What is the relationship between tension force and SHM?

[Peter G.]

Hi,

I am learning about SHM and in order to explain why the acceleration is proportional to the displacement but in opposite directions the book used as an example a particle with mass m, attached to a spring.

He defines the displacement to the right as being positive and to the left as being negative.

When the particle is displaced to the right, x, the extension, will be positive. The tension in the spring is towards the left. But why? Is it a characteristic of the spring to always have its force trying to restore the spring to its equilibrium position?

Thanks,
Peter G.

 

[Doc Al]

Is it a characteristic of the spring to always have its force trying to restore the spring to its equilibrium position?

Absolutely. Ideal springs obey Hooke's law.

And whenever the restoring force is proportional to the displacement from equilibrium, the resulting motion will be simple harmonic motion.

 

[Encephalon]

Yes. This is how Hooke's law is typically expressed $$F=-kx$$
Where x is the displacement from the equilibrium position and k is the spring constant. There are a number of different and non-intuitive ways to come to this conclusion (you will cover springs in differential equations and a course on thermal/statistical physics), but the physically intuitive understanding is that a spring will apply a force to a mass at any point other than it's equilibrium position. If it is moved away from that position, the spring then applies a force to move it in, and, in an undamped case, will continue applying force until the mass is past the equilibrium position in the opposite direction, to which the spring will then a apply a force in reverse. This leads to oscillation, and will be infinitely periodic in the undamped and frictionless case.

 

[Peter G.]

Ok, thanks guys!

 

[rwisz]

Hi Peter,

The relationship between tension force and SHM (simple harmonic motion) is a fundamental concept in physics. In simple terms, SHM is a type of oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium. This means that the further the object is from its equilibrium position, the stronger the restoring force will be, and it will always act in the direction to bring the object back to its equilibrium position.

In the example you mentioned, the tension force in the spring is what provides the restoring force for the particle attached to it. As the particle is displaced to the right, the spring is stretched and the tension force in the spring pulls the particle back towards its equilibrium position. This is because the spring is a type of elastic material, and it has a natural tendency to return to its original shape when stretched or compressed.

So, to answer your question, it is indeed a characteristic of the spring to always have its force trying to restore the spring to its equilibrium position. This is what allows the particle to undergo SHM, as the tension force in the spring is directly proportional to the displacement of the particle from its equilibrium position.

I hope this explanation helps to clarify the relationship between tension force and SHM. Let me know if you have any further questions.

Best,