Study Notes on Angles and Motion in Physics

Angles in Physics

• In physics, angles are measured in radians instead of degrees.
• Radian measurement is defined as:
  $heta ext{ (radians)} ext{ is equal to } \frac{A ext{ (arc length)}}{r ext{ (radius)}}$
• Key conversions:
  • One complete round turn = 360 degrees
  • One complete round turn = $2 heta ext{ radians} = 2 heta = 6.283 ext{ rad}$
  • To convert from radians to degrees:
  • $1 ext{ rad} = \frac{360}{2 heta} ext{ degrees} ext{ (approximately } 57.3 ext{ degrees})$
• The relationship between arc length, radius, and angle is expressed as: $s = r heta$$\frac{s<em>1}{r</em>1} = \frac{s<em>2}{r</em>2}$
  • This holds true irrespective of the size!

Chapter 2: Motion

Part 1: Motion Along a Straight Line

Topics Covered:

• Definition of Motion
• Importance of Understanding Motion
• Description of Motion
  • Position
  • Velocity
  • Acceleration
• One-Dimensional Motion
  • Concepts of Velocity and Acceleration
  • Difference Between Instantaneous and Average Velocity/Acceleration

Understanding Motion

• Definition of Motion:
  • Motion is defined as the change of position with respect to time (position as a function of time).
  • Examples include the movement of:
  • Basketball
  • Missiles
  • Trains

Importance of Studying Motion/Mechanics

• Design of engines and moving components (cars, trains).
• Understanding human motion in sports.
• Designing robots and artificial limbs.
• Describing the motion of electrons in electronic devices.

Description of Motion by Physical Quantities

• Mechanics focuses on motion and its causes (change of position over time).
• Primary objective: Find position as a function of time.
• Other key physical quantities:
  • Velocity
  • Acceleration
• Velocity and acceleration are related to the causes of motion (energy and forces).

Describing Position

• In one dimension:
  • Position described along the x-coordinate.
• Coordinate system setup:
  • Define the origin and use rulers for measurements.
• Position is described in various dimensions:
  • Two Dimensions: Coordinates (x, y) represented by vector formula:
    $extbf{r} = x extbf{i} + y extbf{j}$
  • Three Dimensions: Coordinates (x, y, z) represented by vector formula:
    $extbf{r} = x extbf{i} + y extbf{j} + z extbf{k}$

Describing Motion

• Motion as a change in position with time requires knowledge about:
  • Velocity (rate of change of position)
  • Acceleration (rate of change of velocity)
  • One Dimension:
  • Position represented as a function of time: $x = x(t)$
  • Two Dimensions:
  • $extbf{r}(t) = x(t) extbf{i} + y(t) extbf{j}$
  • Three Dimensions:
  • $extbf{r}(t) = x(t) extbf{i} + y(t) extbf{j} + z(t) extbf{k}$

Displacement (Vector)

• A particle's movement along the x-axis:
  • From initial position $x<em>1$ to final position $x</em>2$,
  • Change in position is defined as:
    $ext{Displacement } riangle x = x<em>2 - x</em>1$
  • Displacement can be positive (right) or negative (left).

Displacement and Average Velocity (Vector)

• Position-time graph visualizing position $x(t)$ changes with time $t$.
• Average x-velocity formula: $v_{ ext{av}-x} = \frac{ riangle x}{ riangle t}$
  • The average velocity describes the speed and direction over time interval $riangle t$.
  • Displacement can also be expressed as:
    $riangle x = v_{ ext{av}-x} riangle t$
  • Average velocity slopes connect positions on the position-time graph.

Instantaneous Velocity

• Average velocity evaluates motion during a time interval $riangle t$; for precise evaluations, instantaneous velocity is required.
• Instantaneous velocity expressed as: $v_x = \frac{dx}{dt}$
  • This involves differentiation of the position with respect to time.

Finding Instantaneous Velocity on an x-t Graph

• Average velocity evaluated within smaller intervals gives a notion of instantaneous velocity:
  • The instantaneous velocity is the slope of the tangent on the x-t curve, thus defined as:
    $v_x = \frac{ riangle x}{ riangle t}$ when $riangle t o 0$

Understanding Distance (Average Speed) vs. Displacement (Velocity)

• Displacement $riangle x$ can be positive or negative, while distance $D$ represents total travel length and is always positive.
  • Example:
  • $riangle x = 2 - 3 = -1$
  • $D = 4 + 3 = 7$
• Average speed:
  $v<em>{ ext{av}} = \frac{D}{ riangle t} \neq \frac{| riangle x|}{ riangle t} = | extbf{v}</em>x |$
• Instantaneous speed's magnitude represented as the absolute value of instantaneous velocity.

Average Acceleration and Instantaneous Acceleration

• Average acceleration defines the velocity change rate over time:
  $a<em>{ ext{av}-x} = \frac{ riangle v</em>x}{ riangle t}$
• Relate instantaneous acceleration to average when $riangle t o 0$:
  $a<em>x = \frac{dv</em>x}{dt}$
• Graphing the average acceleration through slope from a velocity-time graph
  determines the instantaneous acceleration value.

Calculating Acceleration from Velocity-Time Graphs

• Area under the curve from velocity-time graphs gives displacement across time intervals.
• For non-constant velocity, integration over small slices approximates total displacement:
  $riangle x = v_x riangle t$

Motion with Constant Acceleration

• It's noted that constant acceleration changes velocity at uniform rates:
  $riangle v<em>x = a</em>x riangle t$
• Equation of motion specifies velocity and position with time, given consistent acceleration.
  • Equations include:
  • $v<em>x = v</em>{0x} + a_x t$
  • $x = x<em>0 + v</em>{0x} t + \frac{1}{2} a_x t^2$
  • $v<em>x^2 = v</em>{0x}^2 + 2a<em>x (x - x</em>0)$

Example of Calculus in Constant Acceleration

• Calculating instantaneous velocity:
  • Differentiation allows position finding based on velocity, while integration generates position with respect to time vs. acceleration.

Different Bodies with Different Accelerations

• Explore situations involving different acceleration rates from distinct bodies, applying established equations for constant acceleration.

Motion in Two or Three Dimensions

• It extends to describe how vectors represent motion in a space of two or three dimensions, utilizing velocity and instantaneous acceleration vectors while analyzing projectile motion and circular path.

Topics for Chapter 2 Part 2

• Description of motion in 3-D spaces using vectors
  • Velocity and acceleration as vector quantities.
  • Utilizing projectile motion as an example of 2-D motion.

Position and Average Velocity in Multi-Dimensional Spaces

• The position vector defined by coordinates in varying dimensions links back to motion representation.
• Average velocity is derived from position vector changes over time:
  $v_{ ext{av}} = \frac{ riangle extbf{r}}{ riangle t} = \frac{ riangle x}{ riangle t} extbf{i} + \frac{ riangle y}{ riangle t} extbf{j} + \frac{ riangle z}{ riangle t} extbf{k}$

Instantaneous Velocity

• It's the derivative of position vector with regards to time, remaining tangent to paths followed by moving objects.

Differences Between Speed and Velocity

• Speed remains a scalar while velocity as a vector encompasses directionality and changes in magnitude related to motion.

Average and Instantaneous Accelerations

• Compare average acceleration during time intervals with instantaneous accelerations derived directly from velocity changes.

Motion: Projectile and Uniform Circular Motion

• Explore projectile motion characterized by parameters of gravity and paths with specific initial velocities, separating horizontal and vertical motions.

Key equations for projectile motion:

• $v<em>{0x} = v</em>0 ext{cos} heta_0$
• $v<em>{0y} = v</em>0 ext{sin} heta_0$
• Application of gravity to y-direction equations.

Uniform Circular Motion

• Defined with constant speed while direction consistently alters, putting forth centripetal acceleration calculations for objects moving along circular paths.
• Recognize the significance of rotational dynamics and uniform circular trajectories in motion understanding, particularly for satellite movement or turns in automobiles.

Centripetal Acceleration

• It remains perpendicular to movement, acting continuously towards the path's center.
• Changing velocity arises solely due to directional alterations: $a_c = \frac{v^2}{r}$
  • Acceleration correlates with the square of velocity and inversely with radius to reinforce motion dynamics.