Work, Kinetic Energy, and Power: Chapter 7 Study Notes

Work and Kinetic Energy

Overview

This chapter introduces the concepts of work, kinetic energy, and power in physics. These are fundamental ideas for understanding how forces transfer energy to objects and how energy is quantified and utilized in physical systems.

7-1 Work Done by a Constant Force

Definition of Work

• Work is done when a force causes a displacement of an object.

• If the force is parallel to the displacement, work is calculated as: $W = Fd$

• SI unit of work: newton-meter (N·m), also called the joule (J).

Typical Values of Work

Work can vary greatly depending on the activity. The following table provides examples:

Activity	Equivalent work (J)
Annual U.S. energy use	$8 \times 10^{19}$
Mt. St. Helens eruption	$10^{17}$
Burning 1 gallon of gas	$10^{8}$
Human food intake/day	$10^{7}$
Melting an ice cube	$3 \times 10^{3}$
Lighting a 100-W bulb for 1 minute	6000
Heartbeat	1
Turning a page of a book	$10^{-2}$
Hop of a flea	$10^{-6}$
Breaking a bond in DNA	$10^{-19}$

Work at an Angle

• If the force is at an angle $\theta$ to the displacement, only the component of the force in the direction of displacement does work:

• $W = (F \cos \theta)d = Fd \cos \theta$

Dot Product Formulation

• Work can be expressed as the dot product of force and displacement vectors:

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

Sign of Work

• Work can be positive, zero, or negative depending on the angle between force and displacement:

• Positive work: $0^\circ < \theta < 90^\circ$

• Zero work: $\theta = 90^\circ$

• Negative work: $90^\circ < \theta < 270^\circ$

Work by Multiple Forces

• If multiple forces act, the net work is the sum of the work done by each force:

• $W_{net} = W_1 + W_2 + \ldots = F_1 d \cos \theta_1 + F_2 d \cos \theta_2 + \ldots$

• Alternatively, use the net force:

• $W_{net} = F_{net} d \cos \theta$

Example Problems

• Example 7.01: Calculating work done by a tension force at an angle and parallel to the ground.

• Example 7.02: Determining work done by all forces (including friction) on a block being pushed across a rough floor.

Conceptual Questions

• Friction does negative work when opposing motion.

• Tension in a string whirling a ball in a circle does no work (force is perpendicular to displacement).

• When lifting a book at constant speed, the total work done is zero (forces balance, no net change in kinetic energy).

7-2 Kinetic Energy and the Work-Energy Theorem

Kinetic Energy

• Kinetic energy (K) is the energy of motion:

• $K = \frac{1}{2} m v^2$

• It is always non-negative since mass and velocity squared are always positive.

Work-Energy Theorem

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

• $W_{total} = \Delta K = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2$

• Positive work increases speed; negative work decreases speed.

Example Problem

• Example 7.03: Using the work-energy theorem to find the initial speed of a skier given the distance and friction.

7-3 Work Done by a Variable Force

Graphical Interpretation

• If force is constant, work is the area under the force vs. position graph:

• Area = $Fd = W$

Work Done by a Spring

• The force needed to stretch a spring is $F = kx$, where $k$ is the spring constant and $x$ is the displacement.

• Work done in stretching a spring:

• $W = \frac{1}{2} k x^2$

Example Problem

• Exercise 7-12: Calculating work required to stretch a spring and finding compression distance given work and spring constant.

• For compression distance: $x = \sqrt{\frac{2W}{k}}$

7-4 Power

Definition of Power

• Power (P) is the rate at which work is done:

• $P = \frac{W}{t}$

• SI unit: watt (W), where $1 \text{ W} = 1 \text{ J/s}$

• Horsepower: $1 \text{ hp} = 746 \text{ W}$

Typical Values of Power

Source	Approximate power (W)
Hoover Dam	$1.34 \times 10^9$
Car moving at 40 mi/h	$7 \times 10^4$
Home stove	$1.2 \times 10^3$
Sunlight on one square meter	1380
Refrigerator	615
Television	200
Person walking up stairs	150
Human brain	20

Power for Constant Speed Motion

• If an object moves at constant speed $v$ under a force $F$:

• $P = Fv$

Example Problem

• Example 7.04: Calculating average power output during a football throw.

Conceptual Questions

• Work depends only on force and displacement, not on time taken.

• Power depends on both work done and time taken.

• Comparing two people doing the same work in different times: same work, but the one who does it faster produces more power.

Summary Table: Key Equations

Concept	Equation
Work (parallel force)	$W = Fd$
Work (angle)	$W = Fd \cos \theta$
Kinetic Energy	$K = \frac{1}{2} m v^2$
Work-Energy Theorem	$W_{total} = \Delta K$
Work by a spring	$W = \frac{1}{2} k x^2$
Power	$P = \frac{W}{t}$
Power (constant speed)	$P = Fv$

Additional info: These notes expand on the lecture outline by providing definitions, equations, and context for each concept, as well as examples and tables for typical values. The notes are suitable for exam preparation and self-study for college-level physics students.