Unit 3: Newton's Laws of Motion and Applications

Unit 3: Newton's Laws of Motion and Applications

Introduction

  • Investigate and apply Newton's three laws of motion, with practical examples including circular motion and projectile motion in two dimensions.

Chapter 2: Newtonian Theories of Motion

Section 2.1: Newton's Laws of Motion
Newton's First Law

  • Definition: An object will remain at constant velocity unless acted upon by an unbalanced force.

Newton's Second Law

  • Definition: The acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to its mass.

  • Formula: F_{NET} = ma

    • Where:

      • F_{NET} = net force (N)

      • m = mass (kg)

      • a = acceleration (m/s²)

Newton's Third Law

  • Definition: For every action force, there is an equal and opposite reaction force.

Net Force

  • Definition: Net force is the vector sum of all forces acting on an object.

    • Since force is a vector, consider both magnitude and direction.

    • Add forces acting in the same direction and subtract forces acting in opposite directions.

    • For forces acting at angles, resolve them into horizontal and vertical components using trigonometry.

    • Balanced Forces: F_{NET} = 0

    • No acceleration; object remains at rest or moves with constant velocity.

    • Unbalanced Forces: F_{NET}
      eq 0

    • Causes acceleration in the direction of the net force.

Net Force Questions

  1. A 60 kg person in an elevator accelerating upwards at 2 m/s²:

    • Calculate gravitational force acting on the person:

      • F_{gravity} = mg = 60 kg imes 9.8 m/s² = 588 N

    • Calculate normal force exerted by the elevator:

      • F{normal} = F{gravity} + ma = 588 N + (60 kg imes 2 m/s²) = 588 N + 120 N = 708 N

  2. For a 5 kg object pulled with a force of 40 N at an angle of 30°:

    • Determine its horizontal and vertical components:

      • F{horizontal} = 40 N imes ext{cos}(30°) \ F{vertical} = 40 N imes ext{sin}(30°)

    • Calculate net force in the horizontal direction with friction of 10 N:

      • F{net, horizontal} = F{horizontal} - F_{friction}

Applications of Newton's Laws

  • Example of a toddler dragging a cart:

    • Factors include added blocks increasing mass, force of friction at 5.2 N, and handle angle of 30°.

    • Questions:
      a) Calculate the net force on the cart.
      b) Calculate the force exerted by the toddler on the cart.
      c) Determine the force that the cart exerts on the toddler.

Forces on Inclined Plans
Normal Force

  • Always perpendicular to the surface.

  • On inclined planes, normal force and gravitational force act at an angle leading to a net force down the plane.

Problem Examples: Inclined Plane with Skier

  • Skier of mass 85 kg on a slope inclined at 20° (gravity = 9.8 m/s²).

    • Calculate components of gravitational force:

      • F_{gravity, ||} = mg imes ext{sin}( heta)

      • F_{gravity, ot} = mg imes ext{cos}( heta)

    • Determine normal force acting on the skier

    • Calculate acceleration down the slope.

Chapter 2.2: Circular Motion in a Horizontal Plane

Uniform Circular Motion

  • Object moves with constant speed along a circular path, resulting in change in direction (not speed).

  • Centripetal Acceleration: Acceleration towards the center of the circle.

Centripetal Force

  • Definition: Unbalanced force necessary for centripetal acceleration.

  • Formula: F_{NET} = rac{mv^2}{r}

    • Where:

      • m = mass (kg)

      • v = velocity (m/s)

      • r = radius of circular path (m)

Calculating Speed

  • Example: Water wheel blades with 2.0 m length rotating at 10 revolutions per minute. To find speed of the tips in km/h, convert revolutions to meters traveled.

Example in Sports: Hammer Throw

  • Athlete swinging a 7 kg hammer in a circular path:
    a) Calculate acceleration.
    b) Determine tension in the wire holding the hammer.

Chapter 2.3: Circular Motion on Banked Tracks

Banking Concept

  • Banking angle allows a car to make a turn without requiring friction.

    • heta = an^{-1} rac{rg}{v^2}

    • Where the speed is designed not to require sideways friction.

Example Calculations

  1. Highway banking at speed of 110 km/h, radius of 750 m.

    • Find the angle of banking needed to prevent skidding.

  2. Calculate net force on cyclist at design speed of banked track on an Olympic velodrome.

Chapter 2.4: Circular Motion in a Vertical Plane

Acceleration and Forces in Vertical Circular Motion

  • Varies at different points; critical speed at the top is where normal force becomes zero.

  • Example: Toy car track loop problem with mass, height and radius calculations.

Chapter 2.5 and 2.6: Projectile Motion

Definition

  • A projectile is an object that moves through the air affected only by gravity and air resistance.

Motion Characteristics

  • Projectiles follow a parabolic path due to combination of horizontal and vertical components.

  • Horizontal motion remains constant in absence of air resistance.

Air Resistance Factors

  • Opposes motion; depends on factors such as:

    • Object speed

    • Cross-sectional area

    • Aerodynamic shape

    • Density of air

Formulas

  • Various launch scenarios including horizontal and oblique launches, range, time of flight, and maximum height calculations.

Conservation of Momentum

Conservation in Collisions

  • Momentum is conserved during collisions:

    • p{initial} = p{final} and total change in momentum = 0.

    • Includes examples with vehicle collisions, marble dynamics, and explosive events.

Work, Energy & Impulse

Work and Energy Definitions

  • Work: Transformation of energy due to force causing displacement.

  • Work done formula: W = F imes d

  • Kinetic Energy: Energy due to movement. Potential Energy: Energy stored due to position.