What is the correct formula for acceleration for SHM

In summary, there is a discrepancy in the values obtained for the maximum acceleration of a horizontal mass-spring oscillator undergoing SHM when using two different equations. Upon further examination, it is found that the given information is contradictory and suggests a natural period of about a tenth of a second, which is significantly different from the suggested 6 seconds. This leads to doubts about the accuracy of the question and the values obtained.
  • #1
dilton_8000
4
1

Homework Statement


If the mass of a horizontal mass-spring oscillator undergoing SHM is 0.5kg and the force constant is 20N/cm, what is the maximum restoring force of the oscillator? And what is the maximum acceleration? (Time period is 6s and amplitude of oscillation is 12cm)

Homework Equations


Acceleration (a) = -w^2(x) and a = -k(x/m)

The Attempt at a Solution


When I use the given information and plug in the values, the answer is different for each equation. For the first equation, I get an answer of 0.13ms^-2, and for the latter I get 480ms^-2.

I don't seem to understand why they don't match and which one is correct. because I know we had to equate ma=-kx to derive the equation omega(w) = sqr.rt(k/m).

Any replies would be tremendously helpful. Thanks.

PS - This is my first post. :)
 
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  • #2
Welcome to PF

I'm not entirely sure I understand what's confusing you:
You calculated the acceleration using a=-ω2x and got a different answer than when you used a=-kx/m?

But like you said, ω=√(k/m) so how can the equations be any different?
 
  • #3
Thanks for the reply... Actually Yes, I did get two different answers for the different formulae. I think the problem lies with the value to ω. When I use

ω=√(k/m),
with k = 20N/cm, and m = 0.5kg,
I get 6.32 rad s^-1 as the answer.

Plugging in this value of ω in

a=-ω^2x gives me 4.80 ms^-2

But when I use

ω = 2π/T, with T = 6s,
I get the value 1.05 rad s^-1.

Thus, a=-ω^2x now gives me 0.13 ms^-2

I can't figure out why I get two different values. Shouldn't they match? And also, which one is correct?

PS - I used F =-k.x to find the maximum restoring force. I got 240 N which seems to be correct. Its the acceleration bit that's throwing me off.
 
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  • #4
The given spring constant and mass imply a natural period of about a tenth of a second. This is a far cry from the suggested 6 seconds. No mention of damping is made (and it would have to be large indeed), so the given values seem self-contradictory.
 
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  • #5
Thanks so much... I was suspecting something must have been wrong with the question. BTW, when you say the information implies a natural period of a tenth of a second, how do you arrive at that conclusion? I don't think I know of any equations or relationships that helps me calculate that. Can you help? :)
 
  • #6
dilton_8000 said:
Thanks so much... I was suspecting something must have been wrong with the question. BTW, when you say the information implies a natural period of a tenth of a second, how do you arrive at that conclusion? I don't think I know of any equations or relationships that helps me calculate that. Can you help? :)

Angular frequency ω is related to the period. You should know how ω relates to f, and how f relates to T.
 
  • #7
Oh, ofcourse, silly me... :)... Thanks... I think I've figured it out... Cheers
 

1. What is SHM and how does it relate to acceleration?

SHM stands for Simple Harmonic Motion, which is a type of motion in which an object oscillates back and forth around an equilibrium point. Acceleration is a measure of the rate at which an object's velocity changes, and in SHM, the object's acceleration is directly related to its displacement from the equilibrium point.

2. What is the general formula for acceleration in SHM?

The general formula for acceleration in SHM is a = -ω^2x, where a is the acceleration, ω is the angular frequency, and x is the displacement from the equilibrium point. This formula applies to all types of SHM, including simple pendulums, mass-spring systems, and more.

3. How does the amplitude affect the acceleration in SHM?

The amplitude, or maximum displacement, in SHM does not affect the acceleration. This is because the acceleration is directly proportional to the displacement, so as the amplitude increases, the acceleration also increases in proportion.

4. What is the relationship between acceleration and frequency in SHM?

The frequency of SHM is the number of oscillations per unit time. The acceleration in SHM is inversely proportional to the frequency, meaning that as the frequency increases, the acceleration decreases, and vice versa.

5. Can the formula for acceleration in SHM be applied to all types of motion?

No, the formula for acceleration in SHM can only be applied to motion that follows a sinusoidal pattern, where the acceleration is directly proportional to the displacement from the equilibrium point. This type of motion is also known as simple harmonic motion.

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