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Newton's Laws of Motion and Their Applications

CSCA Newton's Laws of Motion and Their Applications study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Newton's Laws of Motion and Their Applications

1. Overview

Newton's laws establish the causal link between "Force" and "Motion".

* **1st Law (Inertia)**: Force is the cause of **changing** motion, not maintaining it.

* **2nd Law (Dynamics)**: Quantifies the relationship ($F=ma$).

* **3rd Law (Interaction)**: Describes the nature of forces between objects.

2. Key Concepts & Exam Focus

#### A. Newton's First Law

* **Content**: An object remains at rest or in uniform motion unless acted upon by a net external force.

* **Focus**:

* **Inertia**: Depends ONLY on **Mass**, not velocity.

* **Equilibrium**: $a=0 \iff F_{\text{net}} = 0$.

#### B. Newton's Second Law

The core for calculation problems.

* **Formula**: $$\vec{F}_{\text{net}} = \sum \vec{F} = m\vec{a}$$

* **Key Properties**:

1. **Vector Nature**: Calculate x and y components independently.

2. **Instantaneity**: If Force changes, Acceleration changes instantly.

3. **Same-Object**: $F, m, a$ must refer to the same system.

#### C. Newton's Third Law

* **Formula**: $\vec{F}_{A \to B} = -\vec{F}_{B \to A}$

* **Trap Alert**: **Action-Reaction** vs. **Balanced Forces**.

* Action/Reaction: Act on **two different** objects (Cannot cancel out).

* Balanced Forces: Act on the **same** object (Can cancel out).

3. Systematic Approach: Free Body Diagram (FBD)

The "5-Step Method" for dynamics:

1. **Object**: Identify the system.

2. **Draw (FBD)**: Draw all forces (Gravity, Normal, Friction, Applied). **Draw forces received, not exerted.**

3. **Axes**: Establish a coordinate system.

* **Tip**: Align the x-axis with the direction of acceleration.

4. **Resolve**: Decompose forces not on the axes.

5. **Equations**:

* x-axis: $\sum F_x = ma$

* y-axis: $\sum F_y = 0$ (usually balanced vertically)

4. Classic Models

#### Model 1: Block on Horizontal Surface

When friction is involved, first find Normal Force $N$.

* y-axis: $N = mg + F_y$

* x-axis: $F_x - \mu N = ma$

#### Model 2: Inclined Plane

A must-know for CSCA. **Focus on decomposing Gravity ($mg$)**.

* **Axes**: x-axis along the slope, y-axis perpendicular to the slope.

* **Decomposition of Gravity**:

* Down-slope component: $F_{g,x} = mg \sin\theta$

* Perpendicular component: $F_{g,y} = mg \cos\theta$

* **Equations**:

* y-axis: $N - mg \cos\theta = 0 \Rightarrow N = mg \cos\theta$

* x-axis (if frictionless): $mg \sin\theta = ma \Rightarrow a = g \sin\theta$

5. Typical Example

**Example: Friction on an Incline**

**Problem**: A block of mass $m=2\text{kg}$ slides down a rough incline (angle $\theta=30^\circ$) at constant velocity. Find the coefficient of kinetic friction $\mu$.

**Solution**:

1. **FBD**: Gravity $mg$, Normal force $N$, Friction $f$ (up the slope).

2. **State**: Constant velocity $\Rightarrow a=0 \Rightarrow F_{\text{net}}=0$.

3. **Equations**:

* Perpendicular: $N = mg \cos30^\circ$

* Parallel: $mg \sin30^\circ - f = 0 \Rightarrow f = mg \sin30^\circ$

4. **Solve**:

$f = \mu N \Rightarrow \mu (mg \cos30^\circ) = mg \sin30^\circ$

$\mu = \tan30^\circ = \frac{\sqrt{3}}{3} \approx 0.577$

**Conclusion**: The condition for sliding down at constant velocity is $\mu = \tan\theta$, independent of mass.