Work-and-Energy-ADA-5thGen-pptx
Work and Energy Overview
Chapters 7 & 8 from Open Stax by Professor Francis Toriello
Focus on the energy of a system, conservative forces, and potential energy.
Scalar Products
Definition: Two methods to multiply vectors, with scalar product being one.
Formula:
In Cartesian Coordinates, the scalar product is defined as:[ A \cdot B = A_x B_x + A_y B_y + A_z B_z ]
Represents projection of one vector onto another.
Interpretation: The scalar product gives a scalar value from two vectors.
Vector Lengths
Finding Length: Calculated using components, equivalently written as:[ |A| = \sqrt{A_x^2 + A_y^2 + A_z^2} ]
Important for understanding energy and work.
Energy and Work
Work Definition: A concrete definition, while energy is more abstract. Work is foundational in deriving energy expressions.
Work Done by a Constant Force
Formula:[ W = F \cdot d \cos(\theta) ]
Work in 1D: If direction aligns with force, it's positive, otherwise negative.
Area Under the Force-Position Curve: Represents net work done on an object.
Direction of Forces and Work Done
Work can be categorized based on the angle between force and displacement:
Positive Work: Force in the same direction as displacement.
Zero Work: Force is perpendicular to displacement.
Negative Work: Force in opposite direction to displacement.
Conservative vs Non-Conservative Forces
Conservative Forces: Work done by these forces is path-independent and related to potential energy.
Example: Gravitational force.
Non-Conservative Forces: Work done is path-dependent and does not store potential energy.
Example: Friction.
Potential Energy**
Definition: Energy stored due to position in a conservative field.
Related to work done by conservative forces, such as gravity.
Formula: [ U = mgh ]
Work-Energy Principle: Relates work done to changes in kinetic and potential energy.
Work-Kinetic Energy Theorem
Indicates that work done by net force results in a change in kinetic energy.
Formula: [ W_{net} = \Delta K ]
Relevant in multiple dimensions (x, y, z).
Law of Conservation of Mechanical Energy
For isolated systems, the total mechanical energy is conserved: [ \Delta K + \Delta U = 0 ]
Useful in solving problems involving energy transitions in a system.
Power
Definition: Rate of doing work or transferring energy.
Measured in Watts (W) where [ P = \frac{dW}{dt} ]
Example: Power required to move objects.
Spring Work and Energy
Work done by springs follows Hooke's Law:
[ F = -kx ]
Work performed during stretching or compressing a spring:[ W_s = \frac{1}{2} k x^2 ]
Potential energy formula for springs:[ U_s = \frac{1}{2} k x^2 ]
Example Problems
Helicopter Example: Work done on an object from horizontal movement.
Frictional Ramp: Analyzes forces on an incline and calculates work done against friction.
Kinematic Examples: Applications of work-energy theorem in scenarios like falling objects and springs.
Review Section
Engage in concept quizzes from Chapter 7 & 8, and read Chapter 9 for continuity.