Can Calculus Validate the Proof of w=(k/m)^(1/2) in SHM Equations?

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In summary, the conversation discusses the use of calculus to prove simple harmonic motion equations and the difficulties in understanding the proof for w=(k/m)^(1/2). The speaker presents their approach of using the x(t)=A*cos(wt+θ) equation and taking the second derivative to find acceleration. They make assumptions and solve for the final equation, but also question the validity of their method and the original x(t) equation. The conversation concludes with a question about the use of calculus on the AP Physics C: Mechanics exam.
  • #1
brendan3eb
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So I am attempting to prove the simple harmonic motion equations with calculus so as to develop a better understanding of shm problems and have more flexibility when it comes to solving them. I am having a bit of trouble understanding the proof of w=(k/m)^(1/2)

Here is how I am doing it. I am taking the x(t)=A*cos(wt+θ) equation and then the second derivative of it to find acceleration

a(t)=-A*w^2*cos(wt+θ)
F=ma=-kx
a=-kx/m
-A*w^2*cos(wt+θ)=-kx/m
so I am guessing that I can make a few assumptions to solve for the final equation. If I assume that the time t=0, thus we x=A, I can solve
-x*w^2*cos(w(0)+0)=-kx/m
x's cancel out
w^2*cos(0)=k/m
w=(k/m)^(1/2)

Is this a valid way to prove the equation? Also, I am assuming that the original x(t) equation is correct, I do not know how to prove it. Is there a way to prove the original x(T) equation? Finally, I am preparing for the AP Physics C: Mechanics exam, so if I were to solve a problem using Calculus, for those of you familiar with the exam, would the graders allow me to start off a proof of an answer with x(t)=A*cos(wt+θ) as a given?
 
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  • #2
Yeah you should be fine, BUT I would recommend recalling that :

x[t] = A Cos wt + B Sin wt

theres two parts depending on your initial conditions.
 
  • #3


First of all, congratulations on taking the initiative to use calculus to better understand simple harmonic motion (SHM) problems. This will definitely give you a deeper understanding and more flexibility when solving SHM problems.

Now, let's take a closer look at your proof for w=(k/m)^(1/2). Your approach is correct, and your assumptions are also valid. However, I would suggest being a bit more careful with your notation. In your first equation, you use lowercase t for time, but in the second equation you use uppercase T. It is important to use consistent notation throughout your proof.

To prove the original x(t)=A*cos(wt+θ) equation, you can use the fact that SHM is a type of periodic motion, meaning that the motion repeats itself after a certain period of time. This can be proven mathematically using the concept of derivatives and integrals, but it may be beyond the scope of the AP Physics C: Mechanics exam.

As for whether the graders would allow you to start off a proof with x(t)=A*cos(wt+θ) as a given, it ultimately depends on the specific problem and the grader's discretion. However, it is always best to show your work and explain your reasoning, rather than just stating a given equation without any explanation.

Overall, your approach to proving w=(k/m)^(1/2) is valid and shows a good understanding of SHM. Keep up the good work and best of luck on your exam!
 

Related to Can Calculus Validate the Proof of w=(k/m)^(1/2) in SHM Equations?

1. What is the meaning of w=(k/m)^(1/2)?

Proof of w=(k/m)^(1/2) is a formula that relates the angular frequency (w) of a system to its spring constant (k) and mass (m). It is commonly used in physics and engineering to calculate the frequency of oscillation of a spring-mass system.

2. How is w=(k/m)^(1/2) derived?

This formula is derived from Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement. By applying Newton's Second Law of Motion and the definition of angular frequency, we can arrive at the equation w=(k/m)^(1/2).

3. Can this formula be applied to all spring-mass systems?

Yes, this formula is applicable to all spring-mass systems as long as the spring is considered to be ideal (no damping or external forces acting on it).

4. What are the units of measurement for the variables in this formula?

The units for angular frequency (w) are radians per second (rad/s), while the units for spring constant (k) are newtons per meter (N/m) and mass (m) is measured in kilograms (kg).

5. Can this formula be used for non-linear spring-mass systems?

No, this formula is only valid for linear spring-mass systems, where the force exerted by the spring is directly proportional to its displacement. Non-linear systems have more complex equations to describe their behavior.

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