Understanding Simple Harmonic Motion: Spring and Hook's Law

[saltrock]

A mass is attached on a spring .When it is stretched and is let free to move then it oscillates.show that this motion is simple harmomic>>>spring obeys Hook's law.
My workings:
As the mass pulls the spring down mg>-kx but when it as it goes further more down -kx is greater than mg and the apring goes up n hence oscillates.
when the mass goes downwards ,displacement is negative,on the extremevelocity is 0 which means acceleration is maxium.so acceleration is proportional to displacement but due to the negative sign of displacement they are in opposite direction
>>> i am a bit confused how acceleration is in opposite direction to the displacement...My working is a bit mixed up ..can you give me some better way of answering this question.
Thanks a lot in advance.

 

[Borxter]

The spring+mass system will reach an equilibrium where the retarding force of the spring will balance out the gravity. ie mg = kx. This new x will be the position of the new equilibrium. If you now stretch the spring any more, it will behave just like any other spring and obey hooke's law and begin oscillating.

 

[wais]

Firstly, let's define simple harmonic motion. It is a type of periodic motion in which the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. In other words, as the object moves away from its equilibrium position, a force is exerted to bring it back towards the equilibrium position.

In the case of a mass attached to a spring, the spring obeys Hooke's law which states that the force exerted by the spring is directly proportional to the displacement from its equilibrium position. This can be represented mathematically as F = -kx, where F is the force, k is the spring constant, and x is the displacement.

When the mass is stretched and then released, it begins to oscillate back and forth around its equilibrium position. As it moves downwards, the displacement is negative and according to Hooke's law, the force exerted by the spring is also negative (in the opposite direction). This negative force causes the mass to slow down and eventually come to a stop at the equilibrium position.

As the mass moves upwards, the displacement becomes positive and so does the force exerted by the spring. This positive force accelerates the mass back towards the equilibrium position, causing it to oscillate back and forth.

Therefore, we can see that the acceleration is in the opposite direction to the displacement, as stated in the definition of simple harmonic motion. This is because the force and displacement are in opposite directions and acceleration is directly proportional to force and inversely proportional to mass (a = F/m). As the mass moves away from the equilibrium position, the force and acceleration act in the opposite direction to bring it back towards equilibrium.

In conclusion, the motion of a mass attached to a spring is an example of simple harmonic motion because it follows the principles of Hooke's law and exhibits a restoring force that is directly proportional to the displacement and acts in the opposite direction.