The graph presented shows the displacement $s$ versus time $t$ for a trapeze bar, which exhibits oscillatory motion. The task is to demonstrate that the amplitude of these oscillations decreases exponentially over time.

Explanation:

1. Oscillatory Behavior: The motion described here seems to be a form of damped harmonic motion, as the oscillations decrease over time.

2. Damped Oscillations: In damped oscillatory systems, the displacement can be modeled by an equation of the form: $s(t) = A_0 e^{-\lambda t} \cos(\omega t + \phi)$ Where:

   • $A_0$ is the initial amplitude,
   • $\lambda$ is the damping coefficient,
   • $\omega$ is the angular frequency of the oscillation,
   • $\phi$ is the phase angle.

3. Exponential Decay: The term $e^{-\lambda t}$ is responsible for the exponential decay of the amplitude. As time $t$ increases, the amplitude $A(t) = A_0 e^{-\lambda t}$ decays exponentially. The graph confirms this behavior: the oscillations start with a larger amplitude and progressively diminish in magnitude.

4. Visual Evidence: From the graph:

   • Initially, the displacement is around 1.5 m.
   • Over time (around 25 to 30 seconds), the displacement reduces to near zero.
   • The reduction in amplitude follows a decaying envelope that resembles an exponential function.

Thus, the observation of the gradually decreasing peak values of the oscillations confirms that the amplitude diminishes exponentially over time.

Would you like to see more detailed equations or further elaboration on any concept?

Follow-up Questions:

1. How do you calculate the time constant $\lambda$ from such a damped motion graph?
2. Can you explain the difference between underdamped, critically damped, and overdamped systems?
3. How would the graph change if there were no damping?
4. What is the physical meaning of the angular frequency $\omega$ in this context?
5. Can you derive the general solution for damped harmonic motion?

Tip:

In damped harmonic motion, the rate of decay of the amplitude is controlled by the damping coefficient $\lambda$. Larger values of $\lambda$ lead to quicker decay.