Chapter 1, Problem 1Q

Suppose a friend told you that the density of a sphere is $\frac{4}{3}\pi r^4$, where r is the radius of the sphere. Compute the dimensions of this expression and show that it cannot be correct.

To determine

The dimension of the expression and evaluate.

Answer to Problem 1Q

The expression $\left(4/3\right)\pi r^4$ cannot be the equation for density.

Explanation of Solution

In the expression $\left(4/3\right)\pi r^4$, only radius, $r$ has dimension and all others are dimensionless quantity. The radius is a dimension of length.  Hence, the expression $\left(4/3\right)\pi r^4$ will have a dimension $L^4$.

Write the equation for density of the sphere.

    $density=\frac{mass}{volume}$        (I)

Hence, from equation (I), the dimension of density is $M/L^3$.

 Conclusion:

Since both the expression has different dimensions, $\left(4/3\right)\pi r^4$ cannot be the equation for density.