Simple Harmonic Motion - Acceleration and Time Period

[ProPM]

Hi,

I am a bit confused regarding the relation between the acceleration and time period in simple harmonic motion.

This questions asks what would be one change to the motion of a car's wing mirror in case the glass was switched for a heavier one. I don't understand for example, how would the period change. I can think that the same force will have to move a greater mass over the same distance but, wouldn't the amplitude decrease if the mass increases?

Thanks.

 

[lightgrav]

the oscillation is going to occur around the _equilibrium_ location
(more massive glass will have a lower equilibrium);
the oscillation Energy depends on Amplitude (squared, times ½k)
but the frequency depends on the spring "stiffness" compared to the inertia
(inside a square root). If you have the same Force, but a bigger mass, what can you tell about the acceleration at these turn-around points?
(that is where it spends most of its time, since its speed is slow there).

 

[ProPM]

Sorry, I couldn't follow very well the beginning. But, for the same force with a bigger mass the acceleration will decrease.

So, from the equation, either omega must decrease (frequency decreases/time period increases) or the displacement will decrease...

How do I know which of these will happen?

 

[lightgrav]

IF the oscillation has the same (total) Energy, it has the same maximum PE (spring),
so it will have the same amplitude (maximum displacement from equilibrium).
. . . . if the mass is doubled . . .
acceleration will be halved, in x = ½at^2 , (here, t is roughly time for ¼ oscillation)
but it has to get just as far, so t^2 is doubled . . . so t is increased by sqrt(2)
that's why the omega formula is squareroot(k/m).

 

[rwisz]

I can provide some clarification on the relationship between acceleration and time period in simple harmonic motion. In simple harmonic motion, the acceleration of an object is directly proportional to its displacement from its equilibrium position. This means that as the object moves further away from its equilibrium, the acceleration increases, and as it moves closer to its equilibrium, the acceleration decreases. This relationship is described by Hooke's Law, which states that the force applied on an object is directly proportional to its displacement from its equilibrium position.

Now, in the context of the car's wing mirror, the switch to a heavier glass would indeed result in a change in the time period of its simple harmonic motion. This is because the heavier glass would require more force to move it, and therefore the acceleration would be lower compared to the lighter glass. This decrease in acceleration would result in a longer time period for the mirror's motion.

Additionally, you are correct in your observation that the amplitude (the maximum displacement from equilibrium) would decrease with the heavier glass. This is because as the mass increases, the force required to move it a certain distance also increases, making it more difficult to achieve a larger displacement.

I hope this helps to clarify the relationship between acceleration and time period in simple harmonic motion. Feel free to ask any further questions if needed.