Work, Kinetic Energy, and Power: Physics with Calculus Study Notes

Work and Kinetic Energy

Energy and Its Conservation

Energy is a fundamental concept in physics, representing the ability of a system to do work. Energy can be transferred or transformed between different forms, but it is always conserved in isolated systems. The main types of energy include kinetic energy, potential energy, internal energy, and electromagnetic energy.

• Kinetic Energy: Energy of motion.

• Potential Energy: Energy due to position or configuration.

• Internal Energy: Energy from random molecular motion or configuration.

• Electromagnetic Energy: Energy carried by electromagnetic waves.

System: A defined portion of the universe whose energy is being studied. The system boundary separates the system from its surroundings, allowing energy to flow in or out as mechanical, heat, or electromagnetic energy.

Work Done by a Constant Force

Work is a measure of energy transfer when a force acts upon an object causing displacement. For a constant force in one dimension, work is calculated as:

• Formula: $W = F_x \Delta x$

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

Work: Positive, Negative, or Zero

The sign of work depends on the direction of the force relative to the displacement:

• Positive Work: Force in the same direction as motion ($W > 0$)

• Negative Work: Force opposite to motion ($W < 0$)

• Zero Work: Force perpendicular to motion ($W = 0$)

Example: Work Calculation

When pulling a sled with a force at an angle, only the component of force in the direction of motion contributes to work:

• Example: Pulling a sled (80 kg) with a force of 180 N at 40° above the horizontal for 5.0 m.

• Calculation: $W = F \cos\theta \cdot \Delta x$

Scalar Product (Dot Product) and Work

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

• Definition: $\vec{A} \cdot \vec{B} = AB \cos\theta$

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

• Work as Dot Product: $W = \vec{F} \cdot \Delta \vec{r} = F \Delta r \cos\theta$

Net Work

Net work is the sum of work done by all forces acting on an object or system:

• Formula: $W_{\text{net}} = W_1 + W_2 + W_3 + \ldots$

• The net work is equivalent to the work done by the net force.

Work Done by Gravity and Kinetic Friction

Gravity and friction are common forces that do work on objects:

• Work by Gravity: $W_g = mg \Delta y$ (positive if moving downward, negative if upward)

• Work by Kinetic Friction: Always negative, as friction opposes motion.

Kinetic Energy and the Work-Energy Theorem

Kinetic energy is the energy of motion, given by:

• Formula: $K = \frac{1}{2}mv^2$

• Kinetic energy is always positive.

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

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

Work Done by a Variable Force

When force varies with position, work is calculated using integration. The area under the force-versus-position curve represents the work done:

• Approximation: $W \approx \sum_{i=1}^N F(x_i) \Delta x_i$

• Exact Calculation: $W = \int_{x_1}^{x_2} F(x) dx$

Work Done by a Spring

A spring exerts a force proportional to its displacement from equilibrium, described by Hooke's Law:

• Spring Force: $F_{\text{spring}} = -kx$

• Work Done by Spring: $W = -\frac{1}{2}k(x_f^2 - x_i^2)$

Power

Power is the rate at which work is done or energy is transferred:

• Average Power: $\bar{P} = \frac{\Delta W}{\Delta t}$

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

• Power by Force: $P = \vec{F} \cdot \vec{v}$

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

Summary Table: Work and Energy Concepts

Concept	Formula	Key Points
Work (constant force)	$W = F_x \Delta x$	Energy transfer by force in direction of displacement
Work (variable force)	$W = \int_{x_1}^{x_2} F(x) dx$	Area under force-position curve
Kinetic Energy	$K = \frac{1}{2}mv^2$	Energy of motion, always positive
Work-Energy Theorem	$W_{\text{net}} = \Delta K$	Net work equals change in kinetic energy
Power	$P = \frac{dW}{dt}$	Rate of doing work, measured in watts

Example Applications: Calculating work done by forces in various scenarios, determining kinetic energy changes, and evaluating power requirements in mechanical systems.

Additional info: Integration and calculus are essential for analyzing work done by variable forces, and understanding energy conservation is foundational for all physics topics.