The article discusses several approaches and some examples on how to calculate mass from force and distance.
Sir Isaac Newton formulated numerous laws and theories that give us various approaches to calculate the body’s mass from a distance traveled when a force acts on it. Plus, the kinematics equations of motion and work-energy formula assist us in determining the mass from the force and distance.
Calculate Mass using Newton’s Law of Gravitation
We can calculate the mass using Newton’s law of gravitation as follows:
To calculate the mass in terms of force and distance, we can utilize Newton’s laws of gravitation, which say “the force of gravity acting between two bodies is directly proportional to their masses and inversely proportional to the square of the distance between centers of the masses.”
The gravity is universal. That means all the objects in the universe attract each other with gravity, and Newton’s law of gravitation explains this universality of gravity. As per the law of gravitation
Fg ∝m1m2/r2
Where, Fg = force of gravity between two objects
m1 is the mass of object 1
m2 is the mass of object 2
r is the distance separating both object’s centers..
(credit: shutterstock)
Since the force of gravity is directly proportional to the masses of both objects, larger objects will attract each other with a more significant force of gravity.
Rewriting the formula in terms of constant of proportionality,
Fg=G*(m1m2/r2)………………(1)
Where, G is universal gravitational constant having constant value 6.67 x 10-11 Nm2/kg2.
In equation (1), like the constant value of G, we also have the constant value of mass of object 2, which is the mass of the earth; as in most cases, we calculate the force of gravity of any object with respect to the earth. Hence, the constant value of the mass of the earth is m2 is, 5.98 x 1024 kg.
From equation (1),
If we determine the distance d and force of gravity on object Fg, we can calculate its mass m1 using Newton’s Law of Gravitation.
The force of gravity between earth and boy is 680N, standing at a distance of 6.38 x 106m from the earth’s center. Determine the mass of a boy standing on the earth.
Given:
Fg = 680 N
r= 6.38 x 106 m
m2= 5.98 x 1024 kg
G= 6.67 x 10-11 Nm2/kg2
To Find: m1 =?
Formula:
Fg=G*(m1m2/r2)
Solution:
The mass of a boy standing on the earth is calculated by using a formula of the law of gravity as,
Fg=G*(m1m2/r2)
Rearranging in terms of m1,
m1=(43.384*108)/(39.904*1013)
m1 = 108.67 kg
The mass of the boy standing on the earth is 108.67 kg.
Calculate Mass using Second Kinematics Equation of Motion
We can calculate the mass using Newton’s second law of motion as follows:
To calculate the object’s mass, we can use the popular Newton’s second law of motion formula that shows a relation between acceleration, force, and mass. Then, we can implement it into the second kinematics equation of motion, which is about distance.
The second law of motion explains that the object is accelerated (a) hardly when the force (F) is employed on the object having mass m.
a=F/m………………… (2)
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The second kinematics equation of motion about distance d is;
d=ut+(1/2)at2………………….(3)
Substituting equation (3) into above equation, we get
d=ut+(Ft2/2m)
If we determine the distance (d) traveled by an object in time (t) when the force (F) applies on it, we can calculate its mass by using the second kinematics equation of motion.
A net force 10N acts on an object which travels a distance of 40m with an initial velocity of 1m/s in 5s. Calculate the mass of an object.
Given:
F = 10 N
d= 40 m
u = 1 m/s
t = 5 s
To find: m =?
Formula:
d=ut+(1/2)at2
Solution:
The mass of an object can be calculated by using the second kinematics equation of motion.
d=ut+(1/2)at2
As per Newton’s second law, a=F/m
Substituting ‘a’ value into the kinematics equation, we get
d=ut+(Ft2/2m)
Substituting all values,
The mass of an object is 3.57 kg.
Calculate Mass using Third Kinematics Equation of Motion
We can calculate the mass using the third kinematics equation of motion.
To calculate the object’s mass, we can implement Newton’s second law of motion formula into the third kinematic equation of motion, which is about velocity.
The third kinematics equation of motion about velocity v is,
v2=u2+2ad………………….(4)
Let’s implement Newton’s second law of motion (2) into the third kinematics equation of motion (4),
v2=u2+(2fd/m)………….(5)
Suppose we determine the distance (d) traveled by an object when its velocity v changes from initial velocity u due to force F. In that case, we can calculate its mass by using the third kinematics equation of motion.
An object travels in a straight path 5m from an initial position with a velocity of 5m/s when force 50N is applied on the same object at rest. Calculate the mass of an object.
Given:
d = 5m
v= 5 m/s
u = 0 since an object is initially at rest.
F =50 N
To Find: m =?
Formula:
v2=u2+2ad
Solution:
The mass of an object can be calculated by the third kinematics equation of motion,
v2=u2+2ad
Substituting ‘a’ value into the kinematics equation, we get
v2=u2+(2fd/m)
Substituting all values,
The mass of an object is 20kg.
Calculate Mass using Work-Energy Formula
We can calculate the mass using the work-energy formula as follows:
To calculate the mass of an object, we can use the work-energy formula, which shows the work done on an object is equal to its kinetic energy conversion when it moves to a certain distance due to applied force.
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The work done is calculated by-product of force applied and distance traveled.
W =F.d ……………………. (6)
The work done is the kinetic energy conversion of an object. Hence, work done equal to kinetic energy,
W = K.E
W=(1/2)mv2………………….(7)
Substituting value of W (6),
Fd==(1/2)mv2 …………….. (8)
If we determine the distance (d) traveled by an object with velocity v when the force (F) applies to it, we can calculate its mass using the work-energy formula.
Note: here, we consider the friction force is negligible.
The box slides about 10m on the horizontal surface with a velocity of 20m/s when we apply a push force of 50N. Calculate the mass of the box.
Given:
d = 10 m
v = 20 m/s
F = 50 N
To find: m=?
Formula:
W=(1/2)mv2
Solution:
The mass of the box can be calculated by work-energy formula,
W=(1/2)mv2
Substituting value of work done W,
Fd=(1/2)mv2
Rearranging equation for ‘m’,
m=2Fd/v2
Substituting all values,
The mass of the box is 2.5 kg.
