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Chapter 1: Problem 160

If the centripetal force is of the form $\mathrm{m}^{\mathrm{a}} \mathrm{v} \mathrm{b}_{\mathrm{r}}^{\mathrm{c}}$ find the values of \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) (a) \(1,2,1\) (b) \(1,2,-1\) \(\begin{array}{ll}\text { (c) } 1,3,-2 & \text { (d) }-1,3,-1\end{array}\)

Short Answer

Expert verified
The correct values for a, b, and c in the centripetal force equation \(F = m^av^br^c\) are a = 1, b = 2, and c = -1, which corresponds to answer choice (b) 1, 2, -1.

Step by step solution

01

Recall the centripetal force formula

The formula for centripetal force is given by \(F = m\frac{v^2}{r}\), where m is the mass of the object, v is its tangential velocity, and r is the radius of the circular path.
02

Compare the given equation to the centripetal force formula

We have the given equation as \(F = m^av^br^c\). Comparing this to the standard formula \(F = m\frac{v^2}{r}\), we can analyze the exponents of the variables m, v, and r.
03

Determine the values of a, b, and c

By comparing the two equations, we can see that: 1. The exponent of m in the given equation is a, which should be equal to 1 (as there is no exponent in the standard formula, and thus it is assumed to be 1). Therefore, a = 1. 2. The exponent of v in the given equation is b, which should be equal to 2 (based on the standard formula). Therefore, b = 2. 3. The exponent of r in the given equation is c, which should be equal to -1 (as r is in the denominator in the standard formula). Therefore, c = -1.
04

Choose the correct answer

With the determined values of a, b, and c, we can see that the correct answer is: (a) 1, 2, 1: Incorrect, since c is not -1. (b) 1, 2, -1: Correct, as this option matches our determined values for a, b, and c. (c) 1, 3, -2: Incorrect, since b is not 3 and c is not -2. (d) -1, 3, -1: Incorrect, since a is not -1, b is not 3, and c is not -1. Therefore, the correct answer is (b) 1, 2, -1.

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