slides_week07_workKE_Feb18

Chapter 7: Work and Kinetic Energy

Page 1: Introduction

• Overview of work and kinetic energy concepts.

Page 2: Dot Product of Vectors

• Definition: Dot product is also known as scalar product or inner product.

• Concept: Projects vector A onto vector B and vice versa.

Page 3: Computing Dot Product Using Cartesian Decomposition

• Vector representations:

  • A = axî + ayĴ + azĥ

  • B = bxî + byĵ + bzĥ

• Dot product formula:

  • A · B = axbx + ayby + azbz

Page 4: Direction of Dot Product

• Question: Which direction does the dot product point?

  • A. South-east

  • B. West

  • C. North-west

  • D. South-west

  • E. No direction (dot product is a scalar).

Page 5: Work Definition

• Work can be positive, negative, or zero, determined by:

  • Dot product.

• Ingredients:

  • Force (𝐹)

  • Displacement (∆𝑟).

Page 6: Units of Work

• Work equations based on angle (θ):

  • If θ = 0° → W = F∆r.

  • If θ = 90° → W = 0 (since cos 90° = 0).

• Work formula:

  • W = F r cos(θ)

  • Units: 1 J = 1 N·m = kg·m²/s².

Page 7: Conditions of Work

• Maximum work occurs when:

  • Force (e.g., wind) acts in the same direction as displacement.

• If force is not parallel to displacement, the work done decreases.

Page 8: Work as Energy Transfer

• Work measures energy transfer:

  • Positive work → energy transferred to the system.

  • Negative work → energy transferred from the system.

Page 9: Example - Work Done by a Vacuum Cleaner

• Problem setup:

  • Force F = 50.0 N at θ = 30.0°

  • Displacement = 3.00 m to the right.

• Work calculation:

  • W = F∆r cos(θ) = 50.0 N × 3.00 m × cos(30°)

  • Result: W = 130 J.

Page 10: Work Done by a Varying Force

• For a varying force F acting in the x direction:

  • Consider force's x component Fx

  • Work in 1 dimension is the area under the force vs. displacement graph.

Page 11: Calculating Work from a Graph

• Problem setup: Analyzing force variation on a particle's movement from x = 0 to x = 6.0 m.

Page 12: Stepwise Work Calculation

• Work segment from:

  • A to B: 20 J (5.0 N × 4.0 m)

  • B to C: 5.0 J (5.0 N × 2.0 m)

• Total work: W = W_A_B + W_B_C = 20 J + 5.0 J = 25 J.

Page 13: Work Done by a Spring

• Spring work involves:

  • Force depending on position.

  • Work done when it is compressed or extended from equilibrium.

Page 14: Work Done by Gravity

• Work done on falling object:

  • Positive work as it gains kinetic energy.

• Work done on an object moving upward:

  • Negative work as it slows down and loses kinetic energy.

Page 15: Forces Doing Work

• Assessment of forces:

  • Some forces do NO work (perpendicular to displacement).

  • Other forces do positive work due to directional components.

Page 16: Kinetic Energy Overview

• Definition: Kinetic energy is energy associated with motion.

• Example: A moving car has kinetic energy, requiring work to stop it.

• Kinetic energy formula:

  • Change in kinetic energy = work done.

Page 17: Energy-Work Theorem

• Connection to Newton’s 2nd law:

  • An object’s acceleration is proportional to net force acting on it.

  • Change in kinetic energy equals total work done by all forces.

Page 18: Example - Work to Stop a Car

• Problem setup:

  • Car mass = 1000 kg,

  • Speed = 8 m/s.

• Work required to stop:

  • KE = ½ mv² = 32,000 J.

Page 19: Units of Work and Kinetic Energy

• Work and kinetic energy are both measured in Joules (J).

• Units breakdown:

  • Work = Newton × meter

  • Kinetic Energy = kg·m²/s².

Page 20: Example - Block on Frictionless Surface

• Problem: A 6.0 kg block pulled by 12 N force over 3.0 m.

• Required: Find the block’s speed using work-energy theorem.

Page 21: Conclusion of Example

• Applying work-energy theorem yields:

  • v_final = 3.5 m/s derived from total work calculation.