Electricity, Energy, Momentum and Dynamics Notes

Voltage and Current

• Voltage (Potential Difference):
  • The change in potential energy per unit charge.
• Current:
  • The rate of flow of charge through the cross-sectional area of a conductor.
  • Formula: $I = \frac{Q}{\Delta t}$
    • I: Current (A)
    • Q: Charge (C)
    • t: time (s)
• Direction of Current:
  • The direction that a positive charge carrier would flow.
  • Current flows in the direction opposite to the flow of electrons.

Resistance and Ohm's Law

• Ohm's Law:
  • The ratio between the voltage and the current through a conductor (load, resistor) is a constant and represents the resistance.
  • Formula: $V = IR$
    • V: Voltage (V)
    • I: Current (A)
    • R: Resistance (Ω)

Example

• Problem: The potential difference across a 5.0 Ω resistor is 1.5 V. Determine the current across the resistor.
• Solution:
  • Using Ohm's Law: $V = IR$
  • $1.5 = I * 5.0$
  • $I = \frac{1.5}{5.0} = 0.3 A$

Circuits

• Circuit Symbols:
  • Battery: (Symbol with + and -)
  • Wire: (Straight line)
  • Junction: (Intersection of wires)
  • Resistor: (Zig-zag line)
  • Bulb: (Circle with a cross)
  • Switch: (Open or closed switch)
  • Voltmeter: (Circle with a V)
  • Ammeter: (Circle with an A)

Series and Parallel Circuits

• Series Circuits:
  • One path for the electrons.
  • Current is the same through all components: $I<em>T= I</em>1 = I<em>2 = I</em>3$
  • Total voltage is the sum of individual voltages: $V<em>T = V</em>1 + V<em>2 + V</em>3$
  • Total resistance is the sum of individual resistances: $R<em>T = R</em>1 + R<em>2 + R</em>3$
• Parallel Circuits:
  • More than one path for the electrons.
  • Total current is the sum of individual currents: $I<em>T = I</em>1 + I<em>2 + I</em>3$
  • Voltage is the same across all components: $V<em>T = V</em>1 = V<em>2 = V</em>3$
  • Reciprocal of total resistance is the sum of reciprocals of individual resistances: $\frac{1}{R<em>T} = \frac{1}{R</em>1} + \frac{1}{R<em>2} + \frac{1}{R</em>3}$
  • Adding more paths always decreases total resistance and increases current.

Examples

• Series Circuit Example:
  • Given: A series circuit with a 12V source, R1 = 12Ω, R2 = 18Ω. Determine the resistance of R3 if the total current I = 0.3A.
  • Solution:
    • Total Resistance: $R<em>T = \frac{V</em>T}{I} = \frac{12}{0.3} = 40 \Omega$
    • $R<em>T = R</em>1 + R<em>2 + R</em>3$
    • $R<em>3 = R</em>T - R<em>1 - R</em>2 = 40 - 12 - 18 = 10 \Omega$
• Parallel Circuit Example:
  • Determine the current through the 5 Ω resistor in a parallel circuit with a 10V source, R1 = 35Ω, and R2 = 5Ω.
  • Solution:
    • In parallel, $V<em>T = V</em>1 = V_2 = 10V$
    • $I<em>2 = \frac{V</em>2}{R_2} = \frac{10}{5} = 2 A$

Power

• Electric Power:
  • The rate at which energy is transferred.
  • Formulas: $P = IV = I^2R = \frac{V^2}{R}$
    • P: Power (W)
    • I: Current (A)
    • V: Voltage (V)
    • R: Resistance (Ω)

Work

• Work Definition:
  • Work is the scalar product between force and displacement.
  • Formula: $W = F \cdot d$
    • W: Work (J)
    • F: Force (N)
    • d: Displacement (m)
• Scalar Product (Dot Product):
  • Two parallel vectors that multiply to a scalar.
• Non-Parallel Vectors:
  • Use the parallel component of force.
  • Formula: $W = Fd \cos\theta$
• Work-Energy Theorem:
  • The work done by all nonconservative forces is equal to the change in the mechanical energy of the system.
  • $W<em>{NC} = \Delta E = \Delta E</em>K + \Delta E_P$
• Positive Work:
  • Force is in the same direction as displacement.
  • Object gains energy.
  • Example: Pushing a crate across the floor.
• Negative Work:
  • Force is in the opposite direction of displacement.
  • Object loses energy.
  • Example: Friction on a sliding object.
• Zero Work:
  • Object does not move.
  • Force and displacement are perpendicular.
  • Examples: Pushing a wall, gravity on an object sliding across a horizontal surface.
• Work as Area Under a Graph:
  • Work is the area under an F-d graph.

Types of Energy

• Mechanical Energy:
  • The sum of the kinetic and potential energy in a system.
  • Energy is a scalar.
  • SI Unit: Joule (J)
• Kinetic Energy:
  • Energy a moving object has because of its motion.
  • Formula: $E_K = \frac{1}{2}mv^2$
    • $E_K$: Kinetic Energy (J)
    • m: mass (kg)
    • v: velocity (m/s)
• Potential Energy:
  • Energy stored in an object.
  • Gravitational Potential Energy: $E_P = mgh$
    • $E_P$: Gravitational Potential Energy (J)
    • m: mass (kg)
    • g: acceleration due to gravity (9.8 m/s²)
    • h: height (m)
• Gravitational Potential Energy Reference:
  • Measured relative to a "zero" such as the ground.
  • Unless specified, this point/height is of your choosing.

Conservation of Energy

• Conservation of Mechanical Energy:
  • If only conservative forces act on a system, the total mechanical energy is constant.
  • $\Sigma E<em>1 = \Sigma E</em>2$
• Conservative Force:
  • Work done depends only on the starting and ending points of motion and not on the path taken.
  • Examples: gravitational force, spring force.
• Nonconservative Force:
  • Work depends on the path taken.
  • Example: Friction.
  • The work done by friction results in a transfer of mechanical energy into thermal energy.
  • $E<em>1 = E</em>2 + Q$
    • Q: Heat (J)
    • $Q = Fd$
      • F: Force of friction (N)
      • d: Displacement (m)

Power

• Power:
  • The rate at which energy is added or used.
  • SI Unit: Watt (W)
  • Formula: $P = \frac{W}{t}$
    • P: Power (W)
    • W: Work (J)
    • t: Time (s)

Momentum and Impulse

• Momentum:
  • Defined as the product of mass and velocity.
  • Momentum is a vector.
  • Formula: $p = mv$
    • p: momentum (kg m/s)
    • m: mass (kg)
    • v: velocity (m/s)
• Impulse:
  • A force acting on an object for a time interval, $\Delta t$.
  • Impulse causes a change in momentum.
• Momentum-Impulse Theorem:
  • $J = F_{NET} \Delta t = m\Delta v = \Delta p$
    • J: Impulse (N s)
    • $F_{NET}$: Net Force (N)
    • $\Delta t$: Time Interval (s)
    • m: Mass (kg)
    • $\Delta v$: Change in Velocity (m/s)
    • $\Delta p$: Change in Momentum (kg m/s)
• Impulse and F-t Graph:
  • Impulse is equal to the area under an F-t graph.

Conservation of Momentum

• System:
  • A collection of two or more objects.
• Closed System:
  • A system on which the net external force is zero.
• Conservation of Momentum:
  • The momentum of a closed system is constant.
  • $\Sigma p<em>i = \Sigma p</em>f$
  • For two objects: $p<em>{1i} + p</em>{2i} = p<em>{1f} + p</em>{2f}$

Dynamics

• Dynamics Definition:
  • The study of forces.
• Force:
  • A push or pull.
  • Force is a vector.
  • SI Unit: Newton (N)
• Newton's Laws of Motion:
  • First Law (Law of Inertia):
    • If all the forces acting on a body are balanced, then the object will not change speed or direction (constant velocity).
    • An object at rest remains at rest; an object in motion remains in motion.
    • $a = 0$
  • Second Law: $F_{NET} = ma$
    • If there is an unbalanced force acting on an object, it will accelerate in the direction of the net force in inverse proportion to its mass.
    • Mass resists acceleration (more mass = more inertia).
  • Third Law (Action-Reaction):
    • If object A exerts a force on object B, then B exerts an equal force back upon A in the opposite direction.
    • Forces always occur in pairs. The action-reaction forces are never on the same object.
• Net Force:
  • The sum of all the forces acting on an object.
  • The net force is not a real force.
• Free-Body Diagrams:
  • Used to show the magnitude and direction of all the forces acting on an object.
  • Use a dot or box to represent the object.

Types of Forces

• Contact Forces:
  • Applied Force ($F_a$): A force used to help move an object (e.g., pushing a box).
  • Force of Friction ($F_f$): A force that opposes sliding motion between surfaces.
  • Normal Force ($F_N$): The force a surface exerts on an object to support it (always acts perpendicular to the surface).
  • Tension ($F_T$): The pull exerted by a string, rope, or cable.
  • Spring Force ($F_s$): The force a spring exerts to restore it to its normal shape.
• At-a-Distance Forces:
  • Gravitational Force (Weight) ($F_g$): The attraction between two objects.
    • $\vec{F_g} = m\vec{g}$
      • g: Gravitational field strength (9.8 m/s²).