Mass And Density Of Earth

What is the Mass of the Earth?

Using Newton’s law of gravitation we can estimate the mass of the earth
$\begin{array}{rl}& M=\frac{g{R}^2}{G}\simeq {10}^{25}\mathrm{kg}\\ & M\to \text{mass of earth}\end{array}$

$G\to$ Gravitational constant
What is the Density of Earth?

$\text{As}g=\frac{4}{3}\pi \rho GR$
Where $\rho$ is the density of the earth

$\begin{array}{rl}& {}_{\mathrm{So}}\rho =\frac{3g}{4\pi GR}\\ & \rho =5478.4\mathrm{kg}/{\mathrm{m}}^3\end{array}$
What is Inertial Mass?

Also known as the mass of material of the body which measures its inertia.

From Newton’s second law of motion

$\begin{array}{rl}& F=m_{i}a\\ & {\text{So}}_i=\frac{F}{a}\\ & \text{Where}\\ & m_i\to \text{inertial mass}\\ & F\to \text{external force}\\ & a\to ac{c}^n\end{array}$

1. Gravity has no effect on inertial mass.

2. Inertial mass is independent of the size, shape, and state of the body.

What is Gravitational Mass?

It is mass which determines the gravitational pull acting upon it.

Let $F=$ gravitational pull on a body of mass applying Newton's law of gravitation

We have

$F=\frac{GM{m}_g}{R^2}$
So we get

$m_g=\frac{F}{GM/R^2}=\frac{F}{I}$
Where $m_g=$ Gravitational mass
$I\to$ Gravitational field intensity

Tip-Spring balance measures gravitational mass.

What is Mass (m)?

1. It is the quantity of matter contained in the body.
2. Its SI unit- Kg
3. Its dimension is $[M]$
4. It is a scalar quantity.
5. It Can never be zero
6. Its value does not change with g.

What is Weight (W)?

1. It is an Attractive force exerted by the earth on anybody.
2. S.I. Unit: Newton or Kg - wt
3. Dimension- $\left[ML{T}^{-2}\right]$
4. It is a vector quantity
5. It changes its value according to the value of $g$
6. At $\mathrm{\infty }$ and at the centre of the earth $\mathrm{g}=0$, So W is equal to zero there.

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Solved Examples Based on Mass and Density of Earth

Example 1: Average density of the earth

1) is directly proportional to g

2) is inversely proportional to g

3) does not depend on g

4) is a complex function of g

Solution:

Density of Earth

$\rho =\frac{3g}{4\pi GR}$

$\rho \to$ density of earth

$g\to 9.8\mathrm{m}/{\mathrm{s}}^2$

$G\to$ Gravitational constant
wherein

$\begin{array}{rl}& \rho =5478.4\mathrm{kg}/{\mathrm{m}}^3\\ & \text{As}g=\frac{GM}{R^2}\Rightarrow M=\frac{g{R}^2}{G}\\ & \rho =\frac{\text{mass}}{\text{volume}}=\frac{M}{\frac{4}{3}\pi R^3}\\ & \text{density}\\ & \rho =\frac{g{R}^2/G}{\frac{4}{3}\pi R^3}\\ & \rho =\frac{3g}{4\pi GR}\Rightarrow \rho \alpha g\end{array}$
Hence, the answer is the option 1.

Example 2: Which one statement is incorrect?

1) Gravity has no effect on the inertial mass of the body.

2) Inertial mass is independent of size, and shape.

3) Inertial mass measured by spring balance.

4) It is conserved when two bodies combine physically or chemically.

Solution:

Inertial mass

Mass of material of body which measures its inertia.

Gravity does not affect inertial mass.
wherein

$\begin{array}{rl}& F=m_{i}a\\ & m_i=\frac{F}{a}\\ & m_i\to \text{the inertial mass}\\ & F\to \text{external force}\\ & a\to ac{c}^n\end{array}$

The inertial mass is measured by physical balance, not by spring balance.

Hence, the answer is the option 3.

Example 3: Which one statement is incorrect?

1) Gravitational mass is defined as the gravitational pull experienced by the body in the gravitational field.

2) Gravitational mass measured by spring balance

3) The dimensional formula of gravitational field intensity is $\left[L{T}^{-2}\right]$

4) Gravitational mass is measured by applying Newton's law of gravitation

Solution:

Gravitational mass

$\begin{array}{rl}{m}_g& =\frac{F}{GM/R^2}=\frac{F}{I}\\ m_g& =\text{Gravitational mass}\end{array}$

$I\to$ Gravitational field intensity
wherein
Mass which determines the gravitational pull acting upon it,

$F=\frac{G{M}_e{m}_g}{R^2}\text{or}{m}_g=\frac{F}{\frac{GM}{R^2}}=\frac{F}{I}$
If $\mathrm{l}=1$, then $m_g=F$

Hence gravitational mass is defined for unit field intensity.

Hence, the answer is the option 1.

Example 4: Which one of the statements is incorrect regarding mass?
1) It is a scalar quantity
2) The value of mass does not depend on gravity
3) The mass of a body is given by

$m=\frac{m_o}{(1-\frac{v^2}{c^2})}$

here symbol have their usual meaning.
4) mass can never be zero.

Solution:

Mass

Quantity of matter contained in the body

$S.I.\to [Kg]$

dimension $[M]$
wherein
(i) Value does not change with g
(ii) Can never be zero
(iii) Scalar quantity

$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$
Hence, the answer is the option 3.

Example 5: Regarding weight (w) which is correct?

(i) Its value changes with g

(ii) At the centre of the earth weight is maximum

(iii) The unit is kg-wt

(iv) It is a vector quantity

1)Only (i) and (iv)

2) Only (i), (iii) and (iv)

3) Only (ii) and (iii)

4) All of the above

Solution:

Weight (W)
The attractive force exerted by the earth on anybody.
S.I. Unit: Newton or Kg - wt

Dimension $\left[ML{T}^{-2}\right]$ wherein
(i) Value changes with g
(ii) at $\mathrm{\infty }$ and at centre of earth $\mathrm{g}=0$
(iii) Vector quantity

At the centre of the earth, $g=0$
Hence, weight $=0$
Hence, the answer is the option (2).

Summary

The Earth contains is estimated to be made up of 5.97 × 10²⁴ kg. A big portion of this mass is iron (nearly 32%), oxygen (30%) and then silicon (15%) followed by magnesium at 14%, while the remaining make up the rest. In its core, there are high concentrations of heavy metals hence it has a high mass density, about 5.52 grams per cm3 as opposed to most planets that fall within our solar system. The reason behind its high density is that there are many heavy elements in the mantle and crust of Earth.