In this article, the topic, “centripetal acceleration and radial acceleration” with 5 important several matters will be discuss in a brief manner.

**Centripetal acceleration is the acceleration conducted in the direction of the centrum of the curve and radial acceleration is in the direction of the radius. Centripetal acceleration and radial acceleration both are similar physical quantities. Tangential component is absent for the both centripetal acceleration and radial acceleration.**

__Formula for the centripetal acceleration and radial acceleration:-__

__Formula for the centripetal acceleration and radial acceleration:-__

The term of centripetal acceleration and radial acceleration are same.

The expression for the centripetal acceleration or radial acceleration is,

[latex]a_r = \frac{v^2}{r}[/latex]

Where,

[latex]a_r[/latex] = Centripetal acceleration and unit is meter per second square.

v = Velocity and unit is meter per second.

r = Radius and unit is meter.

The dimension for the centripetal acceleration and radial acceleration is [latex]ML^1T^-^2[/latex].

__Formula derivation for the centripetal acceleration and radial acceleration:-__

__Formula derivation for the centripetal acceleration and radial acceleration:-__

A substance name M is attached with a string and create to spin round about a particular permanent spot O which is denoted as centre of the spot. When the substance start to spin round fast the string that time it almost like radius of the circle. This signifies that a force is acted on the substance from the spot of the circle. For this reason an acceleration [latex]a_0[/latex] is together with the direction of the radial. (Together with the radius of the circle nears the spot of the circle).

To determine this force, tension is generated towards the string in the direction of the opposite. This force for the tension is derive as, centripetal force.

For this reason the acceleration developed on the substance is called centripetal acceleration or radial acceleration and denoted as, [latex]a_r[/latex].

Performing the property for the similar triangles and we can write,

[latex]\frac{AB}{OA} = \frac{l}{r}[/latex]

A and B both are almost close so we can derive this AB to the length of arc AB and the expression can be write as,

AB = [latex]v \times dt[/latex]

The figure (3) we can observe A and B almost same and the expression we can write as,

[latex]v + dv \approx dv[/latex]

[latex]\frac{v \times dt}{r} = \frac{dv}{v}[/latex]

On rearranging,

[latex]\frac{dv}{dt} = \frac{v^2}{r}[/latex]

Thus,

[latex]\frac{dv}{dt}[/latex]

Under the uniform **circular motion**s the centripetal acceleration or radial acceleration is generated and we can write the formula for the centripetal acceleration or radial acceleration,

[latex]a_r = \frac{v^2}{r}[/latex]

__Problem:-__

__Problem:-__

**A ball which contains mass is 3 kilogram is attached with a ****string and spin round in a circular path. The height of the string is 1.8 meter and when the ball is spin round that time it makes 300 revolutions per minute.**

**Determine,**

**A. linear velocity of the ball.**

**B. Acceleration and the force are exerted upon the ball.**

**Solution:-**

Given data are,

m = 3 kilogram

r = 1.8 meter

N = 300 revolutions per minute

We know that,

[latex]\omega = \frac{2\pi N}{60}[/latex]

[latex]\omega[/latex] = [latex]\frac{2 \pi 300}{60}[/latex]

[latex]\omega[/latex] = [latex]10 \pi[/latex]

[latex]\omega = 31.4 radians per second[/latex]

v = 56.52 meter per second

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

[latex]a = \frac{56.52 ^2}{1.8}[/latex]

a = 1774 meter per second square.

Centripetal force,

F = ma

[latex]F = 3\times 1774[/latex]

F = 5322 Newton

A ball which contains mass is 3 kilogram is attached with a string and spin round in a circular path. The height of the string is 1.8 meter and when the ball is spin round that time it makes 300 revolutions per minute.

So,

A. Linear velocity of the ball is 56.52 meter per second.

B. Acceleration of the ball is 1774 meter per second square.

And the force is exerted upon the ball is 5322 Newton.

__Tangential component:-__

__Tangential component:-__

The tangential component can be derive as, the part of angular acceleration tangential to the way of the circular. The unit is to measure the tangential component is, meter per square second.

The expression for the tangential component can be written as,

[latex]a_t = \frac{v_2 – v_1}{t}[/latex]

Where,

[latex]a_t[/latex] = Tangential component

[latex]v_2[/latex] and [latex]v_1[/latex] = Both are represent the velocities for the two substance in a motion of circular way t = Time period.

**Is radial acceleration same as centripetal acceleration?**

**Is radial acceleration same as centripetal acceleration?**

**Yes, radial acceleration same as centripetal acceleration.**

__Characteristics of Centripetal or radial acceleration:-__

__Characteristics of Centripetal or radial acceleration:-__

The characteristics of the centripetal or radial acceleration are listed below,

**The characteristics of the motion of the pendulum traversing a path in circular way, and centripetal acceleration always notified according to the center of the path in circular way.****The magnitude of the centripetal or radial acceleration can be express as,****[latex]a_c = \frac{v^2}{r} = r\omega^2 = v\omega = \frac{4\pi ^2r}{T^2} = 4\pi^2rn^2[/latex].****The direction for the radial or centripetal acceleration is all time changes.****For, U.C.M. the magnitude of the centripetal or radial acceleration is unchanged.****The centripetal or radial acceleration is identified as a letter. S.I. unit to measure the centripetal or radial acceleration is meter per second square.****The centripetal or radial acceleration is always conducted towards the spot of the circular way along the radius.**

**When radial acceleration and centripetal acceleration are same?**

**Centripetal acceleration is acceleration directed towards the centre of the curve and radial acceleration is acceleration along the radius and these two are exactly the same thing. They are both the same thing.**

Net force is acted in the direction towards the center of a circular path, causing centripetal acceleration. Direction is perpendicular to the matter’s linear velocity.

**What is the relation between radial acceleration and centripetal acceleration?**

**The radial acceleration and centripetal acceleration both are the same term.**

**The expression for the centripetal acceleration or radial acceleration is,**

**[latex]a_r = \frac{v^2}{r}[/latex]**

**Where,**

**[latex]a_r[/latex] = Centripetal acceleration and unit is meter per second square.**

**v = Velocity and unit is meter per second.**

**r = Radius and unit is meter.**

**The radius has an inverse relationship with centripetal acceleration, so when the radius is halved, the centripetal acceleration is doubled**.

**What is the difference between radial and tangential acceleration?**

Though Centripetal and centrifugal forces are same in magnitude and opposite direction these forces do not form action reaction pair because as these both forces act on the same body.

**Centripetal acceleration is the acceleration conducted in the direction of the Centrum of the curve and radial acceleration is in the direction of the radius. Centripetal acceleration and radial acceleration both are similar physical quantities. Tangential component is absent for the both centripetal acceleration and radial acceleration.**

**Problem:**

**A stone is attached with a string and spin round in a circular path. When the stone is spinning that time angular speed is increases from 3 radians per second to 50 radians per second at the time period of 10 second. The radius will be while the string spin round in a circular path is 22 centimeter. Compare the proportions from the centripetal acceleration to tangential acceleration at the 14 second.**

**Solution:-**

**Given data are,**

**Initial angular speed ([latex]\omega_1[/latex]) = 3 radians per second**

**Final angular speed ([latex]\omega_2[/latex]) = 51 radians per second**

**Initial time period ([latex]t_1[/latex]) = 10 second**

**Initial time period ([latex]t_2[/latex]) = 14 second**

**Total taken time period ([latex]t = t_1+ t_2[/latex]) = (10+14) second = 24 second**

**Radius (r) = 22 centimeter = 0.22 meter**

**So,**

**[latex]\alpha = \frac{\omega_2 – \omega_2}{t}[/latex]**

**[latex]\alpha = \frac{51 – 3}{24}[/latex]**

**[latex]\alpha[/latex] = 2 radians per second square**

**Now,**

**[latex]a_t = ra[/latex]**

**[latex]a_t = 0.22 \times 2[/latex]**

**[latex]a_t = 0.44 meter per second square[/latex]**

**The centripetal acceleration to tangential acceleration at the 14 second**

**[latex]a_r = r\omega^2[/latex]**

**[latex]a_r = 0.22 \times 51^2[/latex]**

**[latex]a_r[/latex] = 2601 meter per second square**

**Now, the proportions from the centripetal acceleration to tangential acceleration at the 14 second is,**

**[latex]a_r : a_t = 2601 : 0.44[/latex]**

A stone is attached with a string and spin round in a circular path. When the stone is spinning that time angular speed is increases from 3 radians per second to 50 radians per second at the time period of 10 second. The radius will be while the string spin round in a circular path is 22 centimeter.

So, the proportions from the centripetal acceleration to tangential acceleration at the 14 second are 2601: 0.44.

**Conclusion:-**

Centripetal acceleration is defined as the property of the motion of an object, traversing a circular path. Any object that is moving in a circle and has an acceleration vector pointed towards the center of that circle is known as Centripetal acceleration. Radial acceleration is also known as Centripetal Acceleration. The component of angular acceleration tangential to the circular path is what Tangential Acceleration is.