Question

Two wires A and B, having identical geometrical construction, are stretched from their natural length by small but equal amount. The Young modules of the wires are YA and YB whereas the densities are rhoA and rhoB. It is given that YA > YB and rhoA>rhoB. A transverse signal started at one end takes a time t1 to reach the other end for A and t2 for B.
(a) t1 < t2
(b) t1 = t2
(c) t1 > t2
(d) the information is insufficient to find the relation between t1 and t2.

Answer 1

The relation between t1 and t2 cannot be discovered.

Explanation:

We know that $Y_{a}>Y_{\beta}$, for equal strain tension $\mathrm{T}_{\mathrm{a}}>\mathrm{T}_{\mathrm{\beta}}$. Since $\rho_{\mathrm{a}}>\rho_{\beta}, \mu_{\mathrm{a}}>\mu_{\beta}$. Now$v_{a}=$√ $\left(T_{a} / \mu_{a}\right)$ and  $\vee_{\beta}=$√ $\left(T_{\beta} /\left(\mu_{\beta}\right)\right)$

$\left(v_{a}\right) / v_{\beta}=$√$\left.\left(T_{\beta} /\left(\mu_{\beta}\right)\right) *\left(\left(\mu_{\beta}\right) /\left(\mu_{a}\right)\right)\right\}$  

The ratio $\mathrm{T}_{-} \mathrm{a} / \mathrm{T}_{\beta}>1 \text { but }\left(\mu_{\beta}\right) /\left(\mu_{\mathrm{a}}\right)<1$

So, to ask whether $\mathrm{va}>\mathrm{v} \beta$ or not, we need to know the exact proportions of $\mathrm{T} \mathrm{a} / \mathrm{T}_{\beta}$ and $\left(\mu_{\beta}\right) /\left(\mu_{\mathrm{a}}\right)$ Which is not mentioned here. The relation between t1 and t2 cannot, be discovered.

Answer 2

(d) The information is insufficient to find the relation between t₁ and t₂.

Explanation:

From question, after stretching, the young's modulus is given as:

Y$_a$ > Y$_b$

Thus, the strain tension is given as:

T$_a$ > T$_b$

From question, after stretching, the density is given as:

ρ$_a$ > ρ$_b$

Thus,

μ$_a$ > μ$_b$

Now, volume of wire 'a' is given as:

$\mathrm{v}_{\mathrm{a}}=\sqrt{\left(\mathrm{T}_{\mathrm{a}} / \mu_a\right)}$ ⇒ (equation 1)

Now, volume of wire 'b' is given as:

$\mathrm{v}_{\mathrm{b}}=\sqrt{\left(\mathrm{T}_{\mathrm{b}} / \mu_b\right)}$ ⇒ (equation 2)

On dividing equation 1 by equation 2, we get,

$\frac{v_a}{v_b}=\sqrt{\frac{T_a}{T_b}\times \frac{\mu_b}{\mu_a}}$

Now, the ratio of T$_a$/T$_b$ > 1 and the ratio of μ$_b$/μ$_a$ < 1.

Since, we don't the value of ratio of T$_a$/T$_b$ and μ$_b$/μ$_a$. Thus, we are unable to find vₐ and v$_b$.

Thus, relationship between t₁ and t₂ cannot be found.