The article discusses about several approaches and some problem examples about how to calculate mass from gravitational force.

**Every object with mass in the universe exerts a gravitational force on another. That’s why the gravitational force is directly proportional to the object’s mass. So utilizing different formulas related to gravitational force, we can calculate the object’s non-zero mass. **

Read more about **How to Calculate Mass from Force and Distance**.

**How to Calculate Mass from Gravitational Force using Newton’s Second Law of Motion**

Let’s calculate the mass using newton’s second law of motion as follows:

**In the second law of motion, Newton describes that the force acts on the object with non-zero mass to accelerate it in the same direction. The gravitational force is a natural force that always acts downward on every object to accelerate it, depending on its mass. **

We have already studied two major types of forces that act on the bodies. The gravitational force or gravity, a non-contact force, always acts between every object’s masses.

As per newton’s second law,

F = ma ………………… (*)

When the gravity force is exerted, every object accelerates as per the second law of motion. The acceleration due to gravity force is constant, called **acceleration due to gravity** ‘g’. Since gravity always acts on us, the idea of our ‘weight’ as “mg” originated, which includes our mass m and acceleration ‘a’. That’s the reason the gravitational force is also called the **weight force**.

Hence, newton’s second law formula becomes,

**F _{g} = mg** ………………………. (1)

As per equations (*) and (1),

*We must produce an upward force (ma) greater than the gravitational force (mg) to lift a heavier body.*

Since g has a constant value of 9.8 m/s^{2}, the gravitational force F_{g} only depends on the object’s mass m. The more massive the object, the more force is needed to accelerate it.

** If the gravitational force is applied to an object, we can calculate its mass by the formula of Newton’s second law of motion**.

Read more about **Newton’s Laws of Motion****.**

**The gravitational force acting on a girl that is jogging in the park is 490. Calculate the mass of a girl.**

__Given:__

F_{g} = 490 N

g = 9.8 m/s

** To Find: **m =?

__Formula:__

F =ma

__Solution:__

The mass of a girl is calculated by using **Newton’s second law of motion **formula,

F =ma

The **gravitational force** is given by,

**F _{g} = mg**

m=**F _{g}**/g

m=490/9.8

m = 50 kg …………………………………. (a)

**The mass of a girl running in the park is 50 kg.**

**How do you find Mass with Gravitational Force and Radius?**

Let’s calculate the mass with gravitational force using Newton’s law of gravitation as follows:

**The law of gravitation discovers that the gravitational force between two objects is directly proportional to their masses and inversely proportional to the square of radius between their center of masses. If the second object is the earth with a fixed mass, we can calculate the first object’s mass. **

As per Newton’s law of gravitation,

F_{g}=G(mM/r^{2})……………..(2)

The whole mass of an object is concentrated on one specific point, mostly on its central point, called its **center of mass(CM)**. The radius r measures *the distance or separation between the center of masses of two objects*.

A small mass of 1 kg separated by radius 1 experienced a small gravitation force of 6.67 x 10-11 Nm^{2}/kg^{2}, consistent with every object. Hence, this constant value is the value of the constant of proportionality in the law of gravitation, also called the** Universal gravitational constant G**.

It is easier to calculate the F_{g} between an object and the earth as the planet that has its fixed mass M = 5.98 x 10^{24} kg and also fixed radius r from the center of its earth, r = 6.38 x 10^{6}m

**The gravitational force acting on a girl jogging in the park is 490. Calculate the mass of a girl using Newton’s law of gravitation.**

**Given**:

F_{g} = 490 N

M = 5.98 x 10^{24} kg

r = 6.38 x 10^{6}m

G = 6.67 x 10^{-11} nm^{2}/kg^{2}

**To Find:** m =?

__Formula:__

F_{g}=G(mM/r^{2})

**Solution:**

The mass of a girl is calculated by **Newton’s law of gravitation** is,

F_{g}=G(mM/r^{2})

Rearranging for mass m,

m=F_{g}r^{2}/GM

Substituting all values,

**From (a) and (b), we have noticed the calculated mass using Newton’s second law and law of gravitation formula is the same. **

The law of gravitation can be applied to the two objects having identical or different masses.

**The gravitational force between you and your colleague is 3 x 10**^{-7} N as you both approach at a distance of 1 m apart from each other in the school corridor. Since your mass is 60 kg, calculate the mass of your colleague.

^{-7}N as you both approach at a distance of 1 m apart from each other in the school corridor. Since your mass is 60 kg, calculate the mass of your colleague.

** Given**:

F_{g }= 3 x 10^{-7}** **N

r = 1 m

m_{1} = 60 kg

G = 6.67 x 10^{-11} nm^{2}/kg^{2}

** To Find**: m

_{2}=?

** Formula**:

F_{g}=G(m_{1}m_{2}/r^{2})

__Solution:__

The mass of the colleague is calculated by **Newton’s law of gravitation** is,

F_{g}=G(m_{1}m_{2}/r^{2})

Rearranging for mass m_{2},

m_{2}=F_{g}r^{2}/Gm_{1}

Substituting all values,

**Mass of your colleague is 75 kg.**

**How to Calculate Mass from Gravitational Force using Centripetal Force Formula?**

Let’s calculate the mass with gravitational force using the centripetal force formula as follows:

**When an object moves circularly, its velocity keeps changing due to its direction. The acceleration direction is towards the center, caused due to centripetal force. Since the whole object’s mass is concentrated on its center, we can calculate it from the centripetal force formula. **

The centripetal force is obtained from Newton’s second law of motion.

Since the acceleration is a circular path, we need to consider the radius; that’s why acceleration becomes

v^{2}/r

Therefore, as per equation (*), the centripetal force is given by,

F_{c}=mv^{2}/r

The centripetal force is a *center-seeking force* that acts on an object to move circularly towards its center. The earth exerts this centripetal force on the satellite to sustain its circular motion around.

**The satellite travels in an orbital motion of 20 m/s continuously around the earth. The gravitational force between earth and satellite is 500 N which exerts a centripetal force of about 200 N. Calculate the satellite mass. **

** Given**:

F_{g} = 500 N

F_{c }= 200 N

v = 20 m/s

M = earth’s mass = 5.98 x 10^{24} kg

G = 6.67 x 10^{-11} nm^{2}/kg^{2}

** To find**: m =?

** Formula**:

F_{g}=G(mM/r^{2})

F_{c}=mv^{2}/r

** Solution**:

The mass of the satellite is calculated by **newton’s law of gravitation**,

F_{g}=G(mM/r^{2})

The **centripetal force** on the satellite is,

F_{c}=mv^{2}/r

Solving the formula for radius r,

r=mv^{2}/F_{c}

Substituting above equation into newton’s law of gravitation, we get

F_{g}=F_{c}^{2}[GM/mv^{4}]

Solving for mass m,

Substituting all values,

m=159.4/8

m = 19.94 \approx 20 kg

**The mass of the satellite moving around the earth is 20 kg.**