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Chapter 2: Problem 321

If \(\mathrm{f}\) is the frequency of a body moving in a circular path with constant speed. a is its centrifugal acceleration, so. (A) \(\mathrm{a} \propto \mathrm{f}\) (B) a \(\propto \mathrm{f}^{2}\) (C) \(\mathrm{a} \propto \mathrm{f}^{3}\) (D) a \(\propto(1 / \mathrm{f})\)

Short Answer

Expert verified
The centrifugal acceleration (a) of an object moving in a circular path with constant speed is given by the formula \(a = R \omega^2\). The angular velocity (ω) can be expressed in terms of frequency (f) as \(\omega = 2\pi f\). Substituting the expression for angular velocity into the formula for centrifugal acceleration, we get \(a = R(4\pi^2 f^2)\), which implies \(a \propto f^2\). Therefore, the correct answer is (B) \(a \propto f^2\).

Step by step solution

01

Write down the formula for centrifugal acceleration

The centrifugal acceleration (a) of an object moving in a circular path with constant speed can be expressed as \(a = R \omega^2\), where R is the radius of the circular path, and ω (omega) is the angular velocity of the object (measured in radians per second).
02

Express the angular velocity in terms of frequency

Now we need to express the angular velocity (ω) in terms of frequency (f). The relationship between the two can be described as follows: \(\omega = 2\pi f\) where f is the frequency in Hz (cycles per second).
03

Substitute the expression for angular velocity into the formula for centrifugal acceleration

Next, we will substitute the expression for angular velocity from step 2 into the formula for centrifugal acceleration from step 1: \(a = R (2\pi f)^2\)
04

Simplify the equation and find the relationship between a and f

Now we will simplify the equation from step 3, so we can determine the relationship between a and f: \(a = R(4\pi^2 f^2)\) So, we have: \(a \propto f^2\)
05

Compare the relationship to the given options and select the correct answer

Now let's compare the relationship we found in step 4 to the given options (A), (B), (C), and (D): (A) \(a \propto f\) - Incorrect (B) \(a \propto f^2\) - Correct (C) \(a \propto f^3\) - Incorrect (D) \(a \propto (1 / f)\) - Incorrect The correct answer is (B) \(a \propto f^2\), meaning that the centrifugal acceleration is proportional to the square of the frequency of a body moving in a circular path with constant speed.

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