Both the equations y1 = a sinwt and y2 = a/2( sinwt + coswt) represent SHM. The ratio of amplitudes of the two nations is

To find the ratio of the amplitudes of the two equations representing simple harmonic motion (SHM), we first need to identify the amplitudes from each equation. The equations given are:

Understanding the Equations

The first equation is:

y1 = a sin(wt)

In this equation, the amplitude is simply the coefficient of the sine function, which is a.

The second equation is:

y2 = (a/2)(sin(wt) + cos(wt))

To find the amplitude of this equation, we need to analyze the expression inside the parentheses. The terms sin(wt) and cos(wt) can be combined into a single sinusoidal function using the formula for the amplitude of a sum of sine and cosine functions.

Combining Sine and Cosine

When you have a combination of sine and cosine, you can express it as:

A sin(wt + φ)

where A is the resultant amplitude and φ is the phase shift. The amplitude can be calculated using the formula:

A = √(A1² + A2²)

Here, A1 is the coefficient of sin(wt) (which is 1) and A2 is the coefficient of cos(wt) (which is also 1).

Calculating the Amplitude

Substituting the values:

A = √(1² + 1²) = √(1 + 1) = √2

Now, since the entire expression for y2 is multiplied by a/2, the amplitude of y2 becomes:

Amplitude of y2 = (a/2) * √2

Finding the Ratio of Amplitudes

Now that we have the amplitudes of both equations, we can find the ratio:

Amplitude of y1 = a

Amplitude of y2 = (a/2) * √2

The ratio of the amplitudes is:

Ratio = Amplitude of y1 / Amplitude of y2

Ratio = a / ((a/2) * √2)

Simplifying the Ratio

To simplify this expression:

Ratio = a / (a/2) * (1/√2) = (a * 2) / (a * √2) = 2 / √2

Now, simplifying further:

2 / √2 = √2

Final Result

Thus, the ratio of the amplitudes of the two equations is:

√2

This means that the amplitude of the first equation is √2 times greater than that of the second equation. This relationship highlights how different forms of SHM can have varying amplitudes even when they are derived from similar trigonometric functions.