Relative Velocity and Introduction to Dynamics and Inertia

Inclined Plane Homework & Vector Decomposition

• Vector decomposition on an inclined surface:

  • For a box sitting on an incline with an angle of $37^\circ$, the weight vector acts vertically downward.
  • A standard coordinate system is established relative to the incline surface to split the weight vector into two components:
  • Perpendicular component ($y$-component), acting normal to the surface.
  • Parallel component ($x$-component), acting parallel down the inclined plane.
  • By similar triangles, the angle between the vertical weight vector and the perpendicular ($y$) axis is equal to the incline angle of $37^\circ$.

• Units of Force:

  • The standard unit of force in the SI system is the Newton ($\text{N}$).
  • The standard unit of force in the United States customary system is the pound ($\text{lb}$).

Relative Velocity: Airplane Flying in Wind Example

• Problem Setup & System Parameters:

  • Airspeed of an airplane in still air: $v_{P/A} = 200\,\text{km/h}$.
  • Target relative velocity of the airplane relative to the ground (groundspeed): $\mathbf{v}_{P/G}$ directed due North ($+y$ direction).
  • Wind velocity blowing out of the West towards the East: velocity of air relative to the ground $\mathbf{v}_{A/G} = 50\,\text{km/h}$ in the positive $x$-direction.
  • Coordinate System Assignment:
  • Positive $x$-axis: East
  • Negative $x$-axis: West
  • Positive $y$-axis: North
  • Negative $y$-axis: South

• Vector Addition Equation:

  • The general relative velocity vector equation is:     $\mathbf{v}_{P/G} = \mathbf{v}_{P/A} + \mathbf{v}_{A/G}$
  • To counter the eastward wind drift, the airplane must turn its nose into the wind at an angle $\phi$ West of North.
  • Component forms of the vectors:     $\mathbf{v}_{A/G} = 50 \hat{\mathbf{i}}\,\text{km/h}$$\mathbf{v}_{P/G} = v_{P/G} \hat{\mathbf{j}}$$\mathbf{v}_{P/A} = -v_{P/A} \sin(\phi) \hat{\mathbf{i}} + v_{P/A} \cos(\phi) \hat{\mathbf{j}}$

• Solving Along Component Directions:

  • $x$-direction equation:     $-v_{P/A} \sin(\phi) + v_{A/G} = 0$$-200 \sin(\phi) + 50 = 0$$\sin(\phi) = \frac{50}{200} = 0.25$$\phi = \arcsin(0.25) \approx 14.48^\circ$

  • The plane must direct its nose at an angle of $14.48^\circ$ West of North.

  • $y$-direction equation:     $v_{P/G} = v_{P/A} \cos(\phi)$$v_{P/G} = 200 \cos(14.48^\circ) \approx 193.65\,\text{km/h}$

  • To an observer standing on the ground, the airplane points at $14.48^\circ$ West of North but moves straight North along the ground at a speed of $193.65\,\text{km/h}$.

Lab Feedback & Conceptual Motion Analysis

• Motion Graph Interpretation:
  • Identifying direction changes on position vs. time and velocity vs. time graphs:
  • A change in direction occurs when the sign of velocity transitions from positive to negative or from negative to positive.
  • At the exact instant an object changes its direction of motion, its instantaneous velocity is $0\,\text{m/s}$.

Introduction to Chapter 4: Force and Dynamics

• Kinematics vs. Dynamics:

  • Kinematics: The description of motion without considering its causes.
  • Dynamics: The study of forces, inertia, and the resulting changes in motion.
  • Terminology Origin: The concept and term "inertia" originated from Galileo.

• Operational Definition of Force:

  • Broadly understood as pushes and pulls.
  • Scientifically defined by what it causes: forces produce changes in motion.
  • A force can:
  • Set a stationary object into motion.
  • Cause a moving object to speed up or slow down.
  • Change the direction of motion of an object.
  • Because velocity changes over time ($\frac{\Delta v}{\Delta t}$), force produces an acceleration.

Net Force and Vector Addition of Forces

• Vector Nature of Force:

  • Force is a vector quantity possessing both magnitude and direction.
  • A single applied force will not produce an acceleration if it is balanced by one or more opposing forces.

• Net Force Calculation:

  • The net force $\mathbf{F}_{\text{net}}$ acting on an object is defined as the vector sum of all individual forces $\mathbf{F}_i$ acting upon it:     $\mathbf{F}_{\text{net}} = \sum_{i=1}^{n} \mathbf{F}_i$

• Applied Horizontal Force Scenarios on a Surface:

  • Scenario A: Two forces $F_1$ and $F_2$ applied horizontally in opposite directions with equal magnitudes ($F_1 = F_2$).
  • Net horizontal force is zero ($\mathbf{F}_{\text{net},x} = 0$).
  • The object does not accelerate horizontally.
  • Scenario B: Two forces applied horizontally in opposite directions where $F_2 > F_1$
  • Net force is non-zero in the direction of $F_2$ (positive $x$-direction).
  • The object accelerates horizontally in the positive $x$-direction.

• Vertical Force Equilibrium for Objects Resting on Surfaces:

  • Downward vertical force: Gravitational force exerted by the Earth, equal to the object's weight.
  • Upward vertical force: Normal supporting force exerted by the floor or surface.
  • When an object rests on a floor without vertical motion, the upward floor force balances the downward weight, yielding zero vertical acceleration.

Classification of Forces: Contact vs. Action-at-a-Distance

• Contact Forces:

  • Forces transmitted through physical contact between macroscopic objects.
  • Examples:
  • Normal force (e.g., pushing horizontally on a door).
  • Tension force (e.g., pulling on a rope attached to a mass).
  • Friction force (e.g., interaction between two surfaces sliding or rubbing against each other).

• Action-at-a-Distance Forces (Field Forces):

  • Forces exerted without any physical contact between objects.
  • Examples:
  • Gravity: Causes objects to accelerate downward when unsupported (e.g., falling after jumping off a diving board).
  • Magnetism: Exerted by magnetized material (historically referred to as lodestones).
  • Electricity: Interactions resulting from fundamental charges in matter.
  • Historical Context: Sir Isaac Newton identified the nature of action-at-a-distance forces and established the mathematical equation governing universal gravitational attraction.

Galileo's Experiments and the Concept of Inertia

• Historical Context & Incline Experiments:

  • Galileo conducted motion experiments prior to Newton and established constant-acceleration kinematic relationships.
  • Studied balls rolling up and down inclined planes to analyze gravitational motion.
  • Motivation for using inclines:
  • Acceleration due to Earth's gravity ($g = 9.8\,\text{m/s}^2$) causes objects to speed up too rapidly for precise direct measurement using primitive timekeeping tools like simple pendulums.
  • Rolling balls down inclines reduces ("dilutes") effective acceleration, allowing visual tracking of velocity changes over time.
  • Observed behaviors on inclined planes:
  • Rolling down an incline: object continuously increases speed (accelerates).
  • Rolling up an incline: object decreases speed (decelerates), momentarily stops, and changes direction.
  • Invariance: Altering ball mass or size on the same incline angle results in the exact same acceleration behavior.

• Thought Experiment on Frictionless Planes:

  • If a ball rolls across a flat, perfectly smooth surface (such as ideal ice or a smooth bowling lane) with zero friction and no air resistance:
  • The ball will not slow down or speed up.
  • It will continue moving at a constant speed in a straight line indefinitely.

• Definition of Inertia:

  • Inertia is an object's natural inherent tendency to maintain its state of motion (resist changes in velocity).
  • Two natural states of motion in the absence of a net force ($\mathbf{F}_{\text{net}} = 0$):
  1. An object at rest remains at rest.
  2. An object in motion remains in motion at a constant velocity (constant speed and direction).

• Contrast with Aristotelian / Ancient Greek Views:

  • Ancient Greek physics claimed the natural state of all physical bodies was to be at rest.
  • Greek mechanics postulated that a continuous force was required to maintain motion at a constant velocity ($F \propto v$).
  • Ancient Greeks lacked the concept of acceleration and failed to identify friction as an external force acting to stop moving bodies.