How is simple harmonic motion related to Hooke law?

c0nman56 2022-10-17 Answered
How is simple harmonic motion related to Hooke law?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

espava8b
Answered 2022-10-18 Author has 12 answers
A particle is said to be in simple harmonic motion if it satisfies the following differential equation:
x ¨ + ω 2 x = 0.
where x is the position of the particle, x ¨ is it's second derivative with respect to time (the acceleration) and ω is a constant that gives us the angular frequency of the oscillation.
Hooke's law is a model for the behavior of ellastic materials, such as a spring (and is only valid if the material does not suffer strong deformations). It states that the magnitude of the force F is proportional to a certain displacement x (with respect to an equilibrium position), with the proportinality given by a constant factor k. Additionaly, the force is a restoring force, that is, it's direction is opposite to the direction of the displacement vector:
F=-kx
Newton's second law states that the force F acting over a particle is equal to the product of it's mass m and it's acceleration x ¨ , or:
F = m x ¨
This two equations give us the relation
m x ¨ = k x
Rearranging, we get:
m x ¨ + k x = 0
Divind by the mass m, we have:
x ¨ + k m x = 0
Associating the ω 2 = k k m (which gives us the expression ω = k m for the angular frequency) gives us the equation for simple harmonic motion.
Therefore, any system that satisfies Hooke's law and isn't acted upon by any other forces is in simple harmonic motion.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more