The equation of displacement of a harmonic oscillator is x=3sinωt+4cosωt. The amplitude of the particles will be

1. Break Down the Problem

We need to find the amplitude of a harmonic oscillator given the equation of displacement: $x = 3 \sin(\omega t) + 4 \cos(\omega t)$

2. Relevant Concepts

The amplitude $A$ of a harmonic oscillator described by the equation $x = A \sin(\omega t + \phi)$ can be found using the coefficients of the sine and cosine terms. These coefficients can be thought of as components of a vector, where

• $a = 3$ (the coefficient of $\sin$)
• $b = 4$ (the coefficient of $\cos$)

3. Analysis and Detail

To find the amplitude $A$, we use the formula: $A = \sqrt{a^2 + b^2}$

Substituting the known values:

• $a = 3$
• $b = 4$

Calculating: $A = \sqrt{3^2 + 4^2}$ $A = \sqrt{9 + 16}$ $A = \sqrt{25}$ $A = 5$

4. Verify and Summarize

We have calculated the amplitude by recognizing the coefficients as components of a right triangle and using the Pythagorean theorem to find the hypotenuse, which represents the amplitude.

Final Answer

The amplitude of the particles will be $5$.