Oscillations and Simple Harmonic Motion Detailed Lecture Notes

Course Overview and Administrative Details

Course Code: PHY1031F
Module: Vibrations & Waves
Structure: Consists of 18 Lectures.
Primary Topics (Part I): Oscillations and Simple Harmonic Motion.
Recommended Texts:
- Chapter 13 – OpenStax
- Chapter 14 – Knight, Jones and Field (KJF)
- Instructors: Dr. S.M. Wheaton (Room LT3A, R.W. James Building).

Equilibrium and Equilibrium States

Definition of Equilibrium: An object is in equilibrium when the net force ( $F_{net} = 0$ ) and the net torque ( $\tau_{net} = 0$ ) acting on it are zero.
ConceptTest Analysis - Equilibrium Conditions:
- Statement: "Which of the following statements is/are correct?"
- A: An object in equilibrium is at rest.
- B: An object in equilibrium need not be at rest.
- C: An object at rest must be in equilibrium.
- Correct Answer: B. This is because an object moving with a constant velocity has zero acceleration and is therefore in equilibrium, despite not being at rest.

Methodology for Free-Body Diagrams (FBD)

Visualize: Draw a picture of the physical situation.
Define System: Circle the system of interest on the picture.
Identify Forces: Identify all significant forces acting on the system. Draw them as arrows and label them. Identify forces by asking, "Who, or what, interacts with the system?"
Represent as a Dot: Redraw the system as a single dot and draw the forces as arrows where the lengths represent magnitudes.
Coordinate System: Draw convenient coordinate axes, usually centered on the dot.
Net Force: Draw the net force vector alongside the physical forces.
Summary: An FBD represents a system as a dot and uses arrows to illustrate every force acting on that system.

Dynamics of Oscillatory Systems

Equilibrium Position: The position where an object would naturally come to rest because the net force is zero.
Comparison of Oscillatory Systems: Systems like a mass on a horizontal spring (A), a mass on a vertical spring (B), and a pendulum (D) all exhibit motion around an equilibrium position.
Commonalities in FBDs for Oscillators:
- The origin of the coordinate system is typically placed at the equilibrium position.
- Every FBD for these systems shows a net force ( $F_{net}$ ) that acts toward the equilibrium position when the system is displaced.

Mathematical Description of Simple Harmonic Motion (SHM)

Sinusoidal Nature: By definition, the graphs representing SHM are sinusoidal. This defines the shape (sine, cosine, or shifted sine) but not the specific phase or amplitude details.
Special Case - Starting at Maximum Displacement: If a mass-spring system is released from $x = A$ at $t = 0$ :
- $x(t = 0) = A$
- The position graph starts at a maximum and follows a cosine path.
Equations for Special Case Release at x = A:
- Position: y(t) = A \times \text{cos}\text{(}\frac{2\times\text{\pi}\times t}{T}\text{)}
- Velocity: v_y(t) = -(2\times\text{\pi}\times f) \times A \times \text{sin}\text{(}\frac{2\times\text{\pi}\times t}{T}\text{)}
- Acceleration: a_y(t) = -(2\times\text{\pi}\times f)^2 \times A \times \text{cos}\text{(}\frac{2\times\text{\pi}\times t}{T}\text{)}
Key Kinematic Relationships:
- Arguments for trigonometric functions must be in radians.
- When position ( $y$ ) is at maximum or minimum, velocity ( $v$ ) is zero.
- When position ( $y$ ) is zero (at equilibrium), velocity ( $v$ ) is at its maximum magnitude ( $v_{max}$ ).
- When position is maximum in the positive direction, acceleration is maximum in the negative direction. In Simple Harmonic Motion (SHM), the position of an object oscillating in a harmonic manner can be represented graphically. When the position of the object reaches its maximum displacement in the positive direction, this means the object is farthest from the equilibrium position toward the positive side. At this point, the restoring force, which aims to bring the object back to equilibrium, is at its maximum magnitude and its direction is negative (toward the equilibrium position). Consequently, acceleration, which is directly correlated to this restoring force, is also at its maximum in the negative direction. This relationship highlights that at extremes of displacement, the acceleration is directed opposing the position, which is a fundamental characteristic of SHM.

General Equations for SHM and Phase Constants

Need for Generalization: If the initial starting point is not a maximum displacement ( $x \neq A$ at $t = 0$ ), specialized equations are required.
The Phase Constant (\text{\phi}): This constant is determined by initial conditions ( $t = 0$ ).
- \text{\phi} = 0 recovers the special case starting at maximum amplitude.
Completely General Equations:
- Position: x(t) = A \times \text{cos}\text{(}\frac{2\times\text{\pi}\times t}{T} + \text{\phi}\text{)}
- Velocity: v_x(t) = -(2\times\text{\pi}\times f) \times A \times \text{sin}\text{(}\frac{2\times\text{\pi}\times t}{T} + \text{\phi}\text{)}
- Acceleration: a_x(t) = -(2\times\text{\pi}\times f)^2 \times A \times \text{cos}\text{(}\frac{2\times\text{\pi}\times t}{T} + \text{\phi}\text{)}
Specific Example Scenarios:
- Timing starts at equilibrium moving positive: Use \text{\phi} = -\frac{\text{\pi}}{2}. This converts the cosine into a sine graph.
- Timing starts at equilibrium moving negative: Use \text{\phi} = +\frac{\text{\pi}}{2}.

Defining Properties and Mechanics of SHM

Condition for SHM: SHM occurs specifically when the net restoring force is directly proportional to the object's displacement from its equilibrium position. Not all periodic motion meets this criteria.
Hooke's Law: For a mass-spring system, the restoring force is proportional to the stretch/compression ( $x$ ):
- $F_{spring} = -k \times x$
- $k$ is the spring constant.
- The negative sign indicates the force is always directed opposite to the displacement.
Newton's Second Law Applied to SHM:
- $F_{net} = m \times a \rightarrow -k \times x = m \times a$
- $-k \times x = m \times \frac{d^2x}{dt^2}$
- Result: Acceleration depends on position. Since acceleration is not constant, standard "SUVAT" equations of motion do not apply.
Frequency and Mass-Spring constant relationship:
- f = \frac{1}{2\times\text{\pi}} \times \text{\sqrt{k/m}}

Vertical Mass-Spring Systems

Equilibrium in Vertical Systems: A mass hanging at rest stretches the spring by a distance \text{\Delta}L to reach equilibrium.
Forces at Equilibrium: At $y = 0$ , the upward spring force equals the downward gravitational force (k \times \text{\Delta}L = m \times g).
Dynamics: Moving the block upward from this new equilibrium results in a net downward force, and moving it downward results in a net upward force.

Energy in Simple Harmonic Motion

Conservation Principle: For an isolated mass-spring system (no friction), total mechanical energy ( $E$ ) is conserved.
- $E = K + U = \text{constant}$
- $E = \frac{1}{2} \times m \times v^2 + \frac{1}{2} \times k \times x^2 = \frac{1}{2} \times k \times A^2$
Energy States:
- At equilibrium ( $x=0$ ): Potential energy ( $U$ ) is zero; Kinetic energy ( $K$ ) is at maximum.
- At maximum displacement (x = \text{\pm} A): Velocity is zero; Kinetic energy ( $K$ ) is zero; Potential energy ( $U$ ) is at maximum.
Calculating Speed from Position:
- v = \text{\pm} \text{\sqrt{\frac{k}{m} \times (A^2 - x^2)}}

The Simple Pendulum

Condition for Pendulum SHM: The restoring force for a pendulum is F = -m \times g \times \text{sin}\text{(}\text{\theta}\text{)}. This is not SHM because it is proportional to \text{sin}\text{(}\text{\theta}\text{)} rather than the displacement \text{\theta}.
Small Angle Approximation: For small angles (typically less than 10^{\text{\circ}}), \text{sin}\text{(}\text{\theta}\text{)} \text{\approx} \text{\theta}.
Restoring Force (Small Angle): F \text{\approx} -(\frac{m \times g}{L}) \times x, where x = L \times \text{\theta}.
Period and Frequency:
- T = 2\times\text{\pi} \times \text{\sqrt{L/g}}
- f = \frac{1}{2\times\text{\pi}} \times \text{\sqrt{g/L}}
- Crucially, the frequency/period of a pendulum depends only on its length and gravity, not on its mass.

Damping

Real-world systems: Friction and air resistance reduce total mechanical energy over time, causing oscillations to be "damped."
Damping Decay: The amplitude of the oscillation follows a "dying exponential" envelope.
Time Constant (\text{\tau}): The rate of amplitude decrease is determined by the time constant \text{\tau}.
Calculations: Involving natural logs ( $\text{ln}$ ) and the constant e \text{\approx} 2.7182.

Driven Oscillations and Resonance

Natural Frequency ( $f_0$ ): The frequency at which a system naturally oscillates without external forces.
Driven Oscillator: An oscillator subject to a periodic external force with frequency $f_{ext}$ . The system will oscillate at the frequency of the driver ( $f_{ext}$ ).
Resonance: Occurs when the external driving frequency ( $f_{ext}$ ) matches the natural frequency ( $f_0$ ), resulting in a dramatic increase in the amplitude of oscillation.
Example (Pendula on a Beam): If several pendula of different lengths are attached to a flexible beam and one (Pendulum A) is set in motion, Pendulum C (which has a length matching A) will oscillate with the greatest amplitude because its natural frequency matches that of the driver.

Questions & Discussion (ConcepTests)

Vector Directions in SHM:
- Question: Which pair of vector quantities cannot both point in the same direction?
- Choices: A (position and velocity), B (velocity and acceleration), C (position and acceleration).
- Answer: C. Position and acceleration are always in opposite directions in SHM because acceleration is directed toward equilibrium (restoring force) while position is the displacement away from equilibrium.
Total Distance Traveled:
- Question: What is the total distance traveled by an object in SHM during one period ( $T$ )?
- Answer: 4A (It travels from equilibrium to A, back to equilibrium, to -A, and back to equilibrium).
Effect of Doubling Mass on Spring Period:
- Question: If the mass of a toy car on a spring is doubled, what happens to the period?
- Answer: Period will increase (T \text{\propto} \text{\sqrt{m}}).
Effect of Mass on Pendulum Period:
- Question: Two pendula have same length but different masses. How do periods compare?
- Answer: Period is the same for both cases (independent of mass).
Effect of Doubling Amplitude:
- Question: If amplitude is doubled, which quantity does NOT change?
- Answer: Period (the period of SHM is independent of amplitude).

Worked Examples and Numerical Data

General SHM Example:
- Given: Body executes SHM between two points 10 cm apart ( $A = 5 \text{ cm}$ ). Frequency $f = 0.25 \text{ Hz}$ ( $T = 1/f = 4 \text{ s}$ ). Released from $x = +5 \text{ cm}$ at $t = 0$ .
- a) Position at $t = 1.5 \text{ s}$ : $x = -3.54 \text{ cm}$
- b) Distance traveled in 1.5 s: $|5 - (-3.54)| = 8.54 \text{ cm}$
- c) Velocity at 1.5 s: $-5.55 \text{ cm/s}$
- d) Distance traveled in 5.5 s: One full period (20 cm) + 1.5 s movement (8.54 cm) = $28.54 \text{ cm}$ .
- e) Time to reach $x = +2.5 \text{ cm}$ : $0.67 \text{ s}$
- f) Max speed (v_{max} = 2\times\text{\pi}\times f \times A): $7.85 \text{ cm/s}$
- g) Max acceleration (a_{max} = (2\times\text{\pi}\times f)^2 \times A): $12.3 \text{ cm/s}^2$
Energy Example (Horizontal Spring):
- Given: $m = 500 \text{ g}$ , $A = 10 \text{ cm}$ . At equilibrium, $v = 1.0 \text{ m/s}$ .
- a) Period ( $T$ ): $0.63 \text{ s}$
- b) Speed at $x = 5.0 \text{ cm}$ : $0.87 \text{ m/s}$
Vertical Spring Example:
- Given: $100 \text{ g}$ block stretches spring by $16 \text{ cm}$ ( $0.16 \text{ m}$ ) initially (this identifies $k$ via mg=k\text{\Delta}L). Pulled down additional $26 \text{ cm}$ ( $A = 0.26 \text{ m}$ ).
- Speed at $13 \text{ cm}$ above equilibrium: $1.76 \text{ m/s}$
Damping Decay Example:
- Given: \text{\tau} = 50 \text{ s}; initial $A = 20 \text{ cm}$ .
- a) Time until $A = 10 \text{ cm}$ : $35 \text{ s}$ .
- b) Time until Energy is halved: Since E \text{\propto} A^2, Energy decays faster than amplitude. Time = $17 \text{ s}$ .