Work and Kinetic Energy (Chapter 6): Study Notes

Work and Kinetic Energy

Introduction

This chapter introduces the concepts of work, kinetic energy, and power, which are essential for analyzing physical systems where Newton's laws alone are insufficient. Understanding these concepts allows us to solve a broader range of problems in mechanics, including those involving energy transformations and non-constant forces.

Work

Definition of Work

• Work is done by a force on an object if the object undergoes a displacement.

• Mathematically, for a constant force F acting in the direction of displacement s:

$W = F s$

• Example: Pushing a car along a road involves doing work because a force is applied over a distance.

Units of Work

• The SI unit of work is the joule (J):

$1~\text{J} = 1~\text{N} \cdot 1~\text{m}$

• 1 joule is the work done when a force of 1 newton moves an object 1 meter in the direction of the force.

Work Done by a Constant Force

• If a constant force F acts at an angle $\phi$ to the displacement s:

$W = F s \cos \phi$

• This can be written as the dot product:

$W = \vec{F} \cdot \vec{s}$

Sign of Work

• Positive Work: Force has a component in the direction of displacement ($0 < \phi < 90^\circ$).

• Negative Work: Force has a component opposite to displacement ($90^\circ < \phi < 180^\circ$).

• Zero Work: Force is perpendicular to displacement ($\phi = 90^\circ$), or there is no displacement.

Example: A weightlifter holding a barbell stationary does no work, but lowering the barbell involves both positive and negative work depending on the force and displacement directions.

Total Work

Net Work and Its Effects

• The total work $W_{\text{tot}}$ is the work done by the net force on a particle as it moves.

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

Kinetic Energy

Definition and Properties

• Kinetic energy is the energy of motion of a particle:

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

• Kinetic energy is a scalar quantity and depends only on the mass and speed of the particle, not its direction.

• Kinetic energy is always zero or positive; it is zero only when the particle is at rest.

• The SI unit of kinetic energy is the joule (J).

Dependence on Mass and Speed

• Kinetic energy increases linearly with mass: doubling the mass doubles the kinetic energy (for constant speed).

• Kinetic energy increases with the square of speed: doubling the speed quadruples the kinetic energy (for constant mass).

The Work-Energy Theorem

Statement and Equation

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

$W_{\text{tot}} = K_2 - K_1 = \Delta K$

• This theorem provides a powerful tool for solving problems involving forces and motion.

Work and Kinetic Energy in Composite Systems

Systems with Multiple Parts

• In composite systems, the work done by external forces may be zero, yet the system's kinetic energy can change due to internal forces or redistribution of energy among parts.

• It is sometimes insufficient to model the system as a single point mass.

Work and Energy with Varying Forces

Non-Constant Forces

• When forces are not constant, the work done is calculated by dividing the path into small segments, summing the work for each segment, and taking the limit as the segments become infinitesimal.

• The total work is given by the integral:

$W = \int_{x_1}^{x_2} F_x \, dx$

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

Work Done by a Constant Force (Graphical Interpretation)

• For a constant force, the area under the force vs. displacement graph is a rectangle:

$W = F s$

Stretching a Spring

• The force required to stretch a spring by a distance $x$ is given by Hooke's Law:

$F_x = kx$

• The work done to stretch the spring from $x = 0$ to $x = X$ is:

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

Work-Energy Theorem for Motion Along a Curve

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

$W = \int_{P_1}^{P_2} \vec{F} \cdot d\vec{l}$

• This accounts for both changes in magnitude and direction of the force along the path.

Power

Definition and Units

• Power is the rate at which work is done.

• Average power over a time interval $\Delta t$:

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

• Instantaneous power:

$p = \lim_{\Delta t \to 0} \frac{\Delta W}{\Delta t} = \frac{dW}{dt}$

• The SI unit of power is the watt (W): $1~\text{W} = 1~\text{J}/\text{s}$.

• Another common unit is the horsepower (hp): $1~\text{hp} = 746~\text{W}$.

Power in Terms of Force and Velocity

• In mechanics, power can also be expressed as the dot product of force and velocity:

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

• This form is useful when force and velocity are not necessarily in the same direction.

Examples of Power

• Lifting a box slowly (over 5 s): $P_{\text{av}} = 20~\text{W}$ if $\Delta W = 100~\text{J}$.

• Lifting the same box quickly (over 1 s): $P_{\text{av}} = 100~\text{W}$ for the same work done.

Example: A one-horsepower (746 W) propulsion system is equivalent to the power output of a strong horse.

Additional info: These notes are based on slides from "University Physics with Modern Physics, 15th Edition, Chapter 6: Work and Kinetic Energy." All equations are provided in standard LaTeX format for clarity.