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SHM & Mechanical Waves

CSCA SHM & Mechanical Waves study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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SHM & Mechanical Waves

1. Simple Harmonic Motion (SHM)

#### A. Definition & Features

The most basic periodic motion, defined by a **Restoring Force**.

* **Dynamics**: $F = -kx$ (Force proportional to displacement, opposite direction).

* **Kinematics**: $a = -\omega^2 x$.

#### B. Key Quantities

* **Amplitude ($A$)**: Max displacement from equilibrium.

* **Period ($T$)**: Time for one complete cycle.

* Spring: $T = 2\pi \sqrt{\frac{m}{k}}$

* Pendulum (small angle): $T = 2\pi \sqrt{\frac{L}{g}}$

* **Frequency ($f$)**: $f = 1/T$.

* **Angular Frequency ($\omega$)**: $\omega = 2\pi f = \frac{2\pi}{T}$.

#### C. Vibration Graph ($x-t$ Graph)

Describes the history of **ONE particle**.

* X-axis: Time $t$.

* Y-axis: Displacement $x$.

* Slope: Velocity $v$.

2. Mechanical Waves

#### A. Core Formula

Wave Speed depends on **Medium**; Frequency depends on **Source**.

$$v = \lambda f = \frac{\lambda}{T}$$

#### B. Wave Graph ($y-x$ Graph)

Describes the position of **ALL particles** at **ONE instant**.

* X-axis: Position $x$.

* Y-axis: Displacement $y$.

* **"Same-Side Rule"**: A trick to find direction. On the wave curve, the direction of wave propagation and the direction of particle vibration are on the **same side** of the curve line.

3. Vibration Graph vs. Wave Graph

Most common confusion point!

| Feature | Vibration Graph ($x-t$) | Wave Graph ($y-x$) |

| :--- | :--- | :--- |

| **Subject** | **ONE particle** | **ALL particles** (The medium) |

| **X-Axis** | Time ($t$) | Position ($x$) |

| **Analogy** | A person's "Diary" | A group "Photo" |

| **Read Directly** | Period $T$, Amplitude $A$ | Wavelength $\lambda$, Amplitude $A$ |

4. Typical Examples

**Example 1: Same-Side Rule**

**Problem**: A wave moves RIGHT. Find the vibration direction of particle P at $x=2\text{m}$.

**Solution**:

1. Locate P.

2. Since wave moves RIGHT, imagine the waveform shifting slightly to the right.

3. Or use Same-Side Rule: If P is on an "uphill" slope and wave moves Right, P must move **Down**.

**Example 2: Wave Speed Calculation**

**Problem**: In air, $\lambda_1 = 2\text{m}, v_1 = 340\text{m/s}$. Enters water, $\lambda_2 = 8\text{m}$. Find speed in water.

**Solution**:

Source unchanged $\to$ **Frequency $f$ is constant**.

$f = \frac{v_1}{\lambda_1} = \frac{340}{2} = 170\text{Hz}$.

$v_2 = \lambda_2 f = 8 \times 170 = 1360\text{m/s}$.

**Example 3: Speed from Graphs**

**Problem**: From $x-t$ graph, $T=0.2\text{s}$. From $y-x$ graph, $\lambda=4\text{m}$. Find wave speed.

**Solution**:

$v = \frac{\lambda}{T} = \frac{4}{0.2} = 20\text{m/s}$.