Energy & Work: Fundamental Concepts in Physics

Energy & Work

Introduction

This study guide covers the fundamental concepts of work, kinetic energy, the work-energy principle, and potential energy (both gravitational and elastic) as presented in a college-level introductory physics course. These concepts are essential for understanding how forces cause changes in motion and energy in physical systems.

Work

Definition and Calculation

• Work is defined as the transfer of energy that occurs when a force is applied to an object causing displacement.

• The basic formula for work done by a constant force in the direction of displacement is: $W = Fd$

• If the force is applied at an angle θ to the direction of displacement: $W = Fd\cos\theta$

• For a variable force, work is calculated as: $W = \int F(x)\,dx$

• Units: Joules (J), where 1 J = 1 N·m

Example: Work Done on a Box

• A 20 kg box is pulled across a rough floor (μ = 0.4) with a rope at 45° to the horizontal, moving 10 meters at constant velocity.

• Forces involved: applied force, normal force, gravity, and friction.

• Work done by each force can be calculated using the angle between the force and displacement.

Force	Magnitude (N)	Angle (θ)	Work (J)
Applied Force (FA)	79.2	45°	$(79.2)(10)\cos45° = 560$
Gravity (Fg)	196	90°	$(196)(10)\cos90° = 0$
Normal Force (FN)	140	90°	$(140)(10)\cos90° = 0$
Friction (Ffr)	56	180°	$(56)(10)\cos180° = -560$

Additional info: The net work is zero since the box moves at constant velocity (no change in kinetic energy).

Kinetic Energy (KE)

Definition and Formula

• Kinetic energy is the energy of motion.

• The formula for the kinetic energy of an object of mass m moving at velocity v is: $KE = \frac{1}{2}mv^2$

Work-Energy Principle

• The work-energy principle states that the net work done on an object is equal to its change in kinetic energy:

$W_{net} = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mv_0^2$

• This principle is useful for solving problems involving forces and motion.

Example: Stopping a Runaway Truck

• A 4000 kg truck moving at 25 m/s enters a sandy patch to stop.

• Work required to stop the truck: $W = 0 - \frac{1}{2}(4000)(25^2) = -1.25 \times 10^6$ J$

• If the patch is 100 m long, the friction force required: $F = \frac{W}{d} = \frac{-1.25 \times 10^6}{100} = -12,500$ N

Potential Energy (PE)

Definition and Types

• Potential energy is the energy stored due to an object's position or configuration.

• Main types discussed:

  • Gravitational Potential Energy

  • Elastic (Spring) Potential Energy

Gravitational Potential Energy

• The energy an object has due to its position in a gravitational field.

• Formula: $PE_{grav} = mgh$ where m = mass, g = acceleration due to gravity, h = height above reference point

• Work done to lift an object at constant velocity: $W = mgh$

Example: Carnival Ride Cart

• A cart of mass 500 kg moves through hills of different heights. Calculate gravitational potential energy at each position.

Position	Height (m)	PEgrav (J)
A	40	$(500)(9.8)(40) = 196,000$
B	20	$(500)(9.8)(20) = 98,000$
C	2	$(500)(9.8)(2) = 9,800$

Change in potential energy between positions can be found by subtraction.

Elastic (Spring) Potential Energy

• Energy stored in a stretched or compressed spring.

• Hooke's Law: The force required to stretch or compress a spring by distance x is: $F = kx$ where k = spring constant (stiffness)

• Work done to stretch/compress a spring: $W = \frac{1}{2}kx^2$

• Elastic potential energy stored in the spring: $PE_s = \frac{1}{2}kx^2$

Example: Stretching a Spring

• A perfectly elastic spring requires 0.54 J of work to stretch 6 cm (0.06 m) from equilibrium.

• Find the spring constant k: $0.54 = \frac{1}{2}k(0.06)^2 \implies k = \frac{2 \times 0.54}{0.0036} = 300$ N/m

• Work required to stretch to 12 cm (0.12 m): $W = \frac{1}{2}(300)(0.12)^2 = 2.16$ J

Summary Table: Types of Energy

Type of Energy	Formula	Description
Kinetic Energy	$\frac{1}{2}mv^2$	Energy of motion
Gravitational Potential Energy	$mgh$	Energy due to position in a gravitational field
Elastic Potential Energy	$\frac{1}{2}kx^2$	Energy stored in a stretched/compressed spring

Additional Info

• Other forms of energy (chemical, electrical, electromagnetic, thermal) exist but are not covered in detail here.

• Thermal energy and more advanced topics are typically discussed in later chapters.