Step 1: Understanding the Concept:
A superposition of two Simple Harmonic Motions (SHM) of the same frequency and in the same direction, where one is a sine wave and the other a cosine wave, results in a single SHM.
: Key Formula or Approach:
For an equation of the form \( y = a \sin \omega t + b \cos \omega t \), the resultant amplitude \( R \) is given by:
\[ R = \sqrt{a^2 + b^2} \]
Step 2: Detailed Explanation:
Given the equation: \( y = 3\sin 314t + 4\cos 314t \).
Here, the amplitude of the sine component is \( a = 3 \text{ cm} \).
The amplitude of the cosine component is \( b = 4 \text{ cm} \).
The resultant amplitude \( A \) is:
\[ A = \sqrt{3^2 + 4^2} \]
\[ A = \sqrt{9 + 16} \]
\[ A = \sqrt{25} = 5 \text{ cm} \]
Step 3: Final Answer:
The amplitude of the resulting SHM is 5 cm.