By what factor would the energy carried by the rope change

[looi76]

Homework Statement

The amplitude of a wave in a rope is $$15 \ mm$$. If the amplitude were changed to $$20 \ mm$$. Keeping the frequency the same, by what factor would the energy carried by the rope change?

Homework Equations

$$intensity \propto (amplitude)^2$$

The Attempt at a Solution

$$amplitude_1 = 15 \times 10^{-3} \ m$$
$$amplitude_2 = 20 \times 10^{-3} \ m$$

What to do next

 

[Redbelly98]

First, by what factor does the amplitude change?

One consequence of the equation you wrote is that

$$\frac{I_2}{I_1} = (\frac{A_2}{A_1})^2$$

 

[Moetasim]

To find the factor by which the energy carried by the rope changes, we can use the equation for intensity: intensity \propto (amplitude)^2. This means that the energy carried by the rope is directly proportional to the square of the amplitude.

Therefore, we can set up a proportion:

\frac{energy_1}{energy_2} = \left(\frac{amplitude_1}{amplitude_2}\right)^2

Plugging in the values, we get:

\frac{energy_1}{energy_2} = \left(\frac{15 \times 10^{-3} \ m}{20 \times 10^{-3} \ m}\right)^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16}

This means that the energy carried by the rope changes by a factor of \frac{9}{16} or approximately 0.56. In other words, the energy carried by the rope decreases by about 44% when the amplitude is increased from 15 \ mm to 20 \ mm while keeping the frequency constant.