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Circular Motion & Gravitation

CSCA Circular Motion & Gravitation study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Circular Motion & Gravitation

1. Kinematics of Circular Motion

Quantities describing circular motion:

* **Linear Velocity ($v$)**: How fast it moves along the path. Tangential direction.

* Formula: $v = \frac{\Delta s}{\Delta t} = \frac{2\pi r}{T}$

* **Angular Velocity ($\omega$)**: How fast the angle changes.

* Formula: $\omega = \frac{\Delta \theta}{\Delta t} = \frac{2\pi}{T} = 2\pi f$

* **Relation**: $$v = \omega r$$

* **Centripetal Acceleration ($a_n$)**: Changes direction only.

* Formula: $$a_n = \frac{v^2}{r} = \omega^2 r = \frac{4\pi^2}{T^2}r$$

* Direction: **Always points to the center**.

2. Dynamics of Circular Motion

#### Centripetal Force ($F_n$)

* **Definition**: The force causing centripetal acceleration.

* **Nature**: It is an **Effect Force**, not a new type of force! It is provided by the **net force** or **component** of gravity, tension, friction, etc.

* **Formula**:

$$F_n = m a_n = m \frac{v^2}{r} = m \omega^2 r$$

#### Classic Model: Conical Pendulum

A standard CSCA model for force analysis.

* **FBD**: Gravity $mg$ (down), Tension $T$ (along string).

* **Net Force**: Horizontal, pointing to the center.

* **Equations**:

* Vertical: $T \cos\theta = mg$

* Horizontal: $T \sin\theta = m \frac{v^2}{r} = m \omega^2 r$

* Geometry: $r = L \sin\theta$

3. Gravitation & Spaceflight

#### A. Law of Universal Gravitation

$$F = G \frac{m_1 m_2}{r^2}$$

#### B. Core Equation for Satellites

**"Gravitational Force provides Centripetal Force"** is the key.

$$G \frac{Mm}{r^2} = m \frac{v^2}{r} = m \omega^2 r = m \frac{4\pi^2}{T^2}r$$

**Key Conclusion**: As orbital radius $r$ increases:

* Velocity $v = \sqrt{\frac{GM}{r}}$ $\downarrow$ (Decreases)

* Period $T = \sqrt{\frac{4\pi^2 r^3}{GM}}$ $\uparrow$ (Increases)

#### C. Golden Substitution

On a planet's surface, Gravity $\approx$ Gravitational Force:

$$G \frac{Mm}{R^2} = mg \quad \Rightarrow \quad GM = gR^2$$

* Used to eliminate $M$ when $g$ and $R$ are known.

#### D. First Cosmic Velocity

The orbital speed close to Earth's surface.

$$v_1 = \sqrt{\frac{GM}{R}} = \sqrt{gR} \approx 7.9 \, \text{km/s}$$

4. Typical Examples

**Example 1: Conical Pendulum**

**Problem**: String length $L$, mass $m$, angle $\theta$. Find velocity $v$.

**Solution**:

$F_{net} = m g \tan\theta$.

$m g \tan\theta = m \frac{v^2}{r}$ where $r = L \sin\theta$.

$v = \sqrt{g L \sin\theta \tan\theta}$.

**Example 2: Orbit Change**

**Problem**: Satellite moves from Low Orbit to High Orbit. How do KE and Period change?

**Solution**:

* $r$ increases $\to$ $v$ decreases $\to$ **KE decreases**.

* $r$ increases $\to$ **Period $T$ increases**.

**Example 3: Golden Substitution**

**Problem**: Planet radius is 2x Earth, Mass is 8x Earth. Find ratio of First Cosmic Velocity.

**Solution**:

$v = \sqrt{\frac{GM}{R}}$.

Ratio = $\sqrt{8 \cdot \frac{1}{2}} = \sqrt{4} = 2$.