Question 125: Waves

Question

Two strings $(A, B)$ having linear densities $\mu_A=2 	imes 10^{-4} \mathrm{~kg} / \mathrm{m}$ and, $\mu_B=4 	imes 10^{-4} \mathrm{~kg} / \mathrm{m}$ and lengths $L_A=2.5 \mathrm{~m}$ and $L_B=1.5 \mathrm{~m}$ respectively are joined. Free ends of $A$ and $B$ are tied to two rigid supports $C$ and $D$, respectively creating a tension of 500 N in the wire. Two identical pulses, sent from $C$ and $D$ ends, take time $t_1$ and $t_2$, respectively, to reach the joint. The ratio $t_1 / t_2$ is:

Accepted Answer

When a string with linear mass density $\mu$ (mass per unit length) is under tension $T$, the speed $v$ of a transverse wave is given by :
$$ v=\sqrt{\frac{T}{\mu}} $$
So, the time taken to reach the joint is $t=\frac{	ext { Length }}{	ext { Velocity }}=\frac{L}{v}$
Substituting the velocity formula into the time equation :
$$ \mathrm{t}=\mathrm{L} \sqrt{\frac{\mathrm{~T}}{\mu}}=\mathrm{L} \sqrt{\frac{\mu}{\mathrm{~T}}} $$
The two strings are joined together and tied between two rigid supports, meaning the tension ( $T$ ) is the same throughout both strings $(T=500 \mathrm{~N})$.
For String A (time $\mathrm{t}_1$ ) :
The length of the string $A$ is $L_A=2.5 \mathrm{~m}$ and the linear density is $\mu_A=2 	imes 10^{-4} \mathrm{~kg} / \mathrm{m}$ So, the time taken to reach the end of string A is,
$$ \mathrm{t}_1=2.5 \sqrt{\frac{2 	imes 10^{-4}}{500}} $$
For String B (time $\mathrm{t}_2$ ) :
The length of the string $B$ is $L_B=1.5 \mathrm{~m}$ and the linear density is $\mu_B=4 	imes 10^{-4} \mathrm{~kg} / \mathrm{m}$ So, the time taken to reach the end of string B is,
$$ \mathrm{t}_2=1.5 \sqrt{\frac{4 	imes 10^{-4}}{500}} $$
So, the ratio is,
$$ \frac{\mathrm{t}_1}{\mathrm{t}_2}=\frac{2.5 \sqrt{\frac{2 	imes 10^{-4}}{500}}}{1.5 \sqrt{\frac{4 	imes 10^{-4}}{500}}} $$
$\Rightarrow \frac{t_1}{t_2}=\frac{2.5 \sqrt{2}}{1.5 \sqrt{4}}=\frac{5}{3 \sqrt{2}}$
$$\Rightarrow $$ $$ t_1 t_2 \approx 1.18 $$
Therefore, the ratio of the travel times to be approximately 1.18 .
Hence, the correct option is (B).

Answer

1.90

Answer

1.18

Answer

1.08

Answer

1.67

Waves

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