# Work, Energy, and the Work-Energy Theorem

Physics I · Principles of Mechanics · https://tryals.app/learn/physics-i/work-energy-and-the-work-energy-theorem

## Work & Constant Forces

Energy bookkeeping beats force analysis: you can predict a rollercoaster's final speed without knowing the track's twists, because gravity's work only cares about height.

**Work** is the mechanism by which forces transfer **energy** to or from an object. For a constant force, **Work done by a constant force** is defined as:

$$W = \vec{F}\cdot\vec{d} = Fd\cos\theta$$

where $\theta$ is the angle between $\vec{F}$ and displacement $\vec{d}$. Work is measured in **joules** ($J = N\,m$).

| Quantity | Symbol | SI Unit |
| --- | --- | --- |
| Work | $W$ | J (joule) |
| Kinetic energy | $KE$ | J |
| Power | $P$ | W (watt) |

> **Common pitfall:** A force does zero work when it is perpendicular to the motion ($\theta = 90^\circ$), no matter how large it is. String tension in circular motion and normal force transfer zero energy.

## The Work-Energy Theorem

The **Work-Energy Theorem** states that net work equals the change in **kinetic energy**:

$$W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$

For a variable force, **Work done by a variable force** requires integration:

$$W = \int_{x_i}^{x_f} F(x)\,dx$$

**Power** is the rate of doing work:

$$P = \frac{dW}{dt} = \vec{F}\cdot\vec{v}$$

> **Physics link:** The work-energy theorem is not a new law; it is derived directly from Newton's second law via integration, reframing $F=ma$ into energy terms.

## Practice questions

4 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. Match each quantity to its SI unit.

**Answer:**

- Work → Joule (J)
- Power → Watt (W)
- Force → Newton (N)
- Momentum → kg·m/s

**Why:** Work and energy share the joule. Power is energy per time (W = J/s), and momentum has its own unit, kg·m/s, it is not a form of energy.

Page: https://tryals.app/practice/physics-i/work-energy-and-the-work-energy-theorem/match-each-quantity-to-its-si-unit

### 2. Which of the following are forms of mechanical energy?

A. Kinetic energy
B. Gravitational potential energy
C. Thermal energy
D. Elastic potential energy

**Answer:** A. Kinetic energy; B. Gravitational potential energy; D. Elastic potential energy

**Why:** Mechanical energy = KE + PE (gravitational + elastic). Thermal is internal.

Page: https://tryals.app/practice/physics-i/work-energy-and-the-work-energy-theorem/which-of-the-following-are-forms-of-mechanical-energy

### 3. Carrying a heavy box horizontally at constant speed does no work in the physics sense, though you tire. Why does the definition exclude it?

A. Because work is only defined for objects that change height
B. Because the box’s weight is cancelled by the normal force from your hands
C. Because constant speed means zero net force and so zero work by any force
D. Because the upward force is perpendicular to the motion

**Answer:** D. Because the upward force is perpendicular to the motion

**Why:** The box’s kinetic and potential energy are unchanged, so no work was done ON IT. Your fatigue is real and internal, physiological cost is not mechanical work.

Page: https://tryals.app/practice/physics-i/work-energy-and-the-work-energy-theorem/carrying-a-heavy-box-horizontally-at-constant-speed-does-no-work-in

### 4. You drag a crate 2.0 m with a 10 N force at **60°** above the horizontal. Set the slider to the work done, using $W = Fd\cos\theta$ ($\cos 60° = 0.5$).

**Answer:** 10 (within ±1)

**Why:** $W = 10$ J, half the naive $Fd = 20$ J, because half the force aims uselessly upward. At 90° the cosine kills the work entirely: perpendicular forces move no energy.

Page: https://tryals.app/practice/physics-i/work-energy-and-the-work-energy-theorem/you-drag-a-crate-2-0-m-with-a-10-n-force-at-60-above-the-horizontal