## 17.5 Graphical Representations of Displacement, Velocity and Acceleration for SHM This section describes the mathematical and graphical relationships between displacement, velocity, and acceleration for an object undergoing Simple Harmonic Motion (SHM), starting from an extreme position. ### Equations of Motion for SHM (from extreme position) The motion of a particle in SHM can be described by sinusoidal functions. If the particle starts at its maximum positive displacement ($x = +x_0$) at time $t = 0$, the equations are as follows: 1. **Displacement (x):** The displacement is described by a cosine function. $$x = x_0 \cos(\omega t)$$ - Maximum displacement is $x_0$, also known as the amplitude. 2. **Velocity (v):** The velocity is the first time derivative of displacement. It is represented by a negative sine function. $$v = -x_0 \omega \sin(\omega t)$$ - Velocity is maximum in magnitude ($v_{\text{max}} = \pm x_0 \omega$) at the mean position ($x = 0$). - Velocity is zero at the extreme positions ($x = \pm x_0$). 3. **Acceleration (a):** The acceleration is the first time derivative of velocity. It is represented by a negative cosine function, meaning it is directly proportional to the negative of the displacement ($a = -\omega^2 x$). $$a = -x_0 \omega^2 \cos(\omega t)$$ - Acceleration is maximum in magnitude ($a_{\text{max}} = \mp x_0 \omega^2$) at the extreme positions ($x = \pm x_0$). - Acceleration is zero at the mean position ($x = 0$). ### Graphical Representation The relationship between displacement, velocity, and acceleration over time can be visualized with graphs. ### Phase Differences The graphs show that the three quantities are out of phase with each other. - **Velocity vs. Displacement:** There is a phase difference of $\pi/2$ radians ($90^\circ$). Velocity leads displacement by $\pi/2$. When displacement is at a maximum or minimum, velocity is zero, and vice-versa. - **Acceleration vs. Displacement:** There is a phase difference of $\pi$ radians ($180^\circ$). Acceleration is always in the opposite direction to displacement (in antiphase). When displacement is maximum positive, acceleration is maximum negative. | Quantity | Formula (from extreme pos.) | Maximum Value (Magnitude) | Occurs at... | | :--- | :--- | :--- | :--- | | **Displacement (x)** | $x = x_0 \cos(\omega t)$ | $x_0$ | Extreme positions ($t = 0, T/2, ...$) | | **Velocity (v)** | $v = -x_0 \omega \sin(\omega t)$ | $v_{\text{max}} = x_0 \omega$ | Mean position ($x = 0$) | | **Acceleration (a)** | $a = -x_0 \omega^2 \cos(\omega t)$ | $a_{\text{max}} = x_0 \omega^2$ | Extreme positions ($x = \pm x_0$) | These graphical and mathematical models are fundamental to understanding oscillations in various physical systems. The relationship $a = -\omega^2 x$ forms the basis of Simple Harmonic Motion.