Work and Kinetic Energy: Physics with Calculus 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 or paths are curved. The chapter also covers the calculation of power in physical systems.

• 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 systems with varying forces and curved paths.

• Solve problems involving power.

Motivation for Work and Energy Concepts

Newton's laws are effective for constant forces, but inadequate for variable forces. The introduction of work, energy, and conservation of energy allows for the analysis of more complex systems.

Work in Physics

Definition of Work

In physics, work is defined as the product of force, displacement, and the cosine of the angle between them. It is a scalar quantity, calculated as the dot product of the force and displacement vectors.

• Formula: $W = \vec{F} \cdot \vec{s} = F s \cos\theta$

• 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 ($\theta = 0^\circ$)

• Negative Work: Force opposite to displacement ($\theta = 180^\circ$)

• Zero Work: Force perpendicular to displacement ($\theta = 90^\circ$)

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.

Total work for one repetition (up and down) is zero because 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 is equal to the change in its kinetic energy. This is derived from Newton's second law and the equations of motion.

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

• Work-Energy Theorem: $W = \Delta K = K_2 - K_1$

• Unit: Joule (J)

Example Application

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

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

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

• Upward speed after 2 m: Use $W = \Delta K$ to solve for final velocity.

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.

• Formula: $W_{\text{total}} = W_1 + W_2 + \ldots$

• Application: Tractor pulling a sled with firewood, considering both pulling and friction forces.

Work by Varying Forces

Work with Non-Constant Forces

For forces that vary with position, work is calculated by integrating the force over the displacement.

• Formula: $W = \int_{x_1}^{x_2} F_x(x) \, dx$

• Graphical Interpretation: The area under the force vs. position curve represents the work done.

Work Done by Springs

Hooke's Law and Spring Work

The force required to stretch or compress a spring is proportional to the displacement from equilibrium, described by Hooke's Law.

• Hooke's Law: $F_x = kx$

• Work Done: $W = \frac{1}{2} k X^2$

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

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 Curved Paths

Dot Product Along a Path

When the path is curved, work is calculated by evaluating the dot product of force and displacement along the path.

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

Power

Definition and Units

Power is the rate at which work is done. It can be average or instantaneous.

• Average Power: $P_{\text{avg}} = \frac{\Delta W}{\Delta t}$

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

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

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

Example: Power Climb

A marathon runner climbs the stairs of a tall building. To calculate average power output:

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

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

• Convert to kilowatts or horsepower as needed.

Summary Table: Work and Energy Concepts

Concept	Formula	Unit	Notes
Work	$W = \vec{F} \cdot \vec{s}$	Joule (J)	Dot product; depends on angle
Kinetic Energy	$K = \frac{1}{2} m v^2$	Joule (J)	Energy of motion
Work-Energy Theorem	$W = \Delta K$	Joule (J)	Work changes kinetic energy
Power	$P = \frac{dW}{dt}$	Watt (W)	Rate of doing work
Spring Work	$W = \frac{1}{2} k X^2$	Joule (J)	Area under force vs. position graph

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.