Lecture 24: Oscillations, Simple Harmonic Motion, and Fluid Dynamics

Recap: Fluids and Fluid Dynamics

Fluids: Properties and Types

Fluids are substances that flow, including both liquids and gases. Their behavior is central to many physical and biological systems.

• Liquids are nearly incompressible; their molecules are closely packed but can move freely.

• Gases are compressible; their volume can be easily increased or decreased.

Pressure in Liquids

Pressure in a fluid at rest depends on depth and is described by hydrostatics.

• Hydrostatic Pressure: $p = p_0 + \rho g d$

• Pascal's Principle: A change in pressure at one point in an incompressible fluid is transmitted equally throughout the fluid: $p' = p + \Delta p$

Archimedes' Principle and Buoyancy

Buoyancy is the upward force exerted by a fluid on an immersed object.

• Archimedes' Principle: The buoyant force equals the weight of the fluid displaced: $F_B = \rho_f V_f g$

• Sink: $\rho_{avg} > \rho_f$, $F_B < w_o$

• Float: $\rho_{avg} < \rho_f$, $F_B > w_o$

• Neutral Buoyancy: $\rho_{avg} = \rho_f$, $F_B = w_o$

Equation of Continuity

For incompressible, laminar flow, the equation of continuity relates the speed and cross-sectional area of a fluid:

• $v_1 A_1 = v_2 A_2$

• Volume flow rate: $Q = \frac{\Delta V}{\Delta t}$

Bernoulli Effect

The Bernoulli effect describes how fluid pressure varies with flow speed along a streamline.

• Pressure is higher where fluid moves slower, and lower where it moves faster.

• This principle explains phenomena such as airplane flight.

Poiseuille's Equation

Describes viscous flow through a tube:

• Average speed: $V_{avg} = \frac{R^2}{8\eta L} \Delta p$

• Volume flow rate: $Q = V_{avg} A = \frac{\pi R^4 \Delta p}{8\eta L}$

The Circulatory System

Blood flow in the body is governed by fluid dynamics:

• Small pressure change across large arteries.

• Pressure drops significantly in arterioles and small vessels due to viscosity.

Oscillations and Simple Harmonic Motion (SHM)

Oscillation: Basic Concepts

An oscillation is a repetitive motion about an equilibrium position.

• Amplitude (A): Maximum displacement from equilibrium.

• Period (T): Time for one complete cycle.

• Frequency (f): Number of cycles per second, $f = \frac{1}{T}$

Simple Harmonic Motion (SHM)

SHM occurs when the restoring force is linear and directed toward equilibrium.

• Mass on a spring: $F_{net} = -kx$

• Pendulum: $F_{net} = -\left(\frac{mg}{L}\right)s$

Describing SHM: Position, Velocity, Acceleration

The motion of a mass on a spring can be described by sinusoidal functions:

• Position: $x(t) = A \cos\left(\frac{2\pi t}{T}\right) = A \cos(2\pi f t)$

• Velocity: $v_x(t) = -v_{max} \sin\left(\frac{2\pi t}{T}\right) = -2\pi f A \sin(2\pi f t)$

• Acceleration: $a_x(t) = -a_{max} \cos\left(\frac{2\pi t}{T}\right) = -(2\pi f)^2 A \cos(2\pi f t)$

• Maximum velocity: $v_{max} = 2\pi f A$

• Maximum acceleration: $a_{max} = (2\pi f)^2 A$

Sinusoidal Relationships

Any quantity that oscillates in time can be written as a sinusoidal function:

• $x = A \sin\left(\frac{2\pi t}{T}\right)$ or $x = A \cos\left(\frac{2\pi t}{T}\right)$

• These functions are bounded (between $A$ and $-A$) and periodic (repeat every $T$).

	t = 0	t = T/4	t = T/2	t = 3T/4	t = T
$x = A \sin(2\pi t/T)$	0	A	0	-A	0
$x = A \cos(2\pi t/T)$	A	0	-A	0	A

Connecting SHM to Uniform Circular Motion

Uniform circular motion projected onto one dimension is simple harmonic motion.

• x-component of position: $x = A \cos \phi$

• Angle at time t: $\phi = \omega t$

• Angular velocity: $\omega = 2\pi f$

• Position as a function of time: $x(t) = A \cos(2\pi f t)$

• Velocity: $v_x = - (2\pi f)A \sin(2\pi f t)$

• Acceleration: $a_x = - (2\pi f)^2 A \cos(2\pi f t)$

• The restoring force is the projection of the centripetal force along the x-axis.

Energy in Simple Harmonic Motion

Kinetic and Potential Energy

Energy in SHM alternates between kinetic and potential forms, with total energy conserved (if no friction).

• Potential energy (spring): $U = \frac{1}{2} k x^2$

• Total energy: $E = K + U = \frac{1}{2} m v^2 + \frac{1}{2} k x^2 = \text{constant}$

• At maximum displacement ($x = \pm A$): $E = U_{max} = \frac{1}{2} k A^2$

• At equilibrium ($x = 0$): $E = K_{max} = \frac{1}{2} m (v_{max})^2$

Finding the Frequency for SHM

The frequency and period of SHM depend on the physical properties of the oscillator, not on amplitude.

• From energy conservation: $\frac{1}{2} m (v_{max})^2 = \frac{1}{2} k A^2$

• Maximum velocity: $v_{max} = \sqrt{\frac{k}{m}} A$

• Frequency: $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

• Period: $T = 2\pi \sqrt{\frac{m}{k}}$

Summary Table: SHM Equations

Quantity	Equation
Position	$x(t) = A \cos(2\pi f t)$
Velocity	$v_x(t) = -2\pi f A \sin(2\pi f t)$
Acceleration	$a_x(t) = -(2\pi f)^2 A \cos(2\pi f t)$
Frequency (spring)	$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$
Period (spring)	$T = 2\pi \sqrt{\frac{m}{k}}$
Potential Energy	$U = \frac{1}{2} k x^2$
Total Energy	$E = \frac{1}{2} k A^2$

Additional info:

• These notes cover topics from Chapter 14: Oscillations, SHM, Energy in SHM, and connections to fluid dynamics (Chapter 13).

• For further study, review textbook sections 14.3-14.7 and solve related problems.