Key Takeaways
- Simple harmonic motion (SHM) is periodic motion where the restoring force is proportional to displacement
- The position equation is x(t) = A cos(omega t + phi) where A is amplitude and omega is angular frequency
- Period T and frequency f are inversely related: T = 1/f = 2 pi / omega
- Maximum velocity occurs at equilibrium; maximum acceleration occurs at maximum displacement
- Energy in SHM oscillates between kinetic and potential, but total mechanical energy remains constant
What Is Simple Harmonic Motion? A Complete Explanation
Simple harmonic motion (SHM) is a special type of periodic oscillatory motion where the restoring force acting on the object is directly proportional to its displacement from the equilibrium position and always directed toward that equilibrium. This type of motion is fundamental to understanding everything from pendulum clocks to molecular vibrations and electromagnetic waves.
The defining characteristic of SHM is that the acceleration is always proportional to the negative of the displacement: a = -omega^2 x. This relationship creates the smooth, sinusoidal oscillation pattern that characterizes harmonic motion. Examples of SHM include a mass on a spring, a simple pendulum for small angles, and the oscillation of atoms in a crystal lattice.
Real-World Example: Mass-Spring System
Essential Simple Harmonic Motion Formulas
Understanding the mathematical relationships in SHM is crucial for solving physics problems. Here are the fundamental equations:
x(t) = A cos(omega t + phi)
Velocity and Acceleration Equations
By differentiating the position equation, we obtain velocity and acceleration:
- Velocity: v(t) = -A omega sin(omega t + phi) = omega sqrt(A^2 - x^2)
- Acceleration: a(t) = -A omega^2 cos(omega t + phi) = -omega^2 x
- Maximum Velocity: v_max = A omega (occurs at equilibrium, x = 0)
- Maximum Acceleration: a_max = A omega^2 (occurs at x = plus or minus A)
Period and Frequency Relationships
The period (T) is the time for one complete oscillation, while frequency (f) is the number of oscillations per second:
- Period: T = 2 pi / omega
- Frequency: f = 1/T = omega / (2 pi)
- Angular Frequency: omega = 2 pi f = 2 pi / T
How to Calculate Simple Harmonic Motion (Step-by-Step)
Identify the System Parameters
Determine the amplitude (A), angular frequency (omega), and initial phase (phi). For a mass-spring system, omega = sqrt(k/m). For a pendulum, omega = sqrt(g/L).
Set Up the Position Equation
Write x(t) = A cos(omega t + phi). If the object starts at maximum displacement with zero velocity, phi = 0. If it starts at equilibrium moving in the positive direction, phi = -pi/2.
Calculate Displacement at Time t
Substitute your values into x(t) = A cos(omega t + phi). Example: For A = 0.5 m, omega = 4 rad/s, phi = 0, at t = 1 s: x(1) = 0.5 cos(4) = -0.327 m.
Find Velocity and Acceleration
Use v(t) = -A omega sin(omega t + phi) for velocity and a(t) = -omega^2 x for acceleration. At equilibrium, velocity is maximum; at maximum displacement, acceleration is maximum.
Calculate Period and Frequency
Period T = 2 pi / omega gives you the time for one complete cycle. Frequency f = 1/T tells you how many oscillations occur per second.
Energy in Simple Harmonic Motion
One of the most important aspects of SHM is the continuous exchange between kinetic and potential energy. The total mechanical energy remains constant throughout the motion (assuming no friction or damping).
- Kinetic Energy: KE = (1/2) m v^2 = (1/2) m omega^2 (A^2 - x^2)
- Potential Energy: PE = (1/2) k x^2 = (1/2) m omega^2 x^2
- Total Energy: E = (1/2) k A^2 = (1/2) m omega^2 A^2 (constant)
At equilibrium (x = 0), all energy is kinetic. At maximum displacement (x = plus or minus A), all energy is potential. This energy conservation principle is essential for solving many SHM problems.
Pro Tip: Quick Energy Check
When solving SHM problems, use energy conservation as a check. The total energy at any point should equal (1/2)kA^2. If your calculated kinetic plus potential energy does not equal this value, recheck your calculations.
Mass-Spring Systems: The Classic SHM Example
The horizontal mass-spring system is the prototypical example of simple harmonic motion. When a mass m attached to a spring with spring constant k is displaced from equilibrium, it oscillates with:
- Angular Frequency: omega = sqrt(k/m)
- Period: T = 2 pi sqrt(m/k)
- Frequency: f = (1/2 pi) sqrt(k/m)
Notice that the period depends only on mass and spring constant, not on amplitude. This is a defining characteristic of true SHM - the period is independent of the amplitude.
Vertical Mass-Spring Systems
For vertical springs, gravity causes the equilibrium position to shift, but the oscillation frequency remains the same. The new equilibrium position is where the spring force equals the gravitational force: mg = k delta, where delta is the initial stretch.
The Simple Pendulum and Small Angle Approximation
A simple pendulum (point mass on a massless string) exhibits SHM only for small angular displacements (typically less than 15 degrees). Under this approximation, sin(theta) approximately equals theta, and we get:
- Angular Frequency: omega = sqrt(g/L)
- Period: T = 2 pi sqrt(L/g)
- Frequency: f = (1/2 pi) sqrt(g/L)
Note that the period depends only on length L and gravitational acceleration g, not on mass or amplitude. This property made pendulums essential for early timekeeping devices.
Pro Tip: Measuring Gravity
You can measure local gravitational acceleration by timing a pendulum. Rearranging the period equation: g = 4 pi^2 L / T^2. Time multiple oscillations and divide by the count for better accuracy.
Damped Oscillations: Real-World SHM
In real systems, friction and air resistance cause oscillations to gradually decrease in amplitude. This is called damped harmonic motion. The three types of damping are:
- Underdamped: System oscillates with gradually decreasing amplitude (most common)
- Critically Damped: System returns to equilibrium fastest without oscillating
- Overdamped: System returns to equilibrium slowly without oscillating
For underdamped motion, the amplitude decreases exponentially: A(t) = A_0 e^(-gamma t), where gamma is the damping coefficient. Car shock absorbers are designed to be slightly underdamped for the best ride quality.
Applications of Simple Harmonic Motion
Understanding SHM is crucial across many fields of physics and engineering:
- Mechanical Engineering: Vibration analysis, suspension systems, seismology
- Electrical Engineering: LC circuits oscillate with SHM; resonance in filters
- Acoustics: Sound waves, musical instruments, speaker design
- Molecular Physics: Atomic vibrations in crystals, infrared spectroscopy
- Quantum Mechanics: Quantum harmonic oscillator is a foundational model
- Astronomy: Stellar pulsations, orbital mechanics perturbations
Common Mistakes to Avoid in SHM Problems
Students frequently make these errors when working with simple harmonic motion:
- Confusing omega and f: Remember omega = 2 pi f. Using frequency directly in the cosine function gives wrong answers.
- Wrong phase constant: The initial conditions determine phi. At t=0: if x = A, then phi = 0; if x = 0 and v greater than 0, then phi = -pi/2.
- Calculator in degrees: Angular frequency is in rad/s, so ensure your calculator is in radian mode.
- Sign errors in acceleration: Remember a = -omega^2 x. Acceleration is always opposite to displacement direction.
- Large angle pendulum errors: The SHM equations only apply for small angles (less than 15 degrees).
Frequently Asked Questions
Frequency (f) measures oscillations per second in Hertz (Hz). Angular frequency (omega) measures the rate of change of the phase angle in radians per second. They are related by omega = 2 pi f. Angular frequency is used in the mathematical equations because it simplifies the calculus when working with sinusoidal functions.
For a simple pendulum, both the gravitational force (restoring force) and inertia (resistance to acceleration) are proportional to mass. These effects cancel out, making the period independent of mass. The period T = 2 pi sqrt(L/g) depends only on length and gravitational acceleration. This is why Galileo could use pendulums to study motion regardless of the bob material.
The phase constant is determined by initial conditions at t=0. If the object starts at maximum positive displacement (x = A) with zero velocity, phi = 0. If it starts at equilibrium (x = 0) moving in the positive direction, phi = -pi/2. For arbitrary initial conditions, use: tan(phi) = -v_0 / (omega x_0), where x_0 and v_0 are initial position and velocity.
Yes! Both x(t) = A cos(omega t + phi) and x(t) = A sin(omega t + phi') are valid. The only difference is the phase constant. Since sin(theta) = cos(theta - pi/2), the equations are equivalent with phi' = phi - pi/2. The cosine form is traditional because it starts at maximum displacement when phi = 0, but either form works.
Doubling the amplitude doubles the maximum displacement, maximum velocity (v_max = A omega), and maximum acceleration (a_max = A omega^2). The total energy quadruples since E = (1/2)kA^2. However, the period and frequency remain unchanged - this amplitude independence is a key characteristic of true simple harmonic motion.
SHM is the projection of uniform circular motion onto a diameter. Imagine a point moving in a circle of radius A with angular velocity omega. Its x-coordinate traces out x = A cos(omega t), which is exactly the SHM equation. This connection explains why angular frequency omega is used - it is literally the angular velocity of the corresponding circular motion.
The defining equation a = -omega^2 x means acceleration always opposes displacement. This is because the restoring force always points toward equilibrium. When displaced positively, the force (and acceleration) point negatively (back toward equilibrium). This is what makes the motion oscillatory rather than exponential growth.
Rearrange the period formula T = 2 pi sqrt(m/k) to solve for k: k = (4 pi^2 m) / T^2. For example, if a 2 kg mass oscillates with a period of 0.5 s, then k = (4 pi^2 times 2) / (0.5)^2 = 315.8 N/m. This method is commonly used to measure spring constants experimentally.