Inertia mass and density

Speed and average speed describe how position changes over time, but they do not explain why the same push changes the speed of one body more than another. The inertial mass $m$ of a body measures its resistance to a change in velocity: under the same net force $F$, a larger mass has a smaller acceleration, $a=F/m$. Density $\rho$ connects inertial mass to volume for a uniform substance: $\rho=m/V$, so $m=\rho V$ and $V=m/\rho$. Thus mass and density are different properties: two bodies can have the same mass but different densities if their volumes differ, and the same volume but different masses if their densities differ. In calculations, keep units consistent; in SI, $m$ is in kilograms, $V$ in cubic metres, $\rho$ in kilograms per cubic metre, $F$ in newtons, and $a$ in metres per second squared. For constant net force, a denser body of the same volume has more inertial mass and therefore changes its speed more slowly.

A cart has inertial mass $m=2.0\,\mathrm{kg}$. A net force $F=6.0\,\mathrm{N}$ acts on it along the direction of motion. Calculate its acceleration and state the formula used.

A solid cube has volume $V=1.0\times10^{-3}\,\mathrm{m^3}$ and density $\rho=2500\,\mathrm{kg/m^3}$. A net force $F=5.0\,\mathrm{N}$ acts on it. Calculate its inertial mass and acceleration.

Two spheres have the same volume $V=5.0\times10^{-4}\,\mathrm{m^3}$. One is aluminium with $\rho_A=2700\,\mathrm{kg/m^3}$; the other is plastic with $\rho_P=900\,\mathrm{kg/m^3}$. Both move at initial speed $v_0=2.0\,\mathrm{m/s}$ and are slowed by the same constant resistive force $F=0.90\,\mathrm{N}$. Assuming only this force acts, determine which sphere travels farther before stopping and calculate the stopping distance for each. Use $d=v_0^2/(2a)$.

Use a safe interactive simulation such as PhET Forces and Motion: Basics or Density.

1. In the force-and-motion mode, set a constant applied force, record the acceleration for carts of mass $1\,\mathrm{kg}$, $2\,\mathrm{kg}$, and $4\,\mathrm{kg}$; plot or tabulate acceleration versus mass.
2. In the density mode, choose a material and a fixed volume, read the mass, and verify $m=\rho V$.
3. Repeat for two materials with the same volume but different densities; predict which body will stop later under the same resistive force, then test the prediction in the motion mode.
4. Write a table of $m$, $V$, $\rho$, $F$, and $a$, and check that all units are SI.

To show that inertial mass controls acceleration under a given force, while density connects mass to volume for a material.

A two-panel diagram. Left: two identical carts, one with mass $m$ and one with mass $2m$, each pulled by the same force $F$; acceleration arrows show $a=F/m$ and $a/2$. Right: two cubes of equal volume $V$, one with low density $\rho_1$ and one with high density $\rho_2>\rho_1$; labels show $m_1=\rho_1 V$ and $m_2=\rho_2 V$, and the cubes have the same size but different mass markers.

A football has mass $m_1=0.43\,\mathrm{kg}$ and flies at average speed $v_1=25\,\mathrm{m/s}$. A shot put used in athletics has mass $m_2=7.26\,\mathrm{kg}$ and leaves the hand at $v_2=13\,\mathrm{m/s}$. A goalkeeper catching the football feels a small change of motion; a field judge stopping the shot put must exert a much larger force over a longer time. During uniform slowing over $1.0\,\mathrm{s}$, the football's average speed changes by $25\,\mathrm{m/s}$, while the shot's changes by $13\,\mathrm{m/s}$.

The scenario exercises inertial mass: for the same kind of velocity change, the $7.26\,\mathrm{kg}$ shot resists a change in its state of motion more than the $0.43\,\mathrm{kg}$ football. The average speed relation $v_{\mathrm{avg}}=\Delta s/\Delta t$ tells how quickly each body moves before the catch, so the comparison is about mass, not just speed.

An empty truck has mass $12000\,\mathrm{kg}$ and takes $20\,\mathrm{m^3}$ of sand with density $1600\,\mathrm{kg/m^3}$. The sand mass is $m=\rho V=1600\cdot20=32000\,\mathrm{kg}$, so the total mass of the loaded truck is $12000+32000=44000\,\mathrm{kg}$. Both the empty and the loaded truck travel at $v=15\,\mathrm{m/s}$. During a uniform stop in $5.0\,\mathrm{s}$, the loaded truck requires a much larger braking force; the empty truck changes speed more easily.

Here density $\rho=m/V$ converts the sand volume into mass, and mass acts as the measure of inertia: the loaded truck of mass $44000\,\mathrm{kg}$ resists a change in speed more than the empty one of mass $12000\,\mathrm{kg}$. Average speed $v_{\mathrm{avg}}=s/t$ gives the same initial motion, so the difference during braking is set by mass.

A bottle of volume $1.0\,\mathrm{L}=1.0\times10^{-3}\,\mathrm{m^3}$ is filled with water of density $1000\,\mathrm{kg/m^3}$; the water mass is $1000\cdot1.0\times10^{-3}=1.0\,\mathrm{kg}$. The same bottle is filled with sunflower oil of density $920\,\mathrm{kg/m^3}$; the oil mass is $920\cdot1.0\times10^{-3}=0.92\,\mathrm{kg}$. If both bottles are pushed across a table at average speed $0.50\,\mathrm{m/s}$ and then stopped, the water bottle resists the change in motion slightly more.

The scenario links density and inertial mass: for the same volume $1.0\,\mathrm{L}$, density $\rho=m/V$ gives different masses $1.0\,\mathrm{kg}$ and $0.92\,\mathrm{kg}$. The average speed before stopping is the same, so inertial mass determines which bottle is harder to change in speed.