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Chapter 1: Problem 39

An expression for buoyant force is \(F_{\mathrm{B}}=\rho g V,\) where \(F_{\mathrm{B}}\) has dimensions \(\left[\mathrm{MLT}^{-2}\right], \rho\) (density) has dimensions \(\left[\mathrm{ML}^{-3}\right],\) and \(g\) (gravitational field strength) has dimensions \(\left[\mathrm{LT}^{-2}\right]\). (a) What must be the dimensions of \(V ?\) (b) Which could be the correct interpretation of \(V:\) velocity or volume?

Short Answer

Expert verified
Answer: The dimensions of V are \(\left[\mathrm{L^3}\right]\), and it represents volume.

Step by step solution

01

Write down the expression for buoyant force

The given expression for buoyant force is \(F_{\mathrm{B}}=\rho g V\).
02

Write down the dimensions of the given variables

It is given that: - Dimensions of \(F_{\mathrm{B}}\), the buoyant force: \(\left[\mathrm{MLT}^{-2}\right]\) - Dimensions of \(\rho\), the density: \(\left[\mathrm{ML}^{-3}\right]\) - Dimensions of \(g\), the gravitational field strength: \(\left[\mathrm{LT}^{-2}\right]\)
03

Equate the dimensions of \(F_{\mathrm{B}}\) and the product of dimensions of \(\rho\), \(g\) and \(V\) in the given equation

According to the equation \(F_{\mathrm{B}}=\rho g V\), the dimensions of the left-side and right-side should be the same. So, \(\left[\mathrm{MLT}^{-2}\right] = \left[\mathrm{ML}^{-3}\right] \times \left[\mathrm{LT}^{-2}\right] \times [V]\)
04

Solve for dimensions of \(V\)

Rearranging the equation from Step 3 to solve for \(V\), we get \([V] = \frac{\left[\mathrm{MLT}^{-2}\right]}{\left[\mathrm{ML}^{-3}\right] \times \left[\mathrm{LT}^{-2}\right]} = \frac{\left[\mathrm{MLT}^{-2}\right]}{\left[\mathrm{M^{-1}L^{-1}T^{-4}}\right]}\) By using the laws of exponents, we obtain: \([V] = \left[\mathrm{L^3}\right]\)
05

Identify the interpretation of \(V\) based on its dimensions

Since we found the dimensions of \(V\) to be \(\left[\mathrm{L^3}\right]\), we can now deduce that \(V\) cannot be velocity (as velocity's dimensions are \(\left[\mathrm{LT}^{-1}\right]\)). Instead, \(V\) refers to volume (as volume's dimensions are \(\left[\mathrm{L^3}\right]\)). (a) The dimensions of \(V\) are \(\left[\mathrm{L^3}\right]\). (b) The correct interpretation of \(V\) is volume.

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