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 attached to a spring with spring constant 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 , where is the displacement from the equilibrium position.
Equation of Motion
Using , we have: Rearranging, we obtain the equation of motion: This is a second-order linear differential equation for , describing simple harmonic motion.
Solution of the Equation
To solve this equation, let's define , where is the angular frequency of oscillation. The equation simplifies to:
This differential equation has a general solution: where and are constants determined by the initial conditions of the system.
Alternatively, this can be written in the form: where is the amplitude, and is the phase angle, determined by the initial position and velocity.
This solution represents a simple harmonic oscillation with amplitude , angular frequency , and phase .
Would you like further details on any steps or have additional questions?
Here are five related questions to expand your understanding:
- How would the equation change if damping is introduced?
- What is the period of oscillation in terms of and ?
- How does the solution change if an external force is applied?
- How can the energy of this oscillating system be described?
- What are the initial conditions required to find specific values for and ?
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