A mass is an invariable quantity that is obviously constant. In this article, we will see how to find mass in centripetal force along with examples.
We can find the mass of the object in a centripetal motion by using an equation for centripetal force. The force applied on the object is directly proportional to the mass of the object.
What is Centripetal Force?
The equivalent force to the centripetal force exerted on the body that prevents it from meeting the center of a circle is known as centrifugal force.
An object in centripetal motion experiences a force that keeps the body accelerating in a circular motion called a centripetal force.
Let us see, how to find the mass of an object in centripetal force using different variables and quantities.
Read more on 15+ Centripetal Force Examples, Critical FAQs.
Mass of an Object in a Centripetal Motion
Consider an object of mass ‘m’ in a centripetal motion making a radius ‘r’ from the center.
By Newton’s Second Law of Motion “The force experiencing on a body is equal to the mass time the acceleration of an object” given by the relation,
F=ma
The mass of the object will resist the acceleration produced during its motion.
Acceleration of an object in a centripetal motion is equal to the ratio of a square of the radial velocity and the radius of a circular path.
a=v2/r
Hence, we have centripetal force experienced on the body is given by
F=mv2/r
Hence, mass of the object in a centripetal motion is
m=Fr/v2
Where v is a radial velocity of an object
Read more on How To Find Normal Force In Circular Motion: Several Approaches and Problem Examples.
Problem 1: Supposed a force of 20N is imposed on an object that results in a circular motion of an object making a circle of radius 8 meters. If the object covers 2 meters per second then calculate the mass of the object.
Given: r=8m
F=20N
Radial velocity of an object v=2m/s
We have,
m=Fr/v2
=(20N* 8m)(2m/s)2=40kg
The mass of an object in a circular motion is found to be 40kg.
Mass of an Object with Angular Velocity
The radial velocity of an object is related to the angular velocity by an equation
v=rω
Where v is a radial velocity
ω is an angular velocity
Hence, the force due to the angular velocity of the object in centripetal motion becomes
F=mω2r
We can now write an expression for mass based on the angular velocity as
m=Fω2r
Where F is a centripetal force,
R is a radius
ω is an angular velocity
Read more on Angular Velocity.
Problem 2: A force of 4N is acted on windmill due to wind speed. The angular velocity of a windmill is supposed 4π rad/s and the length of a propeller is 60 cm, then, find the mass of a propeller.
Given: l=60cm=0.6m
ω =4πrad/s
F=4N
We have,
m=F/ω2r
The length of the propeller is equal to the radius of a windmill.
m=4/4π2*0.6=0.042 kg=42.2grams
The mass of a propeller is 42.2 grams.
Relation between Radial and Angular Velocity
Suppose the object was initially at rest, on giving momentum to an object the object sets into motion and covers 20 revolutions in 5 seconds traveling around a cylinder of radius 15cm.
The object completes 20 revolutions in 5 seconds, which implies it covers 4 revolutions per second.
Hence, the angular velocity of an object is
ω =4rps=4* 2π =8π rad/s
The object covers 4 circumferences of a circle formed due to the circular motion.
C=2π r=2π*0.15m=0.3πm
Distance travelled in one second is
x=4* 0.3π m=1.2πm
Hence, radial velocity is
v=1.2πm/s
Read more on How To Find Angular Acceleration From Angular Velocity: Problem And Examples.
Centripetal Force due to Spring
The force due to spring is given as F=-kx by Hook’s Law. Let ‘x’ be the displacement of the object due to the elongation of spring. Hence the radius of a circular path traveled by the object will be the length of spring plus the extension of a spring due to the forces experienced on the object.
If the centripetal force equals the spring force, then
kx=mv2r=mω2r
Since r=l+x,
kx=mω2(l+x)
We can find the mass of the object by calculating the velocity of the object and the displacement of the object due to spring in a centripetal motion.
m=kx/ω2(l+x)
Read more on Is Spring Force Conservative:Exhaustive Insights.
Problems And Solution
Q1. Consider an object attached to a spring of length 80 cms having a spring constant of 1.5. A boy holds another end of a spring and rotates around in a circular path. A length of a spring extends to 20 cms due to the drag force exerting on the object. The object covers one revolution per second. Calculate the mass of the object.
Given: Length of the spring l=80cms= 0.8m
Displacement x=20cm= 0.2m
Spring constant k=1.5
Angular velocity of the object ω= 1rps= 2π rad/s
Hence, the mass of the object is
m=kx/ω2(l+x)
=1.5* 0.2/2π2 * (0.8+0.2)
=0.3/ 39.43* 1=0.0076kg=7.6grams
Q2. A girl of weight 52kg jogging in a circular park of diameter 150 meters completes her one lap in one minute twenty seconds. Determine the force required to maintain the motion of a body in a circular motion by a girl.
Given: m=52kg
t=1 minute 20 seconds= 80 seconds
Diameter of a park d= 150m
the radius of a park r=75m
The circumference of park is
C=2\Pi r=2π* 75m=471m
A girl completes one lap that is a distance of 471m in 80 seconds.
Hence, the velocity of a girl is
v=471m/80s=5.8m/s
The force experience on the body is
F=mv2 / r
F=52 kg * (5.8m/s)2/75m=23.3 N
Frequently Asked Questions
How to Calculate the Radial Velocity?
The radial velocity of the object is given by the equation
v=x2 – x1/ t2-t1
The radial velocity of the object in a centripetal motion is the displacement of the object in a circular track with time.
How to Calculate the Angular Velocity?
The angular velocity of the object is given by the equation
ω =θ2 – θ1/t2 – t1
The angular velocity of the object in a centripetal motion is the rate of change in the angle on the displacement of an object in a circular track with respect to time.
