Work and Kinetic Energy: Principles, Calculations, and Applications

Work and Kinetic Energy

Goals for Chapter 6

• Understand and calculate the work done by a force.

• Comprehend the meaning of kinetic energy.

• Learn how work changes the kinetic energy of a body and how to use this principle.

• Relate work and kinetic energy when forces are not constant or the body follows a curved path.

• Solve problems involving power.

Introduction

Newton's laws are effective for constant forces, but new concepts are needed for variable forces. This chapter introduces work, energy, and the conservation of energy to address such problems in mechanics.

Work

Definition and Calculation

Work is done by a force on a body if the body undergoes a displacement. The work W done by a constant force F is:

• Formula: $W = (F \cos \theta) d$ where $\theta$ is the angle between the force and displacement vectors.

• Work is a scalar quantity.

• SI unit: Joule (J) = Newton-meter (Nm).

• If $F$ is in the same direction as $d$, $\theta = 0$, so $W = Fd$.

Negative Work and Total Work

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

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

• Negative: $90^\circ < \theta \leq 180^\circ$

• Zero: $\theta = 90^\circ$

If multiple forces act, each can do work. The total work is:

• $W_{\text{total}} = (F_{\text{total}} \cos \theta) d = F_{\text{total}} d \cos \theta$

Example: Man Pushing a Chair

• Work done by the man on the chair: positive (force and displacement in same direction).

• Work done by the chair on the man: negative (opposite direction).

• Other forces (e.g., friction) may do negative work.

• Total work on the chair: sum of all individual works.

Work Done by Several Forces

When multiple forces act (e.g., tractor pulling a sled), use a free-body diagram to resolve forces and calculate work for each:

• Sum the work done by each force component along the direction of displacement.

Force vs. Displacement Graph

The work done by a force can be found from the area under the force vs. position graph:

• For constant force: Area = $Fd = W$

Work Done by a Variable Force

If force depends on displacement (e.g., spring force $F = -kx$):

• Work to stretch/compress spring by $x$ from equilibrium:

$W = \frac{1}{2} k x^2$

Work, Force, and Displacement (Vector Formulation)

• Definition: $dW \equiv \vec{F} \cdot d\vec{r} = (F \cos \theta) dr$

• Only the component of force parallel to displacement contributes to work.

Work Along the x-Axis

• If $\vec{F} = i F_x$ and $d\vec{r} = i dx$:

$dW = F_x dx$

• If $F_x$ is constant and displacement is $x_a$ to $x_b$:

$W = \int_{x_a}^{x_b} F_x dx = F_x (x_b - x_a)$

• If displacement is from $x_b$ to $x_a$:

$W' = \int_{x_b}^{x_a} F_x dx = F_x (x_a - x_b) = -W$

Work in the xy-Plane

• If $\vec{F} = i F_x + j F_y$ and $d\vec{r} = i dx$:

$dW = F_x dx$

Block with Constant Velocity on Inclined Plane

• Constant velocity implies zero net force: $\vec{F} = \vec{N} + \vec{W} + \vec{F}_a = 0$

• Work done by net force:

$W = \int_{x_0}^{x} (N + W + F_a) dx = \int_{x_0}^{x} (-mg \sin \theta) dx$

Block on Frictionless Surface Attached to Spring

• Restoring force (Hooke's Law): $\vec{F}_R = -k x$

• Work done by restoring force:

$W = \int_{x_1}^{x_f} (-k x) dx = -\frac{k x_f^2}{2} + \frac{k x_1^2}{2}$

Work on a Curved Path

• Displacement may occur along a curved path; integrate along the path:

$W_c = \int_{r_a}^{r_b} F \cos \phi dr$

where $\phi$ is the angle between $\vec{F}$ and $d\vec{r}$ at each point.

Line Integral

• For variable force along a curve:

$W_c = \int_{r_a}^{r_b} \vec{F} \cdot d\vec{r} = \int_{x_a}^{x_b} F_x dx + \int_{y_a}^{y_b} F_y dy$

Parametrizing Curved Path

• Curve described by $y = f(x)$, use $dy = (df/dx) dx$:

$W_c = \int_{x_a}^{x_b} F_x [x, f(x)] dx + \int_{x_a}^{x_b} F_y [x, f(x)] \frac{df(x)}{dx} dx$

Pendulum and Horizontal Force

• Horizontal force applied so net force is zero: $\vec{F} + \vec{T} + \vec{W} = 0$

• Work done by $\vec{F}$ as pendulum moves between two points:

$W_c = m g l (1 - \cos \theta_f)$

Kinetic Energy

Definition

Kinetic energy is the energy of motion:

$K = \frac{1}{2} m v^2$

• m: mass

• v: speed

• Unit: Joule (J)

• Kinetic energy is a scalar.

Work and Energy

• Work: Effect of a force on an object, causing change in position or state.

• Energy: Ability to do work; property of an object or system.

• Energy can be exchanged but not created or destroyed (conservation).

Kinetic Energy and Net Work

• Net work on a body changes its speed and thus its kinetic energy.

Work-Energy Theorem

The net work done on an object by the total force is equal to the change in its kinetic energy:

$W_{\text{total}} = \Delta K = K_{\text{final}} - K_{\text{initial}}$

Kinetic Energy Theorem (General Form)

• For any force $\vec{F}$:

$W = \int_{a}^{b} \vec{F} \cdot d\vec{r} = K_f - K_i$

Work done by any force equals the change in kinetic energy of the body.

Applications

Mass in Free Fall

• Mass dropped from rest at height $H$:

$v_f = \sqrt{2gH}$

Derived using the work-energy theorem.

Block Moving on Surface (Friction)

• Block of mass $m$, initial velocity $v_0$, coefficient of kinetic friction $\mu_k$.

• Stopping distance:

$d = \frac{v_0^2}{2 \mu_k g}$

Does not depend on mass.

Simple Harmonic Oscillator

• Block attached to spring, moving without friction.

• Work done by spring force:

$W = \frac{k x_i^2}{2} - \frac{k x^2}{2}$

• Velocity at position $x$:

$v^2 = V^2 - \frac{k}{m} x^2$

Conservative Forces

• Forces like spring restoring force are conservative: kinetic energy is restored when returning to original position.

Important Notes

• General form for variable net force:

$\Delta W = \int_{r_{\text{initial}}}^{r_{\text{final}}} \vec{F} \cdot d\vec{s} = \frac{m}{2} v_{\text{final}}^2 - \frac{m}{2} v_{\text{initial}}^2$

• Work can be positive or negative, depending on whether kinetic energy increases or decreases.

• Positive work on one object is often accompanied by negative work on another (Newton's 3rd law).

More Important Notes

• Work can be zero even if net force is zero (e.g., no displacement, perpendicular displacement).

• Normal forces and static friction do not do work; tension and kinetic friction can.

• Kinetic energy is always positive and independent of direction.

• All results are valid only in an inertial system of reference.

Power

Definition and Calculation

• Average Power: Rate of doing work.

$P = \frac{W}{t} = \frac{Fd}{t} = Fv$

• Unit: Watt (W) = Joule/second (J/s)

Instantaneous Power

• Instantaneous rate at which work is done:

$P = \frac{dW}{dt}$

• Electrical energy consumption is measured in kilowatt-hours (kWh), a unit of work/energy, not power.

$1 \text{ kWh} = 3.6 \times 10^6 \text{ J}$

Power, Force, and Velocity

• Express power in terms of force and velocity:

$P = \vec{F} \cdot \vec{v}$

This is the instantaneous rate at which force does work on a particle.

Analogy: Work, Energy, and Power

• Object (system) compared to a person.

• External source of force compared to external source of money.

• Work done compared to cash flow; kinetic energy compared to total cash in pocket.

• Power compared to cash flow per unit time.

Summary Table: Key Concepts

Concept	Definition	Formula	SI Unit
Work	Force applied over displacement	$W = (F \cos \theta) d$	Joule (J)
Kinetic Energy	Energy of motion	$K = \frac{1}{2} m v^2$	Joule (J)
Work-Energy Theorem	Net work equals change in kinetic energy	$W_{\text{total}} = \Delta K$	Joule (J)
Power	Rate of doing work	$P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$	Watt (W)

Additional info: The notes include analogies to financial concepts to help students understand the transfer and conservation of energy. All equations are valid in inertial frames of reference.