Centripetal acceleration is the rate of change of tangential velocity. The net force causing the centripetal acceleration of an object in a circular motion is defined as centripetal force. The derivation of centripetal acceleration is very important for students who want to learn the concept in-depth. The direction of the centripetal force is towards the centre, which is perpendicular to the velocity of the body.
The centripetal acceleration derivation will help students to retain the concept for a longer period of time. The derivation of centripetal acceleration is given in a detailed manner so that students can understand the topic with ease.
The centripetal force keeps a body constantly moving with the same velocity in a curved path. The mathematical explanation of centripetal acceleration was first provided by Christian Huygens in the year 1659. The derivation of centripetal acceleration is provided below.
Centripetal Acceleration Derivation
The force of a moving object can be written as
\(\begin{array}{l} F = ma \,\, \textup{……. (1)} \end{array} \)
\(\begin{array}{l} – v _ { 1 } + v _ { 2 } = \Delta v \end{array} \)
\(\begin{array}{l} \Delta v = v _ { 2 } \, – \, v _ { 1 } \end{array} \)
The triangle PQS and AOB are similar. Therefore,
\(\begin{array}{l} \frac{ \Delta v }{ AB } = \frac{ v }{ r } \end{array} \)
\(\begin{array}{l} AB = arc \,\, AB = v \Delta t \end{array} \)
\(\begin{array}{l} \frac{ \Delta v }{ v \Delta t } = \frac{ v }{ r } \end{array} \)
\(\begin{array}{l} \frac{ \Delta v }{ \Delta t } = \frac{ v ^ { 2 } }{ r } \end{array} \)
\(\begin{array}{l} a = \frac{ v ^ { 2 } }{ r } \end{array} \)
Thus, we derive the formula of centripetal acceleration. Students can follow the steps given above to learn the derivation of centripetal acceleration.
Read More:Centripetal Acceleration
Frequently Asked Questions – FAQs
Q1
What do you mean by centripetal acceleration?
Centripetal acceleration is the rate of change of tangential velocity of a body moving in a circular motion. Its direction is always towards the centre of the circle.
Q2
Give the formula for finding centripetal acceleration.
Let v be the magnitude of the velocity of the body Let r be the radius of the circular path Then centripetal acceleration, a=v^{2}/r
Q3
Which force is responsible for producing centripetal acceleration?
Centripetal force is responsible for producing centripetal acceleration./div>
Is centripetal acceleration a constant or a variable vector?
Q4
Is centripetal acceleration a constant or a variable vector?
Centripetal acceleration has a constant magnitude since both v and r are constant, but since the direction of v keeps on changing at each instant in a circular motion, hence centripetal acceleration’s direction also keeps on changing at each instant, always pointing towards the centre. Hence, centripetal acceleration is a variable vector.
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Comments
Leave a Comment
aziz September 26, 2019 at 11:45 am
nice way to study physics
Reply
Piyush January 14, 2020 at 7:35 am
when we form the triangle why is the v1 vector negative?
Reply
John October 9, 2020 at 4:02 pm
QP is the direction of the +v vector. The derivation uses PQ, and hence -v.
Reply
Safin December 19, 2020 at 10:45 pm
Its because in order to add v1 and v2 vectors vectorially using triangle method ,v1 vector has to attach its head to tail so as to obtain resultant vector delta v
This lets express the arc length c in terms of ω, t and r. And finally, we can use the acceleration = change in velocity / time relationship to derive the formula for centripetal acceleration we a = ω^{2}r.
Centripetal Acceleration Derivation. Where, a= acceleration which is given by the rate of change of velocity with respect to time. Thus, the centripetal acceleration equation is given by a=v^{2}/r or centripetal acceleration is v 2 r. Direction of centripetal acceleration(& force) is towards the centre of the circle.
So is the centripetal acceleration directly or inversely proportional to the radius of the circular path, since ac=v^2/r=w^2*r.In the first scenario, ac is inversely proportional to the radius, whereas, in the second scenario, ac is directly proportional to the radius of the circular path.
Centripetal acceleration a_{c} is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity v and has the magnitude ac=v2r;ac=rω2 a c = v 2 r ; a c = r ω 2 .
Acceleration (a) is the change in velocity (Δv) over the change in time (Δt), represented by the equation a = Δv/Δt. This allows you to measure how fast velocity changes in meters per second squared (m/s^2).
By using the expressions for centripetal acceleration a_{c} from ac=v2r;ac=rω2 a c = v 2 r ; a c = r ω 2 , we get two expressions for the centripetal force F_{c} in terms of mass, velocity, angular velocity, and radius of curvature: Fc=mv2r;Fc=mrω2 F c = m v 2 r ; F c = m r ω 2 .
To calculate centripetal acceleration, the following equation can be used: a c = v 2 r Where is equal to velocity which is measured in meters per second ( ) and is equal to the radius of the circle. The unit of measure for centripetal acceleration is meter per second squared, m / s 2 .
Centripetal force is, simply, the force that causes centripetal acceleration. Objects that move in uniform circular motion all have an acceleration toward the center of the circle and therefore, they must also suffer a force toward the center of the circle. That force is the centripetal force.
Since the direction of the velocity vector is constantly changing towards the center of the circle (for an object moving in a circular path), the acceleration is also directed towards the center.
The centrifugal force in the nonintertial frame of reference of a particle in circular motion is the effect of the acceleration of the frame of reference with respect to an interial frame of reference. therefore, it is called a pseudo or fictitious force.
The acceleration needed to keep an object (here, it's the Moon) going around in a circle is called the centripetal acceleration, and it's always perpendicular to the object's travel. The centripetal acceleration points toward the center of the circle.
Centripetal acceleration is the idea that any object moving in a circle, in something called circular motion, will have an acceleration vector pointed towards the center of that circle. This is true even if the object is moving around the circle at a constant speed. Centripetal means towards the center.
IB Physics Tutor Summary: The formula for centripetal force, F = m(v²/r), is derived by combining Newton's second law (F = ma) with the formula for centripetal acceleration (a = v²/r). This shows that the force keeping an object moving in a circle depends on its mass, speed squared, and the circle's radius.
Centripetal force is the radially inward external force applied to an object to keep it within a circular path. Centripetal acceleration is the direction pointing inward toward the center of the circular path objects subjected to a centripetal force follow.
we have to derive F = mv²/r using dimensional analysis where F is centripetal force, m is mass, v is velocity and r is radius of the circular path of the body. ⇒F = mv²/r , let proportionality constant is 1. hence, it is clear that F = mv²/r from dimensional analysis.
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