Mass

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

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Mass is the amount of matter an object has. It is the property of a body that causes it to have weight in a gravitational field. It is expressed as "lbm" in the English Set of units and grams in the SI system of units.   It is sometimes used interchangeably in place of weight. Weight, is a vector quantity that depends on the gravity at a specific location. Mass on Earth is the same as mass on the moon. However, the weight on the moon is much less than the weight on the Earth.

Mass is a scalar quantity having direction, some of these include area, density, energy, entropy, length, power, pressure, speed, temperature, volume, and work.

Types of Mass

  • Gravitational mass
  • Inertial mass
  • Invariant mass
  • Relativity mass
  • Rest mass

Mass formula

\(m = \rho   V \)          \( mass   \;=\;  density \;\;x\;\;  volume\)

\(m = \frac {F} {a}\)

\(m = \frac {p} {v}\)

\(m = \frac {g r^2} {G} \)

Where:

\(m\) = mass

\(a\) = acceleration

\(F\) = force

\(g\) = gravitational acceleration

\(G\) = universal gravitational constant

\(p\) = momentum

\(\rho \) (Greek symbol rho) = density

\(r\) = radius from the planet center

\(v\) = velocity

\(V\) = volume

Center of Masscenter of mass 1

Center of mass or center of gravity is a specific point in a system in reference to a mass. The center of mass of a rigid object is a fixed point in relationship to the mass.

Center of Mass formula

A system at a specific time, on the same axis, having two masses.

\(x_{cm} = \frac {m_1 x_1 \;+\; m_2 x_2 } {m_1 \;+\; m_2} \)          \(center \; of \; mass  \;=\;  \frac {mass_1 \;\;x\;\;  position_1 \;+\; mass_2 \;\;x\;\;  position_2 } {mass_1 \;+\; mass_2} \)

Where:

\(x_{cm}\) = center of mass

\(m_1\) = mass 1

\(m_2\) = mass 2

\(x_1 \) = position 1

\(x_2\) = position 2

Center of Mass Multiple Masses formula

A system at a specific time, on the same axis, having multiple masses.

\(x_{cm} = \frac {m_1 x_1 \;+\; m_2 x_2 \;+\; \;\cdots } { m_1 \;+\; m_2 \;+\; \;\cdots } \)          \( center \; of \; mass  \;=\;  \frac {mass_1 \;\;x\;\;  position_1 \;+\; mass_2 \;\;x\;\; position_2 \;+\; \;\cdots } { mass_1 \;+\; mass_2 \;+\; \;\cdots } \)

Where:

\(x_{cm}\) = center of mass

\(m_1\) = mass 1

\(m_2\) = mass 2

\(x_1 \) = position 1

\(x_2\) = position 2

Gravitational mass

Gravitational mass is measured by comparing the force of gravity of an unknown mass to the force of gravity of a known mass.  Acceleration of the gravitational mass will always be the same on each object no matter where you are on the planet.

Internal Mass

Internal mass is the mass of an object measured by its resistance to acceleration when a force is applied.  A sphere of the same density has an internal mass to the center.  A sphere of the non-uniform density has a center of mass toward the more dencer of the mass.  A flat mass will have a lower center of gravity with less potential energy.

Mass Diffusivity

Mass diffusivity ( \(D_m\) ) is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species.

Mass Flow Rate

Mass flow rate is the average velocity of an areas density for flow.

Mass Flow Rate formula

\(\dot m_f = \rho v A   \)          \( mass \; flow \; rate  \;=\;  fluid \; density \;\;x\;\;   velocity  \;\;x\;\;   area   \)

Where:

\(\dot m_f\) = mass flow rate

\(A\) = area

\(d_i\) = internal pipe diameter

\(Q\) = volumetric flow rate

\(\pi\) = Pi

\(\rho\) (Greek symbol rho) = fluid density

\(v\) = velocity

Solve for:

\(d_i = \sqrt { \frac { 4 Q }   { \pi v}   }   \)

\(d_i = \sqrt { \frac { 4 \dot m_f}   { \pi \rho v}   }   \)

Relativistic Mass

Relativistic mass refers to the mass of a body which changes with the speed of the body as this speeds approaches close to the speed of light.

Relativistic Mass formula

\(m_r =     \frac { m}  {  \sqrt { \frac {1 - v^2} {c^2} } } \)          \( relativistic \; mass   =     \frac { mass \; at \; rest }  {  \sqrt { \frac {1 - velocity^2} { speed \; of \; light^2} } } \)

Where:

\(m_r\) = relativistic mass

\(c\) = speed of light

\(m\) = mass at rest

\(v\) = velocity

Rest Mass

Rest mass of a body is measured when the body is at rest and motionless, and is also relative to an observer moving or not moving.


Tags: Equations for Mass