Periodic Motion I: Simple Harmonic Motion (SHM) – Study Notes

Periodic Motion I

Introduction to Periodic Motion

Periodic motion refers to any motion that repeats itself at regular time intervals. The most fundamental type of periodic motion in physics is Simple Harmonic Motion (SHM), which describes the oscillatory motion of systems such as springs and pendulums.

Simple Harmonic Motion (SHM)

Definition and Physical Model

• Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that displacement.

• The classic example is a mass attached to a spring, obeying Hooke's Law: $F_x = -kx$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.

• Any system that follows this law is called a harmonic oscillator.

Equation of Motion

• Applying Newton's Second Law to the spring-mass system yields the differential equation:

$\frac{d^2x}{dt^2} = -\frac{k}{m}x$

• This is a second-order linear differential equation, characteristic of SHM.

• General form: $\frac{d^2x}{dt^2} = -\alpha x$, where $\alpha > 0$.

Solution to the Equation of Motion

• The general solution for the position as a function of time is:

$x = A \cos(\omega t + \phi)$

• A: Amplitude (maximum displacement)

• \omega: Angular frequency, $\omega = \sqrt{\frac{k}{m}}$

• \phi: Phase angle, determined by initial conditions

Connection to Circular Motion

• SHM can be visualized as the projection of uniform circular motion onto one axis.

• For a particle moving in a circle of radius $A$ with angular speed $\omega$:

Position: $x = A \cos(\omega t)$ Velocity: $v_x = -A \omega \sin(\omega t)$ Acceleration: $a_x = -A \omega^2 \cos(\omega t)$

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

• Period: $T = \frac{2\pi}{\omega}$

Velocity and Acceleration in SHM

• Velocity: $v_x = \frac{dx}{dt} = -\omega A \sin(\omega t + \phi)$

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

• Acceleration: $a_x = \frac{d^2x}{dt^2} = -\omega^2 A \cos(\omega t + \phi)$

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

Determining Amplitude and Phase Angle

• Given initial conditions $x(0) = x_0$ and $v_x(0) = v_{0x}$:

$v_{0x} = -\omega A \sin \phi$ $x_0 = A \cos \phi$ $A = \sqrt{x_0^2 + \frac{v_{0x}^2}{\omega^2}}$ $\phi = \tan^{-1}\left(\frac{-v_{0x}}{\omega x_0}\right)$ (with quadrant correction as needed)

• Quadrant corrections are necessary to ensure the correct sign and value for $\phi$ (see appendix for trigonometric rules).

Graphical Representation of SHM

• Displacement, velocity, and acceleration are sinusoidal functions of time, with velocity and acceleration graphs shifted in phase relative to displacement.

• Phasor diagrams help visualize the relationship between these quantities.

Energy in Simple Harmonic Motion

Forms of Energy

• The total mechanical energy $E$ in SHM is the sum of kinetic energy ($K$) and potential energy ($U$):

$E = \frac{1}{2}mv_x^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2$

• At maximum displacement ($x = \pm A$), all energy is potential: $U = \frac{1}{2}kA^2$, $K = 0$.

• At equilibrium ($x = 0$), all energy is kinetic: $K = \frac{1}{2}kA^2$, $U = 0$.

• At $x = \pm \frac{A}{\sqrt{2}}$, $K = U = \frac{E}{2}$.

Energy Conservation

• In the absence of non-conservative forces (e.g., friction), the total energy in SHM is conserved.

• Energy oscillates between kinetic and potential forms as the system moves.

Velocity as a Function of Position

• From energy conservation:

$E = \frac{1}{2}mv_x^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2$ $v_x = \pm \sqrt{\frac{k}{m}(A^2 - x^2)}$

Doubling the Total Energy

• To double the total energy of a spring-mass system in SHM, increase the amplitude by a factor of $\sqrt{2}$:

$E_f = 2E_i = 2\left(\frac{1}{2}kA_i^2\right)$ $E_f = \frac{1}{2}kA_f^2$ $A_f = \sqrt{2}A_i$

• The frequency $\omega$ does not change, since $\omega = \sqrt{\frac{k}{m}}$ is independent of amplitude.

Momentum and Energy in SHM: Collisions

Inelastic Collision Example

• Consider a mass $M$ oscillating in SHM. At equilibrium ($x=0$), a putty of mass $m$ lands and sticks to it.

• Conservation of Momentum: Only the x-component is conserved in the collision:

$Mv_1 + 0 = (M + m)v_2$ $v_2 = \frac{M}{M + m}v_1$

• Conservation of Energy: The collision is totally inelastic, so mechanical energy is not conserved.

• Amplitude after collision:

$A_2 = \frac{M}{M + m}A_1$

• The energy of the oscillator decreases due to the inelastic nature of the collision.

Appendix: Trigonometric Relations in SHM

Phase Angle Determination

• The formula $\phi = \tan^{-1}\left(\frac{-v_{0x}}{\omega x_0}\right)$ may require quadrant correction.

• Use the signs of $\sin \phi$ and $\cos \phi$ to determine the correct quadrant:

Quadrant	sin(φ)	cos(φ)
First (0 to π/2)	+	+
Second (π/2 to π)	+	-
Third (π to 3π/2)	-	-
Fourth (3π/2 to 2π)	-	+

• Example: If $v_{0x} = 0.40$ m/s, $x_0 = 0.015$ m, $\omega = 20$ rad/s, then $\phi_1 = \tan^{-1}\left(\frac{0.40}{20 \times 0.015}\right) = -0.93$ rad.

• If $v_{0x} = -0.40$ m/s, $x_0 = -0.015$ m, $\phi_2 = \pi - 0.93 = 2.21$ rad (second quadrant).

Summary Table: Key Equations in SHM

Quantity	Equation	Description
Displacement	$x = A \cos(\omega t + \phi)$	Position as a function of time
Velocity	$v_x = -\omega A \sin(\omega t + \phi)$	Velocity as a function of time
Acceleration	$a_x = -\omega^2 A \cos(\omega t + \phi)$	Acceleration as a function of time
Angular Frequency	$\omega = \sqrt{\frac{k}{m}}$	Depends on spring constant and mass
Period	$T = \frac{2\pi}{\omega}$	Time for one complete cycle
Total Energy	$E = \frac{1}{2}kA^2$	Constant for undamped SHM

Example Applications

• Oscillating spring-mass systems

• Pendulums (for small angles)

• Electrical analogs: LC circuits (not covered here)

Additional info: These notes cover the essential concepts, equations, and problem-solving strategies for Simple Harmonic Motion, as presented in a college-level introductory physics course.