In this article we are going to describe how centripetal acceleration is related to velocity in brief equipped with some numerical problems.
Before beginning with the relation between centripetal acceleration and velocity they should be introduced in simple words. Basically centripetal acceleration is produced due to the change in the direction of velocity of a particle that is moving in a circular path. The direction of centripetal acceleration always remains the same that is radially inwards.
The basic concept of velocity is clear to all. So what is velocity? Velocity is the displacement of a body per unit time. Now if it is not mentioned that is it a linear velocity or an angular velocity then we have to assume that velocity signifies the linear one.
In case of a particle that is moving in a circular path the velocity always acts in tangential direction. This tangential velocity changes its direction from time to time. Due to this change in direction centripetal acceleration is used to produce.
How is centripetal acceleration related to velocity?
Actually velocity is the reason and centripetal acceleration is its result. A force is needed by the particle which is moving around a circular path to continue its motion. Centripetal force is the name of that required force. When the body moves around the circular path with the help of centripetal force its velocity starts to change randomly.
Velocity is a vector quantity. This is the reason behind it. The velocity changes randomly in the direction of a tangent. It is known to all that the rate of change of velocity is termed as acceleration. So the change in direction of velocity produces acceleration which is termed as the centripetal acceleration. In this way centripetal acceleration is related to velocity.
Find centripetal acceleration from velocity
Let’s start with the formula of centripetal force. So the formula of centripetal force is F=m.v2/r ……(1) where v is the linear velocity of the particle,r is the radius of the circular path and m is the mass of the particle. Now we all know that force= mass x acceleration.
Therefore , force F= m.a ……….(2) where, a is the centripetal acceleration of the particle. Comparing the two equations (1) and (2) we get-
F=m.a = m.v2/r
Or, a=v2/r ………..(3)
Or, v=√a.r …….(5)
If the values of linear velocity and radius of the circular path are mentioned in the question then we can calculate the value of centripetal acceleration with the help of equation (3).
We can also use the angular velocity w to calculate the centripetal acceleration. Here the mathematical equation which should be used is v=w.r ……..(6) With the help of this equation (6) we can be able to relate the linear velocity v and the angular velocity w.
If we put the value of v from equation (6) to equation (4) then the value of centripetal acceleration(a) will be, (w.r)^2=a.r
Or, (w2.r2)/r = a
Therefore , a = w2.r …………….(7)
Centripetal acceleration and velocity graph
Centripetal acceleration and angular velocity graph is drawn below:
Angle between centripetal acceleration and velocity
Centripetal force helps a particle to move in a circular path. While moving in that circular path an acceleration gets produced from the effect of this centripetal force. This acceleration is our required centripetal acceleration which is always acts in radially inward direction. Centripetal acceleration also follows the same direction as the centripetal force.
The velocity with which the particle is used to move around the circular path changes randomly from time to time and is always directed towards the tangent of that path. Hence the angle between the centripetal acceleration and velocity is always 90 degree. It concludes that the direction of centripetal acceleration and velocity are orthogonal to each other at any condition.
Can velocity and centripetal acceleration be the same?
No centripetal acceleration and velocity can never be the same. We all know that both the centripetal acceleration and velocity are the vector quantities so both of them have magnitude as well as direction. Centripetal acceleration and velocity can have the same magnitude but their directions are always different from each other.
It has already been clarified earlier that the direction of centripetal acceleration is radially inward whereas the direction of velocity is always in the direction of tangent of that circular path in which the body is moving. Hence it is clear that the centripetal acceleration and velocity are normal to each other at any condition.
When velocity and centripetal acceleration are the same?
It has been clarified already that the magnitudes of centripetal acceleration and velocity may be the same at certain conditions but their directions can never be the same.
For example if the values of velocity(v) and radius(r) of a circular path are 1m/s and 1 m respectively for a particle moving in that path then the value of centripetal acceleration will be a= v2/r = (1)2/1 m/s2 = 1m/s2 in this case magnitudes of both the centripetal acceleration and velocity are the same but their direction will be different as always.
1) A body is moving around a circular path of radius 0.25 m with a linear velocity of 2m/s. What will be the value of centripetal acceleration?
r = 0.25 m
We know that centripetal acceleration(a)= v2/r
= (2)2/0.25 m/s2
= 16 m/s2
2) A body of 50 kg is moving with a centripetal force of 50N around a circular path of radius 1 m. What is the value of linear velocity?
F = m.v2/r = 50 N
r= 1 m
m.v2/r = 50
Or,50.v2/r = 50
Or, v2/r = 1
Or, v2 = r
Or, v2 = 1
Or, v=1 m/s
All the major and minor details related to centripetal acceleration and velocity are covered in this article. How the centripetal acceleration and velocity are related,if they are the same or not – all these questions have been answered in it.