Spring-Mass Systems Calculator

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Example

Created on 2024-06-20Asked by Aria Davis (Solvelet student)

Solve the differential equation for a spring-mass system:

m \frac{d^2x}{dt^2} + kx = 0

Solution

To solve the differential equation for a spring-mass system

m \frac{d^2x}{dt^2} + kx = 0

: 1. **Rewrite the equation:**

\frac{d^2x}{dt^2} + \frac{k}{m}x = 0.

2. **Find the characteristic equation:**

r^2 + \frac{k}{m} = 0.

3. **Solve the characteristic equation:**

r = \pm i\sqrt{\frac{k}{m}}.

4. **General solution:**

x(t) = c_1 \cos\left(\sqrt{\frac{k}{m}} t\right) + c_2 \sin\left(\sqrt{\frac{k}{m}} t\right),

where

c_1

and

c_2

are arbitrary constants. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Camila Carter on Solvelet

1. Determine the period of oscillation for a spring-mass system with a spring constant of 100 N/m and a mass of 0.5 kg.2. Calculate the displacement of a mass attached to a spring with a spring constant of 50 N/m and undergoing simple harmonic motion with an amplitude of 0.2 m.,

DefinitionSpring-mass systems are a common form of such mechanical vibrations or mechanical oscillations systems because they obey Hooke's Law and Newton's Second Law. The equation for the system is mdt2d2x+kx=0 where m is mass and k is the spring constant. Example: Consider a mass-spring system with m=1 kg, k=4 N/m: will oscillate with T=2πm/k.

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