Proving Frequency is Independent of Cord Length in Circular Motion

[devanlevin]

i hope the question is understood,

the question is to prove that two masses hung from 2 strings, from the same point, at the same height from the ground, but on different length cords, can move with the same frequency, despite different lenghts

what i think needs to be done, is prove that the frequency has nothing to do with the radius of the motion but with the velocity


using equation for radial/centrepheutal force
Fr=mar=sinB*t=m(ῳ^2)R
t being tension
R-radius--->tanB*h(height)

sinB*t=m(ῳ^2)*tanB*h
cosB*t=m(ῳ^2)*h

ῳ=sqrt[mh/cosB*t]

now I am not sure what to do, i have proven nothing since B is directly affectd by the lenghth of the cord, am i correct?

 

[anathema]

Hello there,

Thank you for your post and question. I believe I can provide some clarification and additional information to help you prove that two masses hung from two strings, at the same height but different lengths, can move with the same frequency.

First, let's define some terms and concepts. Frequency is a measure of how often something occurs in a given time period. In the context of this scenario, it refers to the number of oscillations or swings per unit of time. The frequency of an object in uniform circular motion is determined by the speed of the object and the radius of its circular path. This means that the frequency is not affected by the mass of the object, as long as the speed and radius remain constant.

Now, let's look at the equation you provided for radial or centripetal force: Fr=mar=sinB*t=m(ῳ^2)R. This equation shows that the force required to keep an object in circular motion is directly proportional to the mass of the object (m), the angular velocity (ῳ), and the radius of the circular path (R). However, it is important to note that the angular velocity is directly related to the frequency (f) through the equation ῳ=2πf. This means that as the frequency increases, so does the angular velocity.

Next, let's consider the effect of the length of the cords on the frequency. As you correctly pointed out, the length of the cord affects the angle (B) at which the mass hangs. However, this angle does not affect the frequency of the motion. This is because the angle only affects the radius of the circular path, not the speed of the object. The speed of the object is determined by the tension in the string, which is the same for both masses in this scenario.

Therefore, we can conclude that the frequency of the motion is not affected by the length of the cords, as long as the speed and radius remain constant. This means that the two masses, hung from two strings of different lengths but at the same height, can indeed move with the same frequency.

I hope this explanation helps you in your research and understanding of this topic. If you have any further questions or need clarification, please let me know. Best of luck with your studies!

 

[baba89]

First of all, it is important to clarify what is meant by frequency in this context. In circular motion, frequency refers to the number of rotations per unit time, which is represented by the symbol ω (omega). This is different from the frequency of a wave, which refers to the number of oscillations per unit time.

Now, let's consider the scenario described in the question. We have two masses, hung from two different length cords, at the same height from the ground and rotating around the same point. We want to prove that despite the different lengths of the cords, the masses will have the same frequency of rotation.

To do this, we can use the equation for the frequency of circular motion, which is ω = v/r, where v is the tangential velocity and r is the radius of the circular motion. Since the masses are hanging from the same point, the radius of their circular motion will be the same. Therefore, the only factor that will affect the frequency of rotation is the tangential velocity.

Now, let's look at the equation provided in the question: ω = √(mh/cosBt). In this equation, m represents the mass of the object, h represents the height of the object from the ground, and cosBt represents the tension in the string. Notice that the radius, which is the only factor that affects the frequency, is not present in this equation.

This means that the frequency of rotation is not dependent on the length of the cord, but on the tangential velocity, which is determined by the tension in the string and the angle at which the string is hanging. As long as the tension and angle remain the same, the tangential velocity and therefore the frequency will remain the same, regardless of the length of the cord.

In conclusion, we can say that the frequency of circular motion is independent of the length of the cord, and is instead determined by the tangential velocity. This is because the radius, which is the only factor that affects the frequency, is not present in the equation for frequency. Therefore, the two masses in the scenario described will have the same frequency of rotation, despite being on different length cords.