that $m$ rests on the plane and $M$ hangs freely vertically downwards. The entire system is in equilibrium and a transverse wave propagates along the wire with a velocity of $100\text{m/s}$. Then:

• A. $m=20\text{kg}$
• B. $M=5\text{kg}$
• C. $\frac{m}{M}=\frac{1}{2}$
• D. $\frac{m}{M}$=2

Answer 1

Answer: (A), (D)

$\frac{m}{M}$=2

For equilibrium, $Mg=mg\sin {30}^{\circ }=T$
$\Rightarrow M=m/2$

Velocity of transverse waves on a string is given by
$v=\sqrt{\frac{T}{\mu }}$
$\Rightarrow 100=\sqrt{\frac{Mg}{9.8\times {10}^{-3}}}=\sqrt{\frac{M\left(9.8\right)}{9.8\times {10}^{-3}}}$
$\Rightarrow 100=\sqrt{M\left(1000\right)}$
$\Rightarrow M=10\text{kg}$ and $m=20\text{kg}$

Option (a) and (d) are correct.