Question

A 200 Hz wave with amplitude 1 mm travels on a long string of linear mass density 6 g m−1 kept under a tension of 60 N. (a) Find the average power transmitted across a given point on the string. (b) Find the total energy associated with the wave in a 2⋅0 m long portion of the string.

Answer 1

Energy is 9.4 mJ

Explanation:

Given,

Frequency of the wave, f = 200 Hz

Amplitude, A = 1 mm =  $10^-^3$ m

Linear mass density, m = 6 $gm^-^3$

Applied tension, T = 60 N

Now,

Let the velocity of the wave be v.

Thus, we have:

v = $\sqrt \frac {T}{m} = \sqrt \frac {60}{60 \times 10^-^3}$

= $10^2 = 100\ ms^-^1$

(a) Average power is given as

$P_a_v_e_r_a_g_e = 2\pi^2m \nu A^2f^2$

= $2\times (3.14)^2 \times (6 \times 10^-^3) \times 100 \times (10^-^3) \times 200^2$

= $473 \times 10^-^3$ = 0.47 W

(b) Length of the string = 2 m

Time required to cover this distance:  

t = $\frac {2}{100}$ = 0.02 s

Energy = $Power \times t$

= $0.47 \times 0.02$

= $9.4 \times 10^-^3$ J = 9.4 mJ