Mass — Question 8

Question 8

Density and volume can predict an inertial mass ratio

Two uniform cubes A and B have the same density. Their side lengths are 2 cm and 3 cm. A has mass 0.40 kg. Use volume proportional to the cube of side length.

1. Find B’s mass.

2. Find their accelerations under separate equal 2.7 N net forces.

3. Why is the mass ratio not the side-length ratio?

Question 8 - Solution

Density and volume can predict an inertial mass ratio

(a) Scale the volume.

$$\frac{m_B}{m_A}=\frac{\rho L_B^3}{\rho L_A^3}=\left(\frac32\right)^3=\frac{27}{8},\quad\boxed{m_B=1.35\,\mathrm{kg}}$$ $m_B/m_A=(3/2)^3=27/8$. Thus $m_B=0.40(27/8)=1.35$ kg.

(b) Use inverse mass response.

$$a_A=\frac{2.7}{0.40}=6.75,\quad a_B=\frac{2.7}{1.35}=2.00\quad\mathrm{m/s^2}$$ $a_A=2.7/0.40=6.75$ m/s$^2$ and $a_B=2.7/1.35=2.00$ m/s$^2$.

(c) Count three dimensions.

$$V=L_xL_yL_z,\quad\frac{V_B}{V_A}=\frac32\frac32\frac32\ne\frac32$$ At equal density each of the three linear dimensions changes by $3/2$. Their product controls volume and mass. A factor of $3/2$ alone would apply to a length, not a three-dimensional material amount.