# What are the ways in which the mass of a body can be determined without using a laboratory balance?

Dec 9, 2014

There are many equations that can be used to calculate the mass of a body. You can use the equation for density , the equation from Newton's second law of motion, the equation for kinetic energy , and the equation for specific heat capacity . Which equation you use depends on what data you have.

DENSITY EQUATION
The density equation is $\text{density}$ = $\text{mass"/"volume}$.

You can use the density equation to calculate mass if you know the density and volume of a substance. The way to solve the density equation for mass is to multiply density x volume:

$\text{mass}$ = $\text{density x volume}$.

Example.

Q. What is the mass of a piece of aluminum that has a density of ${\text{2.70 g/cm}}^{3}$and a volume of ${\text{8.0 cm}}^{3}$?

A. $\text{mass}$ = $\text{density x volume}$ = $\text{2.70g/cm"^3}$ x ${\text{8.0cm}}^{3}$= $\text{21.6 g}$

NEWTON'S SECOND LAW:

The equation for Newton's 2nd law is usually written as $\text{force = mass x acceleration}$, or $\text{F}$ = $\text{ma}$.

You can use Newton's 2nd law of motion if you know the force (in Newtons) and the acceleration . To solve for mass, divide force by acceleration:

$\text{mass}$ = $\text{force"/"acceleration}$.

Example 1. Determining the mass of a dog from its weight in Newtons. Weight is the force in Newtons, that is caused by the gravity of the earth acting on a body's mass. The acceleration due to gravity is ${\text{9.8m/s}}^{2}$.

Q. What is the mass of a dog that weighs 323 N?

A. $\text{mass}$ = $\text{force"/"acceleration}$ = ${\text{323 N"/"9.8 m/s}}^{2}$ = $\text{33 kg}$

Example 2.

Q. What is the mass of a truck if it produces a force of 14,000 N while accelerating at a rate of ${\text{5.0 m/s}}^{2}$?

A. $\text{mass}$ = $\text{force"/"acceleration}$= ${\text{14000 N"/"5.0 m/s}}^{2}$ = $\text{2800 kg}$

Note: Because $\text{1 N}$ = $\text{1 kg m/s"^2}$, the unit for mass will always be in kilograms.

KINETIC ENERGY

The kinetic energy (KE) equation is:

$\text{KE}$ = $\text{1/2 mv"^2}$, where KE is in Joules, m is mass in kg, and $\text{v"^2}$ is speed squared. $\text{1 Joule}$ = ${\text{1 kg m"^2"/s}}^{2}$.

If you know KE and speed, you can use the KE equation and solve for m.

$\text{m}$ = $\text{2KE"/"v"^2}$

Example.

Q. What is the mass of a baseball that has a kinetic energy of 105 J and is traveling at 10 m/s?http://www.biologycorner.com/physics/energy/problemset_kinetic_potential_energy.html

A. $\text{m}$ = $\text{2KE"/"v"^2}$ = $\text{(2)(105J)}$$\div$${\text{100m"^(2)"/s}}^{2}$ = $\text{2.1 kg}$

SPECIFIC HEAT

The equation for specific heat is $\text{Qm}$$\Delta$T, where Q is heat energy in Joules or calories gained or lost, c is specific heat capacity, and $\Delta$T is temperature change. If you know Q, c, and $\Delta$T, then you can find mass:

$\text{m}$ = $\text{Q}$$/$$\text{(c)}$($\Delta$T)

Example.

Q. The water in a swimming pool gives up $\text{1.09 × 10"^10 "J}$ of energy to the cool night air. If the temperature of the water, which has a specific heat of $\text{4186 J/kg K}$, decreases by $\text{5.0 K}$. What is the mass of the water in the pool?http://go.hrw.com/resources/go_sc/ssp/HK1MSW78.PDF

A. $\text{m}$ = $\text{Q}$$/$$\text{(c)}$($\Delta$T) = $\text{1.09 × 10"^10 "J}$$/$$\text{(4186 J/kg K)(5.0K)}$ = $\text{5.2 x 10"^5 "kg}$

SUMMARY

If you know density and volume , you would use the density equation to find mass.

If you know force in Newtons and acceleration , you would use the equation from Newton's second law to determine mass.

If you know KE and speed , you can use the equation for KE to calculate mass.

If you know heat energy gained or lost, specific heat capacity , and temperature change , you can use the specific heat equation to calculate mass.

There are probably more ways to calculate mass depending on what data you have. However, at the moment, these are the ones I can think of.