Question

Two simple harmonic motions are represented by the equations \( x_1 = 5 \sin(2\pi t + \frac{\pi}{4}) \) and \( x_2 = 5\sqrt{2}(\sin(2\pi t) + \cos(2\pi t)) \). The amplitude of second motion is __________ times the amplitude in first motion.

Hint:

Remember the R-formula for combining sine and cosine functions: \( a\sin\theta + b\cos\theta = \sqrt{a^2+b^2} \sin(\theta + \arctan(b/a)) \). This is extremely useful for finding the amplitude and phase of SHMs that are not given in the standard form.

Answer 1

Correct Answer: 2

Step 1: Understanding the Question:
We are given two equations representing simple harmonic motions. We need to find the ratio of the amplitude of the second motion to the amplitude of the first motion.
Step 2: Key Formula or Approach:
1. The standard form for a simple harmonic motion is \( x = A \sin(\omega t + \phi) \), where A is the amplitude.
2. The first equation is already in this standard form.
3. The second equation needs to be converted to the standard form using the trigonometric identity: \( a\sin\theta + b\cos\theta = R\sin(\theta + \alpha) \), where \( R = \sqrt{a^2 + b^2} \) and \( \tan\alpha = b/a \).
Step 3: Detailed Explanation:
First Motion:
The equation is \( x_1 = 5 \sin(2\pi t + \frac{\pi}{4}) \).
Comparing this with the standard form \( x = A \sin(\omega t + \phi) \), the amplitude of the first motion is \( A_1 = 5 \).
Second Motion:
The equation is \( x_2 = 5\sqrt{2}(\sin(2\pi t) + \cos(2\pi t)) \).
Let's simplify the term in the parenthesis: \( \sin(2\pi t) + \cos(2\pi t) \).
Here, \( a=1 \) and \( b=1 \). The resultant amplitude of this part is \( R = \sqrt{1^2 + 1^2} = \sqrt{2} \).
The phase angle \(\alpha\) is given by \( \tan\alpha = 1/1 = 1 \), so \( \alpha = \pi/4 \).
Therefore, \( \sin(2\pi t) + \cos(2\pi t) = \sqrt{2} \sin(2\pi t + \pi/4) \).
Substitute this back into the equation for \(x_2\):
\[ x_2 = 5\sqrt{2} \left[ \sqrt{2} \sin(2\pi t + \frac{\pi}{4}) \right] \] \[ x_2 = 5 (\sqrt{2} \cdot \sqrt{2}) \sin(2\pi t + \frac{\pi}{4}) \] \[ x_2 = 10 \sin(2\pi t + \frac{\pi}{4}) \] Comparing this with the standard form, the amplitude of the second motion is \( A_2 = 10 \).
Ratio of Amplitudes:
The question asks for the ratio \( \frac{A_2}{A_1} \).
\[ \frac{A_2}{A_1} = \frac{10}{5} = 2 \] Step 4: Final Answer:
The amplitude of the second motion is 2 times the amplitude of the first motion.