Source: ENGR 216 Lectures, Lab 6
Tags: harmonic motion, simple harmonic motion, SHM, SHO, oscillation, sine function, spring, pendulum, angular frequency, period, amplitude, ordinary differential equation, ODE, ENGR 216, PHYS 206
Simple harmonic motion (SHM) occurs whenever a restoring force is proportional to displacement. The motion is sinusoidal, characterised by amplitude, frequency, and phase. Springs and pendulums are the standard examples. The underlying ordinary differential equation (ODE) d²x/dt² = −ω²x produces sine and cosine solutions, and recognising this pattern is key to analysing oscillatory systems in engineering.
Simple harmonic motion (SHM)
Periodic motion where the restoring force is directly proportional to displacement from equilibrium and directed toward equilibrium: F = −kx. The resulting motion is sinusoidal.
Amplitude (A)
The maximum displacement from equilibrium, measured in metres. The object oscillates between +A and −A.
Period (T)
The time for one complete oscillation, measured in seconds. T = 1/f = 2π/ω.
Frequency (f)
The number of complete oscillations per second, measured in hertz (Hz). f = 1/T.
Angular frequency (ω)
The rate of oscillation in radians per second: ω = 2πf = 2π/T. For a mass-spring system, ω = √(k/m). For a simple pendulum, ω = √(g/L).
Phase angle (φ)
The initial angle that determines where in the cycle the motion starts. In x(t) = A cos(ωt + φ), the value of φ sets the starting position.
Restoring force
Any force that pulls a displaced object back toward its equilibrium position. For SHM, this force must be proportional to displacement.
Spring constant (k)
The stiffness of a spring, measured in N/m. Hooke's law: F = −kx. A larger k means a stiffer spring and a higher oscillation frequency.
Damping
Energy dissipation that causes oscillations to decay over time. Underdamped systems oscillate with decreasing amplitude; overdamped systems return to equilibrium without oscillating; critically damped systems return fastest without oscillating.
Ordinary differential equation (ODE)
An equation involving derivatives of a function with respect to one variable. The SHM equation d²x/dt² + ω²x = 0 is a second-order linear ODE with constant coefficients.
A mass m attached to a spring with spring constant k on a frictionless surface:
Restoring force: F = −kx
Applying Newton's second law: ma = −kx, so d²x/dt² = −(k/m)x
This is the SHM differential equation with ω² = k/m
The general solution is:
x(t) = A cos(ωt + φ)
where A and φ are determined by initial conditions (initial position and velocity).
The velocity and acceleration follow:
v(t) = −Aω sin(ωt + φ)
a(t) = −Aω² cos(ωt + φ) = −ω²x(t)
Maximum speed occurs at equilibrium (x = 0): v_max = Aω. Maximum acceleration occurs at the extremes (x = ±A): a_max = Aω².
A mass on a light string of length L, swinging through small angles:
Restoring torque: τ = −mgL sin θ ≈ −mgLθ (small angle approximation, θ in radians)
This gives: d²θ/dt² = −(g/L)θ
Angular frequency: ω = √(g/L)
Period: T = 2π√(L/g)
The period of a simple pendulum depends on length and gravitational acceleration, not on mass. This is a useful experimental fact.
The small-angle approximation (sin θ ≈ θ) is valid for θ less than about 15° (0.26 rad). Beyond that, the motion deviates from true SHM.
Total mechanical energy is constant (in the undamped case) and oscillates between kinetic and potential forms:
Potential energy: U = ½kx² = ½kA² cos²(ωt + φ)
Kinetic energy: KE = ½mv² = ½kA² sin²(ωt + φ)
Total energy: E = ½kA² (constant)
At maximum displacement, all energy is potential. At equilibrium, all energy is kinetic.
The sine and cosine functions describe SHM. Key properties:
Period of sin(ωt) is 2π/ω
sin and cos are phase-shifted by π/2: cos(ωt) = sin(ωt + π/2)
Derivatives cycle: d/dt[sin(ωt)] = ω cos(ωt), d/dt[cos(ωt)] = −ω sin(ωt)
Squared forms average to ½ over one cycle
The equation d²x/dt² + ω²x = 0 is a "universal" ordinary differential equation (UODE) in the sense that it appears across physics and engineering whenever a system has a linear restoring mechanism: mechanical springs, LC electrical circuits, vibrating beams, acoustic resonance.
Recognising that an equation has this form tells you immediately that the solution is sinusoidal with angular frequency ω, without needing to solve it from scratch.
For a damped system, the equation becomes:
d²x/dt² + 2γ(dx/dt) + ω₀²x = 0
where γ is the damping coefficient. The solution is an exponentially decaying sinusoid when underdamped (γ < ω₀).
Hooke's law: F = −kx
SHM solution: x(t) = A cos(ωt + φ)
Angular frequency (spring): ω = √(k/m) , T = 2π√(m/k)
Angular frequency (pendulum): ω = √(g/L) , T = 2π√(L/g)
Max speed: v_max = Aω
Max acceleration: a_max = Aω²
Total energy: E = ½kA²
SHM ODE: d²x/dt² + ω²x = 0
⚠️ The period of a mass-spring system depends on m and k, not on amplitude. The period of a simple pendulum depends on L and g, not on mass or amplitude (small-angle regime).
⚠️ Be careful with the distinction between frequency (f in Hz) and angular frequency (ω in rad/s). They differ by a factor of 2π.
⚠️ Energy conservation in SHM is a powerful alternative to force analysis. If you know the amplitude and spring constant, you can find the speed at any position without solving the differential equation.
⚠️ Lab 6 (Harmonic, virtual) will ask you to verify the period-mass or period-length relationship experimentally. Expect to plot T² vs. m or T² vs. L and extract k or g from the slope.
⚠️ The UODE framing means exam questions may present an unfamiliar system (e.g. an LC circuit or a floating object bobbing in water) and ask you to identify ω from the governing equation.
Q: A 0.5 kg mass on a spring (k = 200 N/m) is displaced 0.1 m and released. What is the period of oscillation?
A: ω = √(k/m) = √(200/0.5) = 20 rad/s. T = 2π/ω = 2π/20 ≈ 0.314 s.
Q: What is the maximum speed of the mass in the problem above?
A: v_max = Aω = 0.1 × 20 = 2 m/s.
Q: A pendulum has a period of 2 seconds. How long is it?
A: T = 2π√(L/g), so L = gT²/(4π²) = 9.81 × 4 / (4π²) ≈ 0.993 m, roughly 1 metre.
Q: At what position in SHM is the speed greatest?
A: At the equilibrium position (x = 0), where all energy is kinetic.
Q: How do you recognise that a physical system will undergo SHM?
A: If the restoring force (or torque) is proportional to displacement (or angle) from equilibrium and directed back toward equilibrium, the system undergoes SHM. Equivalently, if the equation of motion can be written as d²x/dt² = −ω²x.
simple harmonic motion, SHM, SHO, oscillation, spring-mass system, Hooke's law, spring constant, amplitude, period, frequency, angular frequency, phase angle, simple pendulum, small angle approximation, damping, underdamped, critically damped, overdamped, energy in SHM, ordinary differential equation, ODE, UODE, sine function, cosine function, Lab 6, ENGR 216, PHYS 206, mechanics, Texas A&M