Oscillations and Electromagnetism: Study Notes for PHY60A (Grade 11 ASP Physics)

Unit 7: Oscillations

7.1 Defining Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force acting on an object is directly proportional to its displacement from the equilibrium position and is directed towards that position.

• Restoring Force: Acts opposite to displacement; $F = -k\Delta x$

• Equilibrium Position: The point where the net force on the object is zero.

• Newton's Second Law Application: $F = ma_x = -k\Delta x$

Example: A mass attached to a spring oscillates back and forth when displaced from equilibrium.

7.2 Frequency and Period of SHM

The period (T) is the time for one complete cycle of motion, and the frequency (f) is the number of cycles per second. The angular frequency ($\omega$) relates to these as:

• $T = \frac{2\pi}{\omega} = \frac{1}{f}$

• For a mass-spring system: $T_s = 2\pi\sqrt{\frac{m}{k}}$

• For a simple pendulum (small angle): $T_P = 2\pi\sqrt{\frac{l}{g}}$

Example: Calculate the period of a 0.5 kg mass on a spring with $k = 200$ N/m.

7.3 Representing and Analyzing SHM

The displacement, velocity, and acceleration in SHM can be described mathematically and graphically:

• Displacement: $x = A\cos(2\pi ft)$ or $x = A\sin(2\pi ft)$

• Differential Equation: $\frac{d^2x}{dt^2} = -\omega^2 x$

• General Solution: $x = A\cos(\omega t + \phi)$

• Acceleration: $a = -\omega^2 x$

• Maximum Velocity: $v_{max} = A\omega$

• Maximum Acceleration: $a_{max} = A\omega^2$

• Resonance: Occurs when an external force matches the system's natural frequency, increasing amplitude.

• Changing amplitude does not affect the period.

Example: A pendulum exhibits resonance when pushed at its natural frequency.

7.4 Energy of Simple Harmonic Oscillators

The total mechanical energy in SHM is conserved and is the sum of kinetic and potential energies:

• $E_{total} = U + K$

• For a spring-mass system: $E_{total} = \frac{1}{2}kA^2$

• Kinetic energy is maximum when potential energy is minimum, and vice versa.

• Minimum kinetic energy is zero (at maximum displacement).

• Increasing amplitude increases total energy.

Example: At the equilibrium position, all energy is kinetic; at maximum displacement, all energy is potential.

7.5 Simple and Physical Pendulums

A physical pendulum is a rigid body oscillating about a fixed axis. For small amplitudes, the period is:

• $T_{phys} = 2\pi\sqrt{\frac{I}{mgd}}$

• Restoring torque: $\tau = -mgd\sin\theta$

• Small angle approximation: $\sin\theta \approx \theta$

• Newton's second law (rotational): $I\alpha = -mgd\theta$

• Differential equation: $\frac{d^2\theta}{dt^2} = -\omega^2\theta$

• Simple pendulum (point mass): $T_P = 2\pi\sqrt{\frac{l}{g}}$

• Torsion pendulum: $I\alpha = -k\Delta\theta$

Example: A meter stick pivoted at one end acts as a physical pendulum.

Unit 8: Magnetic Fields and Electromagnetism

8.1 Magnetic Fields

Magnetic fields are regions where magnetic forces can be detected. Field lines show the direction and strength of the field.

• Field lines around a wire are circular; use the right-hand rule to determine direction.

• Field strength around a straight wire: $B = \frac{\mu_0 I}{2\pi r}$

• Superposition principle applies to fields from multiple wires.

• Field lines around a loop resemble those of a bar magnet; right-hand rule #2 applies.

• A solenoid produces a strong, uniform field inside and a weak, divergent field outside: $B = \mu_0 n I$ where $n = \frac{N}{l}$

• Electromagnets are solenoids with enhanced field strength; strength depends on current and number of turns.

Example: Two parallel wires with current in the same direction attract; in opposite directions, they repel.

8.2 Magnetic Forces

Magnetic forces act on current-carrying wires and moving charges in a magnetic field.

• Force on a wire: $F_{m,wire} = I l B \sin\theta$

• Force on a charge: $F_{m,charge} = q v B \sin\theta$

• Direction found using the right-hand rule or Fleming’s left-hand rule.

• Charged particles move in circular or helical paths in uniform fields.

• Current loops in fields experience torque, leading to rotation (basis for electric motors).

Example: An electron moving perpendicular to a magnetic field follows a circular path.

8.3 Electromagnetic Induction and Faraday’s Law

Changing magnetic flux through a loop induces an electromotive force (emf) and possibly a current.

• Magnetic flux: $\phi = B_\perp A = B A \cos\theta$

• Change in flux: $\Delta\phi_B = \phi_f - \phi_i = \Delta(B A \cos\theta)$

• Faraday’s Law: $\mathcal{E} = -N \frac{\Delta\phi}{\Delta t}$

• Induced current: $I = \frac{\mathcal{E}}{R}$

• Lenz’s Law: Induced current opposes the change in flux.

• Motional emf: $\mathcal{E}_{motional} = B l v \sin\theta$

• Power: $P = \frac{\mathcal{E}^2}{R}$

Example: A conducting bar moving through a magnetic field induces a current in a closed loop.

8.4 Generators and Transformers

Generators convert mechanical energy to electrical energy; transformers change the voltage of alternating currents.

• AC Generator: Rotating coil in a magnetic field induces alternating emf; slip rings maintain contact.

• Transformer: Two coils on a core; voltage ratio relates to turns ratio: $\frac{V_p}{V_s} = \frac{N_p}{N_s}$

• Ideal transformer: $V_p I_p = V_s I_s$

• Step-up transformer: More turns in secondary, increases voltage.

• Step-down transformer: Fewer turns in secondary, decreases voltage.

• Power losses reduced by laminated cores and low-resistance coils.

• National Grid: High voltage, low current transmission minimizes losses; step-up at power stations, step-down near consumers.

Step-up Transformer	Step-down Transformer
Increases output voltage	Decreases output voltage
Input voltage < output voltage	Input voltage > output voltage
More turns in secondary	Fewer turns in secondary
Low current in secondary	High current in secondary

Example: A transformer with 100 turns in the primary and 200 in the secondary doubles the voltage.