Explain why, when a body is moving in SHM, an equation of the form a=-kx must be satisfied. In this equation, k is a positive constant, and a and x have their usuak meanings.

Question

anonymous · Answer

An oscillation follows simple harmonic motion if it fulfils the following two rules:

Acceleration is always in the opposite direction to the displacement from the equilibrium position
Acceleration is proportional to the displacement from the equilibrium position

mani_jha · Answer

What do you mean by a harmonic motion? It is esentially a periodic motion which can be represented by a sine/cosine function.
For the motion to be periodic, you've got to ensure that the body in motion stops after some time, and returns to its original position.
And the best way to ensure that is that a force continously acts opposite to its motion, so that at a point the body stops(because of deceleration) and then this so-called restoring force reverses its direction, and brings it back to its original position.
In an analogy to this, think of a car that is situated at a particular place(mean postion) where you are standing. Suddenly it starts runnning away from you, you take hold of the car, and try to pull it backwards so that it returns to the mean position. As the car moves further and further, you pull harder and harder. At a point, the velocity of the car becomes zero(extreme position), and then you pull it easily towards the mean position. Meanwhile, its velocity increases. When it reaches the mean position, it doesnt stop there and it moves forward because it has speed. Again, you try to pull the car towards the mean position.
Did you get this analogy?

anonymous · Answer

Ok so what is your answer?

anonymous · Answer

This relationship comes out of Hooke's law, i think. Due to the elasticity of the material, when there is elongation or compression in it, there is a restoring force set up. According to Hooke's law this force would directly proportional to this displacement.

anonymous · Answer

?

anonymous · Answer

you know Hooke's law

 force is directly proportional to the elongation or compression in material

anonymous · Answer

Yes i know Hookes law

anonymous · Answer

well you could just derive it from equation of motion. 

$$a = -\omega ^{2}x$$

anonymous · Answer

would be the answer