Math Problem Statement

Write down the equation of motion for a mass on a spring. Find the solution of the equation

Solution

The equation of motion for a mass mm attached to a spring with spring constant kk is derived from Newton's second law. Assuming no damping or external forces, the only force acting on the mass is the restoring force of the spring, given by Hooke's Law as F=kxF = -kx, where xx is the displacement from the equilibrium position.

Equation of Motion

Using F=ma=md2xdt2F = ma = m \frac{d^2x}{dt^2}, we have: md2xdt2=kxm \frac{d^2x}{dt^2} = -kx Rearranging, we obtain the equation of motion: d2xdt2+kmx=0\frac{d^2x}{dt^2} + \frac{k}{m} x = 0 This is a second-order linear differential equation for x(t)x(t), describing simple harmonic motion.

Solution of the Equation

To solve this equation, let's define ω=km\omega = \sqrt{\frac{k}{m}}, where ω\omega is the angular frequency of oscillation. The equation simplifies to: d2xdt2+ω2x=0\frac{d^2x}{dt^2} + \omega^2 x = 0

This differential equation has a general solution: x(t)=Acos(ωt)+Bsin(ωt)x(t) = A \cos(\omega t) + B \sin(\omega t) where AA and BB are constants determined by the initial conditions of the system.

Alternatively, this can be written in the form: x(t)=Ccos(ωt+ϕ)x(t) = C \cos(\omega t + \phi) where C=A2+B2C = \sqrt{A^2 + B^2} is the amplitude, and ϕ\phi is the phase angle, determined by the initial position and velocity.

This solution represents a simple harmonic oscillation with amplitude CC, angular frequency ω\omega, and phase ϕ\phi.

Would you like further details on any steps or have additional questions?


Here are five related questions to expand your understanding:

  1. How would the equation change if damping is introduced?
  2. What is the period of oscillation in terms of mm and kk?
  3. How does the solution change if an external force is applied?
  4. How can the energy of this oscillating system be described?
  5. What are the initial conditions required to find specific values for AA and BB?

Tip: In problems involving oscillations, identifying if the system is undamped, damped, or driven will help you choose the correct form of the differential equation to model it.

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Simple Harmonic Motion
Newton's Second Law

Formulas

F = -kx (Hooke's Law)
m * d²x/dt² = -kx (Equation of Motion)
d²x/dt² + (k/m) * x = 0 (Equation of Motion)
ω = sqrt(k/m) (Angular Frequency)
x(t) = A * cos(ωt) + B * sin(ωt) (General Solution)
x(t) = C * cos(ωt + φ) (Alternative Solution)

Theorems

Hooke's Law
Second-Order Linear Differential Equations

Suitable Grade Level

Grades 11-12