Simple Harmonic Motion: Mass-Spring Systems and Pendulums

Simple Harmonic Motion (SHM)

Introduction to SHM

Simple Harmonic Motion (SHM) describes the periodic oscillatory motion of an object where the restoring force is proportional to and opposite its displacement from equilibrium. SHM is foundational in physics, underlying many mechanical and wave phenomena.

• Restoring Force: The force always acts to return the object to equilibrium and is described by Hooke's Law for springs: $F = -kx$.

• Energy Exchange: Energy oscillates between kinetic energy (KE) and potential energy (PE), but the total mechanical energy remains constant.

• Periodicity: The motion repeats in cycles, with each cycle taking a fixed period $T$.

Connection to Uniform Circular Motion (UCM)

SHM can be visualized as the one-dimensional projection of uniform circular motion (UCM). The mathematical relationships for period and frequency in UCM apply directly to SHM.

• Period ($T$): Time for one complete cycle (measured in seconds).

• Frequency ($f$): Number of cycles per second (measured in Hertz, Hz).

• Relationship: $f = \frac{1}{T}$ and $T = \frac{1}{f}$

Mass-Spring System

Horizontal Mass-Spring System

A mass attached to a spring oscillates horizontally when displaced from equilibrium. The system demonstrates SHM if the force is linear (Hooke's Law).

• Displacement ($x$): Measured from equilibrium ($x=0$).

• Amplitude ($A$): Maximum displacement from equilibrium.

• Restoring Force: $F = -kx$

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

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

Vertical Mass-Spring System

When the spring is vertical, gravity shifts the equilibrium position, but the oscillatory motion remains SHM about the new equilibrium.

• Equilibrium: Occurs where spring force balances gravity: $k x = mg$

• Oscillation: About the new equilibrium position, with the same period formula as the horizontal case.

Key Properties at Specific Positions

Position	Displacement	Velocity	Force	Acceleration
At $x = \pm A$	Maximum	0	Maximum	Maximum
At $x = 0$	0	Maximum	0	0

Energy in the Mass-Spring System

The total mechanical energy in a mass-spring system is conserved and is the sum of kinetic and elastic potential energy:

• Kinetic Energy: $KE = \frac{1}{2}mv^2$

• Elastic Potential Energy: $EPE = \frac{1}{2}kx^2$

• Total Energy: $E_{total} = \frac{1}{2}kA^2$ (constant)

Simple Pendulum

Definition and Conditions for SHM

A simple pendulum consists of a mass (bob) attached to a lightweight, inextensible string. For small angles ($\theta < 15^\circ$), the restoring force is proportional to displacement, and the motion approximates SHM.

• Restoring Force: $F = -mg\sin\theta \approx -mg\theta$ for small $\theta$

• Displacement: $x = L\theta$

Period and Frequency of a Simple Pendulum

• Period: $T = 2\pi \sqrt{\frac{L}{g}}$

• Frequency: $f = \frac{1}{2\pi} \sqrt{\frac{g}{L}}$

• Note: The period is independent of mass and depends only on the length $L$ and gravitational acceleration $g$.

Energy in the Simple Pendulum

The energy in a simple pendulum oscillates between gravitational potential energy (GPE) and kinetic energy (KE), with total energy conserved.

• Gravitational Potential Energy: $GPE = mgh$

• Kinetic Energy: $KE = \frac{1}{2}mv^2$

• Total Energy: $E_{total} = mgh_{max}$ (constant)

Graphical Analysis of SHM

Energy vs. Position Graphs

Energy graphs for oscillating systems (mass-spring or pendulum) show how kinetic and potential energy vary with position. The total energy remains constant, while KE and PE are complementary.

• At maximum displacement ($x = \pm A$): All energy is potential.

• At equilibrium ($x = 0$): All energy is kinetic.

Bullet-Block Oscillating System

Conservation of Momentum and Energy

When a bullet embeds in a block attached to a spring, the system's motion can be analyzed using conservation of momentum (for the collision) and conservation of energy (for subsequent oscillations).

• Initial Kinetic Energy: $KE = \frac{1}{2}mv^2$

• Final Velocity (after collision): $v' = \frac{m_{bullet}v_{bullet}}{m_{bullet} + m_{block}}$

• Maximum Compression: All kinetic energy converts to spring potential energy: $\frac{1}{2}k x_{max}^2 = KE_{after}$

• Period of Oscillation: $T = 2\pi \sqrt{\frac{m_{total}}{k}}$

Summary Table: Key Equations in SHM

System	Period ($T$)	Frequency ($f$)	Total Energy
Mass-Spring	$2\pi \sqrt{\frac{m}{k}}$	$\frac{1}{2\pi} \sqrt{\frac{k}{m}}$	$\frac{1}{2}kA^2$
Pendulum	$2\pi \sqrt{\frac{L}{g}}$	$\frac{1}{2\pi} \sqrt{\frac{g}{L}}$	$mgh_{max}$

Example Problems

• Find the period of a 0.5 kg mass on a spring with $k = 120$ N/m: $T = 2\pi \sqrt{\frac{0.5}{120}}$

• Find the maximum speed of a 1.4 kg mass on a spring ($k = 75$ N/m, $A = 0.05$ m): $v_{max} = A \sqrt{\frac{k}{m}}$

• Find the period of a 2.0 m pendulum: $T = 2\pi \sqrt{\frac{2.0}{9.8}}$

Additional info: This guide covers the core concepts, equations, and applications of simple harmonic motion, including mass-spring systems, simple pendulums, and energy analysis. It is suitable for college-level introductory physics courses.