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Work, Energy, and Conservation of Mechanical Energy

CSCA Work, Energy, and Conservation of Mechanical Energy study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Work, Energy, and Conservation of Mechanical Energy

1. Overview

Work and Energy are the core languages for describing "interactions" and "state changes". Derived from Newton's laws, the **Work-Energy Theorem** and **Conservation of Mechanical Energy** provide simpler methods than Newton's Second Law for solving non-uniform motion problems (like curved motion or variable forces).

* **Work**: The measure of energy transfer (Process Quantity).

* **Energy**: The capacity to do work (State Quantity).

2. Work

#### A. Definition & Formula

Work is the accumulation of force over space.

* **Formula**: $$W = F s \cos\theta$$

* $F$: Magnitude of Force.

* $s$: Magnitude of **Displacement** (relative to ground).

* $\theta$: Angle between Force and Displacement vectors.

#### B. Positive vs. Negative Work

* **$0^\circ \le \theta < 90^\circ$**: **Positive Work** (Driving force), promotes motion.

* **$\theta = 90^\circ$**: **No Work** (e.g., Lorentz force, Centripetal force).

* **$90^\circ < \theta \le 180^\circ$**: **Negative Work** (Resisting force), hinders motion.

3. Work-Energy Theorem

The preferred tool for relating force and displacement for a **single object**, especially when **time is not involved**.

* **Statement**: The net work done on an object equals the change in its kinetic energy.

* **Formula**:

$$W_{\text{net}} = W_1 + W_2 + ... = \Delta E_k = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2$$

**CSCA Exam Tip**:

To calculate $W_{\text{net}}$, you have two options:

1. Find Net Force $F_{\text{net}}$ first, then calculate Work (for constant forces).

2. Calculate work for each force individually, then sum them up algebraically (for variable forces or multi-stage motion).

4. Conservation of Mechanical Energy

#### A. Potential Energy (PE)

* **Gravitational PE**: $E_p = mgh$. Note that $h$ is height relative to a **Reference Plane** (usually the lowest point).

* **Work-PE Relation**: Gravity does positive work $\to$ PE decreases. $$W_G = -\Delta E_p = mgh_1 - mgh_2$$

#### B. The Law

* **Condition**: **Only Gravity or Elastic Force does work**.

* *Note: Other forces (like Normal force) can exist, but if they do zero work, energy is still conserved.*

* **Formula**:

1. **State Equation**: $$E_{k1} + E_{p1} = E_{k2} + E_{p2}$$

2. **Transformation Equation**: $$\Delta E_k = -\Delta E_p$$ (Gain in KE = Loss in PE)

5. Strategy: Work-Energy Theorem vs. Conservation?

| Scenario | Preferred Method | Reason |

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

| Friction or Air Resistance does work | **Work-Energy Theorem** | Mechanical energy is not conserved. This theorem applies to ALL processes. |

| Only Gravity/Spring force does work (Smooth slopes, pendulum, projectiles) | **Conservation Law** | Simpler. No need to calculate work; just equate states. |

| Finding work done by a variable force | **Work-Energy Theorem** | Use $\Delta E_k$ to solve for $W$ indirectly. |

6. Typical Examples

**Example 1: Work-Energy Theorem (with Friction)**

**Problem**: A $2\text{kg}$ block moves $5\text{m}$ under a pull $F=10\text{N}$ and friction $f=2\text{N}$. Find final speed ($v_0=0$).

**Solution**:

Net Work: $W_{\text{net}} = W_F + W_f = Fs - fs = 10\times5 - 2\times5 = 40\text{J}$.

Theorem: $40 = \frac{1}{2}mv^2 - 0 \Rightarrow 40 = v^2 \Rightarrow v = \sqrt{40} \approx 6.32\text{m/s}$.

**Example 2: Conservation (Pendulum)**

**Problem**: A ball on a string of length $L$ is released from rest at an angle of $60^\circ$. Find speed at the lowest point.

**Solution**:

1. **Check**: Only Gravity does work (Tension is perpendicular) $\to$ Conserved.

2. **Ref**: Lowest point is $h=0$.

3. **Height**: Initial height $h = L - L\cos60^\circ = \frac{1}{2}L$.

4. **Calc**:

$$mgh = \frac{1}{2}mv^2$$

$$g(\frac{1}{2}L) = \frac{1}{2}v^2 \Rightarrow v = \sqrt{gL}$$