Work and Kinetic Energy: Mechanics Chapter 6 Study Notes

Work and Kinetic Energy

Goals of Chapter 6

This chapter introduces the concepts of work, kinetic energy, and their relationship, providing tools to analyze systems where forces are not constant. It also covers the calculation of work, the work-energy theorem, and the concept of power.

• Understand and calculate work done by a force.

• Define and interpret kinetic energy.

• Apply the work-energy theorem to relate work and kinetic energy.

• Analyze work for varying forces and curved paths.

• Solve problems involving power.

Motivation for the Chapter

Newton's laws are limited when dealing with non-constant forces. The introduction of work and energy allows for a broader analysis of physical systems, including those with variable forces and complex motion.

Work in Physics

Definition and Calculation

Work is a measure of energy transfer when a force acts upon an object causing displacement. It is defined mathematically as:

• Work formula: $W = F s \cos\theta$

• Scalar product: Work is the dot product of force and displacement vectors.

• SI unit: Joule (J), where $1\,\text{J} = 1\,\text{N} \cdot 1\,\text{m}$

The Sign of Work

Work can be positive, negative, or zero depending on the direction of force relative to displacement:

• Positive work: Force in the direction of displacement.

• Negative work: Force opposite to displacement.

• Zero work: Force perpendicular to displacement.

Negative Work Example

When a weightlifter lowers a barbell, the force exerted by their hands is opposite to the direction of displacement, resulting in negative work.

Example: Total work for one repetition (lifting and lowering) is zero, as the positive work done lifting is canceled by the negative work lowering.

Work-Kinetic Energy Theorem

Statement and Derivation

The work-kinetic energy theorem states that the work done on an object equals the change in its kinetic energy:

• Kinetic energy: $K = \frac{1}{2} m v^2$

• Theorem: $W = \Delta K = K_2 - K_1$

• Units: Joules (J)

Example Application

Consider a truck of mass 3000 kg lifted by a crane exerting an upward force of 31 kN over 2 m:

• Work by crane: $W_{\text{crane}} = F \cdot s$

• Work by gravity: $W_{\text{gravity}} = -mg \cdot s$

• Final speed: Use $W = \Delta K$ to solve for $v$ after 2 m.

Work Done by Several Forces

Superposition Principle

When multiple forces act on an object, the total work is the sum of the work done by each force:

• Example: A tractor pulls a sled with firewood, with forces from the tractor, friction, and gravity.

• Calculation: Add the work from each force to find the net work.

Work by Varying Forces

Concept and Calculation

For forces that change with position, work is calculated by integrating the force over the displacement:

• Approximation: Divide displacement into small segments and sum the work for each.

• Exact calculation: $W = \int_{x_1}^{x_2} F_x \, dx$

Springs and Elastic Work

Hooke's Law and Work

The force required to stretch or compress a spring is proportional to the displacement:

• Hooke's Law: $F_x = kx$

• Spring constant: $k$ measures stiffness (N/m).

• Work done: $W = \frac{1}{2} k X^2$

Spring Examples and Applications

• Slinky: $k \approx 1\,\text{N/m}$

• Car suspension: $k \approx 10^5\,\text{N/m}$

Notes on Spring Work

• Formulas apply to both stretching and compression.

• Force on the spring: $F = kx$; force by the spring: $F = -kx$.

• Work done by the spring is opposite in sign to work done on the spring.

• Choose $x=0$ at equilibrium position.

Work Along a Curved Path

Dot Product Along Path

When an object moves along a curved path, work is calculated by evaluating the dot product of force and displacement at each segment:

• Formula: $W = \int \vec{F} \cdot d\vec{s}$

• Requires knowledge of force direction and path geometry.

Power

Definition and Units

Power is the rate at which work is done:

• Average power: $P_{\text{av}} = \frac{\Delta W}{\Delta t}$

• Instantaneous power: $P = \frac{dW}{dt}$

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

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

Example: Power Climb

A marathon runner climbs the stairs of the Sears Tower. To calculate average power output:

• Find work done against gravity: $W = mgh$

• Divide by time to find power: $P = \frac{W}{t}$

• Convert to kilowatts or horsepower as needed.

Summary Table: Work and Energy Concepts

Concept	Formula	SI Unit
Work (constant force)	$W = F s \cos\theta$	Joule (J)
Kinetic Energy	$K = \frac{1}{2} m v^2$	Joule (J)
Work-Kinetic Energy Theorem	$W = \Delta K$	Joule (J)
Work (variable force)	$W = \int F_x \, dx$	Joule (J)
Work (spring)	$W = \frac{1}{2} k X^2$	Joule (J)
Power (average)	$P_{\text{av}} = \frac{\Delta W}{\Delta t}$	Watt (W)
Power (instantaneous)	$P = \frac{dW}{dt}$	Watt (W)

Additional info: Academic context and examples were expanded for clarity and completeness. All images included are directly relevant to the explanation of the adjacent paragraph.