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Centripetal Acceleration Derivation Explained Step-by-Step

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Why Understanding Centripetal Acceleration is Essential in Physics

Acceleration that acts on the object in a circular motion is called the centripetal acceleration. It is a property of the motion of the body traversing a circular path. It acts radially towards the centre of the circle.


Centripetal Acceleration

The property of a moving body traversing a circular path is called centripetal acceleration. This acceleration is directed radially in the circular path's centre. The magnitude of centripetal acceleration is directly proportional to the square of the body's speed along the curve divided by the distance between the moving body and the circle's centre. Centripetal force is the force that creates centripetal acceleration and it is directed towards the centre of this circular route.


When an object follows a circular course, its trajectory alters at every point along the way. An object might feel centripetal acceleration even when tracing an arc or a circle at a constant velocity.


There are two perpendicular accelerations at each point of the circular path the object is moving: centripetal acceleration directed towards the body or inward acceleration and tangential acceleration directed directly along the tangent of the circular path.


According to Newton's law, a body in motion has its acceleration in the same direction as the force exerted to move it.


Consider a particle of mass, ‘m’ moving with a constant speed, ‘v’’ having a uniform angular velocity, ‘ω,’  around a circular path of radius, ‘r’ with centre O. 


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Let on time t, the particle be at point P where OP = r1 and on time  t + ∆t, the particle is at Q i.e., OQ = r2

 

Where ∠POQ equals to  ∆Ө (∆Ө  = arc traced by the particle from P to Q or simply angular displacement)

 

Since, |r1| = |r2| = r

 

Angular velocity \[\omega = \frac{\Delta \theta}{\Delta t} \]…(1)

 

Let v1 and v2 be the velocity vectors at P and Q respectively.

 

So, velocity along the tangent to the circular path  at a location is represented by the tangents

 

Now the change in velocity with time from t to t + ∆t is represented by,

|O’a| = v1 and |O’b| = v2

 

Clearly, ∠aO’b = Ө

 

Applying  ∆ law of vectors:

 

O’a + ab  = O’b or ab = O’b - O’a  = v2 - v1 =  ∆v

 

At  ∆t -> 0,  a lies close to b. 

 

Now taking arc ab = radius r.

 

Then O’a = |v|......(2)

 

Therefore,  \[\Delta \theta = \frac{ab}{O’a} = \frac{| \Delta v |}{| v |}  \]  

 

From eq(1), 

 

\[\omega \Delta t = \frac{| \Delta v |}{| v |} \]   

 

\[\frac{\Delta v}{\Delta t} = \omega | v | \]......(3)

 

Since, v = rω putting in eq(3)

             

= (rω)r = ωr² …(4)

 

When  ∆t -> 0, 

 

\[\frac{\Delta v}{\Delta t} = \text{magnitude of centripetal acceleration,} \] |a| at P given by,

                           

\[|a| = \frac{|\Delta v|}{\Delta t} = ωr² = \frac{v}{\frac{r^2}{r}} \]

 

Thus

 

\[|a| = \frac{v}{r^2} \]               


Centripetal Acceleration Formula Proof

Consider a particle traversing a circular path of radius r with centre C. 

 

The Initial particle is at P with linear velocity v and angular velocity ω. 

 

Since,   v = r x ω …(a)

 

Differentiating both the sides w.r.t:

 

\[\frac{dv}{dt} = \omega \frac{dr}{dt} + r \frac{d \omega}{dt} \]......(b)

 

Here,\[\frac{dv}{dt} = a\] (resultant acceleration of particle at P)....(c) 

 

\[\frac{d \omega}{dt}  =  \alpha \] (angular acceleration at P)....(d)

 

Where, angular acceleration is the time rate of change in the angular velocity of an 

object traversing a circular path.

 

\[\frac{dr}{dt} = v\] (linear velocity at P)....(e)

 

Putting values of (c), (d), (e) in (b)

 

\[\overline{a} = \overline{\omega} \times \overline{v} + \overline{\alpha} \times \overline{r} \].....(f)

 

Here, we can see the resultant acceleration has two components:

(i) ω x v  and (ii) r x α

  

|a(c)| = |v x ω|

 

a(c) =  radial or centripetal acceleration.

 

Both are perpendicular to each other.

 

|a(c)| = |v x ω| = v ω Sin 90° =vω 

 

Putting v = rω, we get,

 

Centripetal acceleration, \[a(c) = \omega ^2 r = \frac{v^2}{r} \]

 

Derive an expression for centripetal acceleration in uniform circular motion    

 

 As we know that resultant acceleration of the particle at P is given by,

 

 a  = ω x v + r x α

 

Where the component, a(t)  = r x α 

 

When r and α are perpendicular to each other , then,

 

a(t) =  r x α =  r α Sin 90° = r α

 

a(t) is called the tangential acceleration acting along the tangent to the circular path at point P.

 

Since, In case of uniform circular motion, the object moves with a constant speed (v),

Therefore,  \[α  = \frac{dω}{dt} = 0\]

 

So a(t) = 0

 

While a(c) ≠ 0 

 

Thus in a circular motion, only centripetal acceleration acts on the body which is given by,

 

a(c) = ω x v

  

This expression can be zeroed when ω = 0.

 

This is possible only when a particle moves in a straight line.


Centrifugal Acceleration

In Newtonian mechanics, a kind of fictitious acceleration (appears to) acts in a body having a circular motion. It is always directed away from the centre around which the body moves.

              

Centripetal Force Derivation

The force that acts on a body moving in a circular path and is directed towards the centre around which the body is moving is called the centripetal force.

 

The circle represents the orbit of any satellite of radius R moving from point A to B with speed v in time t.

 

Now, draw vector AP to represent the initial velocity of the satellite at A, which is along a tangent at A, and second vector, BQ, to represent new velocity at B.

 

Redraw the initial and new vectors, both starting from the same point D. 

 

They both have a magnitude equal to v.

 

FG represents the change in velocity, and must be added to the old velocity v to generate a new velocity, having the same magnitude i.e., v.

 

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∆AOB and ∆FDG

 

AO = DF, OB = DG and  ∠AOB = ∠FDG = X

 

Hence,  ∆AOB ~  ∆FDG 

 

So, \[\frac{\text{change in velocity}}{v} = \frac{AB}{R}\]

 

Acceleration, \[a = \frac{\text{change in velocity}}{\text{time taken A to B}}\]

 

= AB x v = R x time to A to B = \[\frac{v²}{r}\]

 

Using the relation, F = ma, we get,

 

Fc = \[\frac{mv^2}{r} = mr\omega ^2\]

 

This expression is for the centripetal force.

 

Conclusion:

Haven’t we already established that it is important for the Earth to revolve around the Sun? Well, being able to understand what causes that movement is equally important. This article introduces you to Centripetal Acceleration and its derivation. Go through it thoroughly for a better understanding. 

FAQs on Centripetal Acceleration Derivation Explained Step-by-Step

1. What is the fundamental concept of centripetal acceleration in physics?

Centripetal acceleration is the acceleration experienced by an object moving in a circular path. It is always directed towards the centre of the circle. While the object's speed might be constant (as in uniform circular motion), its velocity is continuously changing because its direction is changing. This change in velocity is what defines centripetal acceleration.

2. What is the primary formula for centripetal acceleration and what do the terms represent?

The primary formula for centripetal acceleration (a_c) is a_c = v²/r. In this equation:

  • a_c is the centripetal acceleration, measured in m/s².
  • v is the tangential or linear speed of the object along the circular path.
  • r is the radius of the circular path.
This formula shows that acceleration increases with the square of the speed and decreases as the radius of the path increases.

3. Why is centripetal acceleration not constant even in uniform circular motion?

Even though the magnitude (speed) of the centripetal acceleration might be constant in uniform circular motion, the acceleration itself is not constant. This is because acceleration is a vector quantity, having both magnitude and direction. Since the object is continuously moving around the circle, the direction of the centripetal acceleration vector is also continuously changing to always point towards the centre. A change in direction means the vector is changing, and thus, the acceleration is not constant.

4. What causes centripetal acceleration, and how does it relate to centripetal force?

Centripetal acceleration is caused by a centripetal force. According to Newton's Second Law of Motion (F = ma), an acceleration can only exist if there is a net force. In circular motion, this net force is directed towards the center and is called the centripetal force. This force is responsible for continuously changing the object's direction, thereby producing the centripetal acceleration. Without this force, the object would move in a straight line.

5. How can the formula for centripetal acceleration be derived using a geometric approach?

The derivation using a geometric approach involves analysing the change in velocity vectors over a small time interval (Δt). By considering the initial and final position vectors and the corresponding velocity vectors, two similar isosceles triangles are formed. One triangle is formed by the two position vectors and the displacement vector, and the other is formed by the two velocity vectors and the change in velocity vector (Δv). By comparing the ratios of the sides of these similar triangles and taking the limit as Δt approaches zero, we arrive at the expression a_c = v²/r.

6. What are some real-world examples that demonstrate the importance of centripetal acceleration?

Centripetal acceleration is crucial in many real-world scenarios. Key examples include:

  • A car turning a corner: The friction between the tires and the road provides the necessary centripetal force that causes the car to accelerate towards the center of the turn.
  • A satellite orbiting Earth: Earth's gravitational pull acts as the centripetal force, continuously accelerating the satellite towards the Earth and keeping it in a stable orbit.
  • A centrifuge: In a lab, a centrifuge spins samples at high speeds. The centripetal acceleration forces denser particles to the bottom of the tube, separating components of a mixture.

7. How is centripetal acceleration expressed using angular velocity (ω)?

Centripetal acceleration can also be expressed in terms of angular velocity (ω). The relationship between linear speed (v) and angular velocity is v = ωr. By substituting this into the primary formula (a_c = v²/r), we get a_c = (ωr)²/r, which simplifies to a_c = ω²r. This form is particularly useful in problems involving rotational motion where the angular velocity is known.

8. What is a common misconception about the relationship between centripetal acceleration and an object's mass?

A common misconception is that centripetal acceleration depends on the object's mass. The formula a_c = v²/r shows that the acceleration depends only on the object's speed and the radius of its path. However, the centripetal force (F_c = ma_c = mv²/r) required to produce this acceleration is directly proportional to the mass. Therefore, a more massive object requires a greater centripetal force to follow the same circular path at the same speed, even though it experiences the same centripetal acceleration.