Chapter 9: Work and Kinetic Energy – Physics with Calculus Study Notes

Work and Kinetic Energy

Energy Overview

Energy is a fundamental concept in physics, representing the ability to cause change or do work. It is central to understanding physical systems and their interactions.

• Kinetic Energy (K): Energy of motion; depends on mass and speed.

• Potential Energy (U): Stored energy associated with position (e.g., gravitational potential energy).

• Thermal Energy (E_{th}): Energy from random motion of atoms and molecules; increases with temperature.

• Energy Conservation: Energy can be transferred or transformed but is conserved within a closed system.

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

The System and the Environment

Energy is analyzed by distinguishing between the system (the object or collection of objects under study) and the environment (everything else).

• Energy Transfer: Occurs via work (mechanical means) or heat (thermal means).

• Energy Transformation: Energy can change form within a system (e.g., kinetic to potential).

Forms of Energy

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

• Potential Energy: For gravity, $U = mgh$

• Thermal Energy: Sum of microscopic kinetic and potential energies of atoms and bonds.

The Energy Principle

The energy principle states that the change in a system's energy equals the work done on it:

• $\Delta E = W_{ext}$

• For kinetic energy: $\Delta K = W$

• For systems with friction: $\Delta K + \Delta E_{th} = W_{ext}$

Work

Work is the process of energy transfer to or from a system by means of a force acting through a displacement.

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

• Positive Work: Force in direction of displacement; increases energy.

• Negative Work: Force opposite to displacement; decreases energy.

• Zero Work: Force perpendicular to displacement; no energy change.

Work Done by a Variable Force

When force varies with position, work is calculated using integration:

• $W = \int_{s_i}^{s_f} F(s) ds$

• Area under the force vs. displacement curve represents work.

Dot Product and Work

The dot product is used to calculate work when force and displacement are vectors:

• Dot Product: $\vec{A} \cdot \vec{B} = AB \cos\alpha$

• Component Form: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

• Work is a scalar quantity resulting from the dot product of force and displacement.

Examples and Applications

• Pulling a Suitcase: Work done by a force at an angle: $W = F \Delta s \cos\theta$

• Launching a Rocket: Work done by thrust and gravity; use energy principle to solve for speed.

• Calculating Work Using Dot Product: For inclined motion, use vector components to find work.

• Zero-Work Situations: Forces perpendicular to motion (e.g., centripetal force in circular motion) do no work.

Restoring Forces and Hooke’s Law

Elastic objects like springs exert restoring forces proportional to displacement from equilibrium.

• Hooke’s Law: $F_{sp} = -k \Delta s$

• Spring Constant (k): Measures stiffness; higher k means stiffer spring.

• Work Done by a Spring: $W_{sp} = \frac{1}{2}k(s_i^2 - s_f^2)$

Dissipative Forces and Thermal Energy

Dissipative forces, such as friction, convert mechanical energy into thermal energy.

• Kinetic Friction: $f_k = \mu_k N$

• Increase in Thermal Energy: $\Delta E_{th} = f_k \Delta s$

• Thermal energy increases in both objects involved in friction.

Power

Power quantifies the rate at which work is done or energy is transferred.

• Definition: $P = \frac{dW}{dt}$

• For constant force and velocity: $P = \vec{F} \cdot \vec{v}$

• Units: Watt (W), where $1\,\text{W} = 1\,\text{J}/\text{s}$; Horsepower (hp) is also used ($1\,\text{hp} = 746\,\text{W}$).

Summary Table: Key Formulas and Concepts

Concept	Formula	Description
Kinetic Energy	$K = \frac{1}{2}mv^2$	Energy of motion
Potential Energy (gravity)	$U = mgh$	Energy due to height
Work (constant force)	$W = F \Delta s \cos\theta$	Energy transfer by force
Work (variable force)	$W = \int F(s) ds$	Integral for non-constant force
Hooke's Law	$F_{sp} = -k \Delta s$	Restoring force of spring
Work by Spring	$W_{sp} = \frac{1}{2}k(s_i^2 - s_f^2)$	Work done as spring changes length
Power	$P = \frac{dW}{dt}$	Rate of energy transfer
Power (force & velocity)	$P = \vec{F} \cdot \vec{v}$	Instantaneous power
Dot Product	$\vec{A} \cdot \vec{B} = AB \cos\alpha$	Scalar product of vectors

Examples and Applications

• Pulling a suitcase: Calculate work using force, displacement, and angle.

• Launching a rocket: Use energy principle to find final speed after work by thrust and gravity.

• Stick-slip motion: Earthquakes occur when elastic restoring force overcomes friction.

• Increasing kinetic and thermal energy: Work done by tension and friction increases both kinetic and thermal energy.

• Lifting a motor: Power output calculated from work and time.

Important Concepts

• Energy is a property of the system.

• Work transfers energy between system and environment.

• Dissipative forces transform mechanical energy into thermal energy.

• Power measures how quickly energy is transferred or transformed.

Quick Concept Checks

• Work is positive when force and displacement are in the same direction.

• Friction always does negative work (removes energy from system).

• Forces perpendicular to displacement do zero work.

• Power is measured in watts; higher power means faster energy transfer.

Additional info:

• Thermal energy is only defined for extended objects, not point particles.

• Energy conservation is foundational for all physical processes.

• Dot product is essential for calculating work when force and displacement are not parallel.