Chapter 14, Problem 1CQ

To determine

Whether the object’s density change and by what factor.

Answer to Problem 1CQ

Solution: (a) The density $\rho$remains constant. (b) The density decreases by a factor of 8.

Explanation of Solution

Given:

An object has density $\rho$ . (a) Each of the objects 3 dimensions is increased by a factor of 2 without changing the material of which the object is made. (b) The dimensions are increased by a factor of 2 without changing the object’s mass.

Formula used:

The mass of an object with uniform density $\rho$ is given by $m=\rho V$ , where $V-$ volume of the object.

(a) When the volume increases, the mass also proportionally increases. Since the object’s material does not change, the density $\rho$ remains constant. (b) Since, the linear dimensions change by a factor of 2, the density decreases by a factor of 8.

Calculation:

The mass remains constant. Hence, $m=\rho V=\rho \text{'}V\text{'}$ . Now, since the linear dimensions change by a factor of 2, new volume $V\text{'}$ is $2\times 2\times 2V=8V$ .

Substituting,

 $\begin{array}{l}m=\rho V=\rho \text{'}\left(8V\right)\\ \rho \text{'}=\frac{\rho }{8}\end{array}$

Conclusion:

Since the object’s material does not change, the density $\rho$ remains constant. Since, the linear dimensions change by a factor of 2, the density decreases by a factor of 8.