Fundamental Units and Quantities

• The course covers units and dimension, space and time, kinematics, fundamental laws of mechanics, statics and dynamics; work and energy; conservation laws, elasticity, Hooke's law, Young's shear and bulk moduli, hydrostatics; pressure; buoyance, Archimedes' Principles, surface tension; adhesion, cohesion, capillarity, drops, bubbles, temperature, heat, gas laws, laws of thermodynamics, kinetic theory of gases, and sound application.

• A fundamental quantity cannot be expressed in terms of other physical quantities.

• Examples of fundamental quantities include Length, Mass, Time, and Electric Current.

• A unit of measurement is a magnitude of a quantity defined and adopted by convention or law.

• It's used as a standard for measuring quantities of the same kind, where any other quantity can be expressed as a multiple of this unit.

Fundamental Quantities and Units:

• Length is measured in meters (m).
• Mass is measured in kilograms (kg).
• Time is measured in seconds (s).
• Electric current is measured in amperes (A).
• Thermodynamic temperature is measured in kelvin (K).
• Amount of substance is measured in moles (mol).
• Luminous intensity is measured in candelas (cd).

SI Unit Prefixes:

• tera (T) = 10¹²
• giga (G) = 10⁹
• mega (M) = 10⁶
• kilo (k) = 10³
• pico (p) = 10⁻¹²
• nano (n) = 10⁻⁹
• micro (µ) = 10⁻⁶
• milli (m) = 10⁻³
• atto (a) = 10⁻¹⁸
• femto (f) = 10⁻¹⁵

Dimensions:

• Length: L
• Mass: M
• Time: T

Speed, Velocity, and Acceleration:

• Speed is the rate of change of distance with time.
• average speed = distance / change in time, unit is ms⁻¹, dimension is LT⁻¹
• Velocity is the rate of change of displacement with time.
• average velocity = displacement / change in time, unit is ms⁻¹, dimension is LT⁻¹
• Acceleration is the rate of change of velocity with time.
• acceleration = change in velocity / change in time, unit is ms⁻², dimension is LT⁻²

Scalars and Vectors:

• Scalars are physical quantities with magnitude but no direction.
• Examples include distance, speed, work, and energy.
• Vectors are quantities with both magnitude and direction.
• Examples include displacement, velocity, acceleration, force, and momentum.

Equations of Uniformly Accelerated Motion:

• velocity = x/t or v = dx/dt or ∫v dt = ∫dx
• average velocity = (u+x)/2
• acceleration = (v-u)/t or v = dv/dt or ∫a dt = ∫dv
• v = u + at
• x = ut + (1/2)at²
• v² = u² + 2ax
• x = ((u+v)/2)t
• u = initial velocity (in ms⁻¹)
• v = final velocity (in ms⁻¹)
• x = displacement (in m)
• a = acceleration (in ms⁻²)
• t = time of travel through distance x (in s)
• Acceleration (+a) is the positive/increasing rate of change of velocity with time.
• Deceleration (-a or Retardation) is the negative/decreasing rate of change of velocity with time.
• If accelerating, assign a + (plus) sign; if decelerating, assign a - (minus) sign.

Distance–Time and Velocity–Time Graphs:

• Uniform velocity represents the slope of the x-t graph.
• Uniform velocity, acceleration, deceleration, and slope are concepts represented under the v-t graph.
• "Brought from rest" implies u = 0; "brought to rest" implies v = 0; so also u = 0 and u > 0 under the v - t graph
• Total distance travelled equals the total area under the v-t graph.
• Slope of the displacement-time graph indicates the velocity of the body.
• Slope of segment 1 of the velocity-time graph is +ve, hence acceleration.
• Slope of segment 2 of the velocity-time graph is zero (0), hence uniform velocity.
• Slope of segment 3 of the velocity-time graph is -ve, hence deceleration.
• Area under each segment of the velocity-time graph represents the distance travelled during that segment.
• Total area under the velocity-time graph represents the total distance travelled by the body.
• Final velocity for one stage of motion serves as the initial velocity for the next stage of motion.

Motion Under Gravity:

• vy = uy + gt
• h = uyt + (1/2)gt²
• vy² = uy² + 2gh
• h = ((uy+vy)/2)t
• uy = initial velocity (in ms⁻¹)
• vy = final velocity (in ms⁻¹)
• g = acceleration due to gravity (in ms⁻²)
• h = height attained at a particular instant of time (in m)
• t = time of travel (in s)
• g is a constant at 9.8 ms⁻² (≈10 ms⁻²)
• For upward motion, g is -ve; v = 0 at the maximum height.
• For downward motion, g is +ve; u = 0 at the maximum height.
• t is the time taken for the body to travel from the level of projection to the maximum height
• Never use T = 2t to find the total time of travel T because of the three situations that can possibly occur.
• Calculate t₁ = time to reach the maximum height, and t2 = time taken to fall back from the maximum height to the level of interest.
• Then, use T = t₁ + t2

Equations of Parabolic (Projectile) Motion:

• Velocity components:
• ux = u cos θ
• uy = u sin θ
• ux is the component of u in the horizontal plane (in ms⁻¹), responsible for horizontal motion.
• uy is the component of u in the vertical plane (in ms⁻¹), responsible for vertical motion.
• t is the time taken for the body to travel from the level of projection to the maximum height.