Work and Kinetic Energy – University Physics Chapter 6 Study Notes

Work and Kinetic Energy

Learning Outcomes

This chapter introduces the fundamental concepts of work, kinetic energy, and power in classical mechanics. Students will learn how forces transfer energy to objects, how to calculate work, and how energy changes in various physical scenarios.

• Work: Understanding what it means for a force to do work on an object and how to calculate the amount of work done.

• Kinetic Energy: Defining kinetic energy (energy of motion) and relating work to changes in kinetic energy.

• Work-Energy Principle: Using the relationship between total work and changes in kinetic energy, including cases with non-constant forces and curved paths.

• Power: Solving problems involving power, the rate at which work is done.

Introduction to Work, Energy, and Conservation

In many physical problems, Newton's laws alone are insufficient. The concepts of work, energy, and conservation of energy provide powerful tools for analyzing systems where forces transfer energy and change the state of motion.

• Example: A baseball pitcher does work on the ball, giving it kinetic energy.

• Conservation of Energy: Energy can be transferred between objects but is conserved in isolated systems.

Work

Definition of Work

A force does work on an object if the object undergoes a displacement due to that force.

• Formula for Work (Constant Force):

$W = F s$

• Direction: The force must have a component in the direction of the displacement.

• Example: Pushing a car: The force exerted by people on the car as it moves is work.

Units of Work

The SI unit of work is the joule (J), named after James Prescott Joule.

• Definition: $1 ext{ J} = 1 ext{ N} imes 1 ext{ m}$

• Example: Lifting a 1 N apple by 1 meter requires 1 J of work.

Work Done by a Constant Force at an Angle

When a constant force acts at an angle to the displacement, only the component of the force in the direction of displacement does work.

• Formula:

$W = F s \, \cos \theta$

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

Dot Product and Work

The dot product projects one vector onto another, quantifying how much of one vector lies along the direction of the other.

• Formula:

$\vec{A} \cdot \vec{B} = |A||B| \cos \theta$

• Application: In work calculations, this allows us to sum work from multiple forces without decomposing vectors.

Positive, Negative, and Zero Work

• Positive Work: Force component is in the direction of displacement.

• Negative Work: Force component is opposite to displacement.

• Zero Work: Force is perpendicular to displacement or object does not move.

• Example: A weightlifter holding a stationary barbell does no work; lowering the barbell involves negative work by the lifter's hands.

Work as Energy Transfer

Work is a transfer of energy between two objects. According to Newton's third law, forces come in pairs, and work done by one object on another is matched by negative work in the opposite direction.

• Key Point: Objects do not 'have' work; work is an exchange of energy.

Kinetic Energy

Definition of Kinetic Energy

Kinetic energy is the energy of motion of a particle.

• Formula:

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

• Properties: Scalar quantity, depends only on mass and speed, always non-negative.

• SI Unit: Joule (J).

• Frame Dependence: Kinetic energy depends on the reference frame.

Dependence on Mass and Speed

• Mass: Kinetic energy increases linearly with mass.

• Speed: Kinetic energy increases with the square of speed.

• Example: Doubling mass doubles kinetic energy; doubling speed quadruples kinetic energy.

The Work-Energy Theorem

Statement and Derivation

The work-energy theorem states that the work done by the net force on a particle equals the change in its kinetic energy.

• Formula:

$W_{\text{net}} = \Delta K = K_f - K_i$

• Interpretation: If $W_{\text{net}} > 0$, the particle speeds up; if $W_{\text{net}} < 0$, it slows down; if $W_{\text{net}} = 0$, speed remains constant.

• Example: Calculating the speed of an object after work is done on it.

Application to Composite Systems

In systems with multiple parts, external forces may do zero net work, but internal energy changes can still occur. The work-energy theorem applies to the system as a whole.

Work and Energy with Varying Forces

Work Done by a Variable Force

When the force varies with position, work is calculated by integrating the force over the path of motion.

• Formula:

$W = \int_{x_i}^{x_f} F_x \, dx$

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

Work Done by a Constant Force (Graphical)

For a constant force, the work is the area of a rectangle under the force vs. displacement graph.

• Formula: $W = F s$

Stretching a Spring (Hooke's Law)

The force required to stretch a spring is proportional to the displacement: $F = kx$.

• Work Done:

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

• Application: Calculating energy stored in a spring.

Nonideal Springs and Hysteresis

Real springs and tendons may not follow Hooke's Law perfectly, exhibiting hysteresis and nonlinear behavior. However, Hooke's Law is a reasonable approximation for many cases.

Work-Energy Theorem for Curved Paths

Motion Along a Curve

For a particle moving along a curved path under a varying force, work is calculated using the line integral:

$W = \int_{P_i}^{P_f} \vec{F} \cdot d\vec{l}$

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

Power

Definition of Power

Power is the rate at which work is done.

• Average Power:

$P_{\text{avg}} = \frac{\Delta W}{\Delta t}$

• Instantaneous Power:

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

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

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

Power in Terms of Force and Velocity

• Formula:

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

• Application: Calculating the power delivered by engines, motors, or humans during physical tasks.

Examples of Power Output

Power Output	Time Interval
2 hp (1500 W)	6 s
1 hp (750 W)	60 s
0.35 hp (260 W)	35 min
0.2 hp (150 W)	5 h
0.1 hp (75 W)	8 h (average daily level)

Example: Power Delivered by an Elevator Motor

• Calculation: The power required to lift an elevator car and load at constant speed can be found using $P = F v$.

• Application: Engineering design for motors and lifting systems.