Step 1: Understanding the Concept:
In Simple Harmonic Motion (S.H.M.), the restoring force is proportional to the negative of displacement.
The shapes of the displacement and force graphs must directly reflect this negative proportionality. Step 2: Key Formula or Approach:
Equation of S.H.M.: $F = -ky$.
If the displacement graph is $y = A\sin(\omega t)$, then the force graph is $F = -kA\sin(\omega t)$. Step 3: Detailed Explanation:
The given displacement $y$-$t$ graph is a positive sine wave, which can be mathematically modeled as:
\[ y = A\sin(\omega t) \]
Using the S.H.M. restoring force equation:
\[ F = -ky \]
Substitute the displacement $y$:
\[ F = -k(A\sin(\omega t)) = -F_{\text{max}}\sin(\omega t) \]
This resulting function describes a negative sine wave (an inverted sine wave).
Looking at the options provided in the image:
- Graph (a) starts at a negative maximum, representing $-\cos(\omega t)$.
- Graph (b) starts at a positive maximum, representing $+\cos(\omega t)$.
- Graph (c) starts at zero and initially goes negative, representing $-\sin(\omega t)$.
- Graph (d) starts at zero and initially goes positive, representing $+\sin(\omega t)$.
Therefore, graph (c) perfectly represents the force. Step 4: Final Answer:
Graph (c) is the correct representation.
