Answer to Question #99509 in Optics for Gift

Question

A body moving in a progressive wave pattern has acceleration of 10cm/s² when the displacement is 6cm and a speed of 12cm/s² when the displacement is 8cm. What is the amplitude of the wave?

Answer 1

Amplitude of the Wave

We need to find the Amplitude of the wave.

Solution:

Given,

Acceleration = $a =10 \space cm/s²$

Displacement = $x = 6 \space cm$

Frequency = $\omega$

We know the formula,

$$a = \omega^2 \times x$$

Plug the values in this,

$$10 = \omega ^2 \times 6$$

$$\omega^2 = \frac {10}{6}$$

$$\omega = \sqrt {\frac {10} {6} }$$

Here,

$$Speed = v = 12 \space cm/sec \\ Displacement = x = 8 \space cm$$

$$v = \omega \space \sqrt { A^2 - x^2}$$

Squaring on both sides,

$$v^2 = \omega ^2 ( A^2 - x^2)$$

Here, A is the Amplitude,

Plug the values in the formula,

$$144 = (\frac {10}{6}) ( A^2 - 64)$$

$$A^2 - 64 = 144 \times \frac {6}{10} = 86.4$$

$$A^2 = 86.4 + 64 = 150.4$$

So,

$$A = \sqrt {150.4} = 12.26 \space cm$$

Answer: Amplitude of the Wave = A = 12.26 cm