Question

Derive the differential equation of motion for SHM using Hooke’s law.​

Answer 1

Answer:

The force is a restoring force because it tends to restore the object back to its original position. This relationship is called Hooke's Law. If a mass is attached to a spring and then displaced from its rest position and released, it will oscillate around that rest position in simple harmonic motion.

Explanation:

A particle is said to be in simple harmonic motion if it satisfies the following differential equation:

..

x+ω2x=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=−kx

Rearranging, we get:

m..x+kx=0

Divided by the mass

m

, we have:

..x+mx=0

Associating the

ω2=kk m

(which gives us the expression

ω=√km

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.

Answer 2

Answer:

The differential equation of motion for SHM using Hooke's law is

d²x/dt² +  (ω² ) x = 0.

Explanation:

The restoring force due to SHM can be written as,

Using Hooke's law,

F ∝ x

F = -kx

and, according to Newton's second law of motion,

F = ma

Equating both the equation of forces,

ma = -kx

a = - (k/m) x

Let, ω² = - (k/m)

a = - (ω² ) x

Acceleration can be defined as the double differentiation of the distance,

a = d²x/dt²

So, the differential equation will be

d²x/dt²  = - (ω² ) x

d²x/dt² +  (ω² ) x = 0

Hence, the differential equation of motion for SHM using Hooke's law is d²x/dt² +  (ω² ) x = 0.

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