Unit 1 Physics: Errors and Uncertainty Study Notes

Unit 1 Physics Notes: Errors and Uncertainty

Types of Errors

• Every measurement has an associated uncertainty or error.

  • Random Errors

  • Caused by environmental conditions or the person taking measurements.

  • Examples include parallax error and fluctuations in temperature.

  • Measurements are often taken multiple times, and averages are used to reduce the effect of random errors.

  • Systematic Errors

  • Caused by faults in instruments or experimental procedures.

  • Examples include zero error and procedural faults in conducting experiments.

  • These require recalibration of instruments or review of experimental procedures.

Measurement Fundamentals

• Reading: The value assigned to a measurement when using instruments.

• Accuracy: The degree to which a series of readings are close to one another.

• Measurement: A reading taken from an instrument for the size or amount of a base unit.

Error Analysis

• Definition of Error: The uncertainty in a measured value due to various factors such as wrongly calibrated instruments, incorrect measuring conditions, or failure to follow experiment rules.

• Accuracy of Measurement: Refers to how well a measured value agrees with the standard value.

  • Example: If a measured weight of a stone is 0.5 kg, while the actual weight is 1 kg, then the measurement is not accurate.

• Precision: Indicates how closely a set of measurements agree with each other.

  • Example: Measurements of a cord: 2.3m, 2.2m, 2.4m, 2.2m, 2.23m are considered precise because they are closely grouped together.

Error Propagation

• Errors in Addition:

  • For the equation: $y = x + w$

  • Associated error: $ext{δY} = ext{±}[ ext{δX} + ext{δW}].$

• Errors in Subtraction:

  • For the equation: $y = m - u$

  • Associated error: $ext{δY} = ext{±}[ ext{δm} + ext{δu}].$

• Errors in Multiplication:

  • For the equation: $Y = X imes P$

  • Associated error: rac{ ext{δY}}{Y} = ext{±}igg[ rac{ ext{δX}}{X} + rac{ ext{δP}}{P} igg].

• Errors in Division:

  • For the equation: $Y = rac{X}{P}$

  • Associated error: rac{ ext{δY}}{Y} = ext{±}igg[ rac{ ext{δX}}{X} + rac{ ext{δP}}{P} igg].

• Errors Involving Powers:

  • For the equation: $Y = x^{n/mk}$

  • Associated error: rac{ ext{δY}}{Y} = ext{±} igg[n rac{ ext{δx}}{x} + k rac{ ext{δm}}{m}igg].

• Percentage Errors:

  • Calculated as:
    ext{Percentage Error} = ext{Fractional Error} imes 100 ext{%}

Forces

• Force:

  • Defined as anything that keeps a body at rest or in uniform motion in a straight line.

• Types of Forces:

  1. Contact Force: eg. Reaction forces

  2. Electric Forces

  3. Magnetic Forces

  4. Gravitational Forces

  5. Electrostatic Forces

  6. Nuclear Forces

Diagrammatic Representations

• Forces at Rest:

  • $R = W$ (At rest the reaction force equals weight)

• Forces in Motion:

  • An unbalanced force occurs if other conditions change.

Triangle Law of Vector Addition

• Definition: States that when two vectors are represented as two sides of a triangle, the resultant vector is represented by the third side in magnitude and direction.

Equations of Motion

• Recall: Speed is defined as distance/time. Average speed is given by the equation: $ext{Average Speed} = rac{ ext{Total Distance}}{ ext{Total Time Taken}}$

• Equations:

  • $S = rac{(V + U)}{2} imes t$

  • $a = rac{(V - U)}{t}$ where acceleration is defined as the rate of change of velocity.

  • Other equations derived from basic definitions include:

    • $S = ut + rac{1}{2} at^2$

    • $V^2 = U^2 + 2 a s$

Momentum

• Definition: The product of an object’s mass and its velocity.

  • $p = mv$

• Newton's Laws of Motion:

  1. 1st Law: A body remains in state of rest or uniform motion unless acted upon by an external force.

  2. 2nd Law: The force acting on an object equals rate of change of momentum, $F = rac{(mv - mu)}{t}$ or $F = ma$.

  3. 3rd Law: For every action, there is an equal and opposite reaction.

Circular Motion

• Definition of Centripetal Force: The force that keeps a body in circular motion, directed towards the center of the path.

• Equations:

  • $F = rac{mv^2}{r}$

• Newton's Law of Gravitation:

  • $F = G rac{m_1 m_2}{r^2}$. Here, $G$ is the universal gravitational constant ($6.67 imes 10^{-11} ext{Nm}^2/ ext{kg}^2$).

Heat Transfer

• Heat Flow:

  • Described by their gradient and can be represented as: $rac{ ext{ΔQ}}{t} = -K A rac{ ext{Δθ}}{ ext{ΔX}}$

  • Where $K$ is thermal conductivity, $A$ is cross-sectional area, and $rac{Δθ}{ΔX}$ is the temperature gradient.

• First Law of Thermodynamics:

  • Change in internal energy is equal to heat added to the system minus work done by the system.

  • $ext{ΔU} = Q - W.$

Simple Harmonic Motion (SHM)

• Definition: Motion in which the restoring force is directly proportional to the displacement from the equilibrium position.

• Equation of SHM:
  $F = -kx$

• Energy in SHM: The total mechanical energy is the sum of kinetic and potential energies. $E_{total} = KE + PE$

• Damping: The reduction in amplitude of oscillations, which can be light, heavy, or critical damping.

Resonance

• Definition: Occurs when a system is driven at its natural frequency, leading to large amplitude of oscillation.

• Coherence: When two waves have the same frequency, amplitude, and phase leading to resonance.

Blackbody Radiation

• Definition: A perfect absorber and emitter of radiation, with its efficiency dependent on temperature.

  • Higher temperatures lead to higher emission intensity.