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

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:

\[v = \sqrt{\left( \frac{T}{m} \right)} = \sqrt{\frac{\left( 60 \right)}{\left( 6 \times {10}^{- 3} \right)}}\] 

\[ =  {10}^2  = 100  m/s\]
(a) Average power is given as 

\[P_{average}  = 2 \pi^2 m\nu A^2  f^2 \] 

\[= 2 \times  \left( 3 . 14 \right)^2  \times \left( 6 \times {10}^{- 3} \right) \times 100 \times \left( {10}^{- 3} \right) \times  {200}^2 \] 

\[  = 473 \times  {10}^{- 3}  = 0 . 47  W\] 
(b) Length of the string = 2 m
Time required to cover this distance:

\[t = \frac{2}{100} = 0 . 02  s\] 

\[Energy = Power \times t\] 

\[ = 0 . 47 \times 0 . 02\] 

\[ = 9 . 4 \times  {10}^{- 3}   J = 9 . 4  mJ\]