How do I explain WHY centripetal force is directly proportional to frequency squared...not relying on information based on the graph of Fc vs F squared solely
thanks
Graph of centripetal force against frequency Squared at constant radius
To explain why centripetal force is directly proportional to frequency squared, we can use some fundamental principles and equations related to circular motion. Here is a step-by-step explanation:
1. Start with the equation for centripetal force: Fc = m * (v^2 / r), where Fc is the centripetal force, m is the mass of the object in circular motion, v is the linear velocity of the object, and r is the radius of the circular path.
2. Recognize that the frequency (f) of an object in circular motion is related to its linear velocity (v) and the circumference of the circular path (C) through the following equation: f = v / C.
3. Substitute the expression for v in the centripetal force equation using the relationship from step 2. This gives: Fc = m * ([(f * C)^2] / r).
4. Simplify the equation by canceling out the common terms. Now we have: Fc = (m * C^2) / r * f^2.
5. As we can see, the centripetal force (Fc) is directly proportional to the square of the frequency (f^2). This means that if the frequency doubles, the centripetal force will increase by a factor of four (2^2). Similarly, if the frequency is tripled, the centripetal force will increase by a factor of nine (3^2), and so on.
By using these steps and the equations for circular motion, we can explain why the centripetal force is directly proportional to the frequency squared without solely relying on the graph of Fc vs. F squared.
To explain why centripetal force (Fc) is directly proportional to the square of frequency (f), we can start by understanding the formula for centripetal force.
Centripetal force is given by the equation Fc = m * (v^2 / r), where m is the mass of the object in circular motion, v is the linear velocity of the object, and r is the radius of the circular path.
Frequency is defined as the number of revolutions or cycles made by an object in one unit of time and is given by the equation f = 1 / T, where T is the period (time taken for one complete revolution).
Now, to establish the relationship between Fc and f, we need to make some assumptions:
1. The mass (m) and the radius (r) of the object are constant.
2. The linear velocity (v) remains constant as the frequency (f) changes.
With these assumptions in mind, let's analyze the relationship between Fc and f:
1. The linear velocity (v) is given by the formula v = 2πr / T. As we assume that the radius remains constant, we can rewrite this equation as v = 2πr * f.
2. Substituting the value of v into the centripetal force formula, we get Fc = m * [(2πr * f)^2 / r] = 4π^2 * m * r * f^2.
From this equation, we can clearly see that Fc is directly proportional to f^2. The constant of proportionality is 4π^2 * m * r.
Therefore, regardless of the graphical representation, when the frequency (f) of an object in circular motion increases, the centripetal force (Fc) acting on the object also increases, and it follows a squared relationship.
centripetal force is related to velocity squared.
but....
velocity= distancearound*frequency.