Work and Kinetic Energy

Introduction

This chapter explores the concepts of work, kinetic energy, and power in classical mechanics. It covers how work is calculated for both constant and varying forces, the relationship between work and kinetic energy, and the application of these principles to springs and real-world scenarios.

Work and Energy with Varying Forces

Non-Constant Forces

• Many forces in nature, such as the force required to stretch a spring, are not constant.

• When stretching a spring, the force required increases with displacement, making it a variable force.

• Examples include springs in vehicles and pogo sticks.

Calculating Work for Varying Forces

• Suppose a particle moves along the x-axis from to under a changing force .

• The total work done by the force is the integral of the force over the displacement:

• On a graph of force vs. position, the area under the curve between the initial and final positions represents the total work done.

Work Done by a Constant Force

Definition and Calculation

• If the force is constant and acts along the direction of displacement , the work done is:

• Graphically, this is the area of a rectangle under the force vs. position graph.

Example: Varying Force on a Cow

• A force with x-component is applied as a cow moves from to m.

• Work is calculated by integrating the force over the displacement:

• Result: J (negative because the force opposes the cow's motion).

Example: Work from a Force-Position Graph

• For a force that varies with position, the work done as the object moves from m to m is the area under the curve.

• Answer: J

Calculating Total Work Done from a Graph

Area Under the Curve

• For a force that is constant over some intervals and varies over others, split the area into geometric shapes (rectangles, triangles) and sum their areas.

• Example:

• Example:

• Total:

Stretching a Spring: Hooke's Law

Spring Force and Spring Constant

• The force required to compress or stretch a spring is given by Hooke's Law:

• k is the spring constant, measuring the stiffness of the spring (units: N/m).

• The negative sign indicates the force is a restoring force, acting opposite to displacement.

Work Done by a Spring Force

• To stretch a spring from to :

• If and , then (work done by the spring).

Work-Kinetic Energy Theorem

Statement and Derivation

• The work-kinetic energy theorem relates the net work done on a particle to its change in kinetic energy.

• If a particle of mass moves along the x-axis under net force :

• Thus,

Example: Spring Compression

• A canister of mass kg slides at m/s and compresses a spring ( N/m) until it stops.

• Set kinetic energy equal to spring potential energy:

• Solve for :

m = 1.2 cm

Power

Definition and Units

• Power is the rate at which work is done.

• Average power:

• Instantaneous power:

• SI unit: watt (W), where

• US Customary unit: horsepower (hp),

Power in Terms of Force and Velocity

• Average power can also be written as:

• Useful for calculating power required to move objects at constant speed against forces like gravity or friction.

Kilowatt-Hours

• Electric companies bill by the kilowatt-hour (kWh), a unit of energy.

Examples and Applications

• If you lift a 100 N barbell 1 m, you do 100 J of work.

• Doing the same work in less time increases power output.

• Example: A 700 N marine climbs a 10 m rope in 8 s. Power output:

• Example: Elevator motor lifts 1800 kg (elevator + load) at 3 m/s against 4000 N friction:

Summary Table: Key Equations and Concepts

Key Points to Remember

• Work is the product of force and displacement in the direction of the force.

• For variable forces, work is the area under the force vs. position graph.

• Hooke's Law describes the force in an ideal spring; work done by a spring is related to the change in its potential energy.

• The work-kinetic energy theorem connects the net work done to the change in kinetic energy.

• Power quantifies how quickly work is done; higher power means more work in less time.

• Units of power and energy are crucial for understanding real-world applications, such as electricity billing.