# Is Centripetal Acceleration Constant: Different Use Cases And Problems With Facts

Centripetal acceleration is a vector quantity as it has direction along with magnitude.

An object that has centripetal acceleration is always in a circular motion, which results in constantly changing direction. Therefore, centripetal acceleration is not constant.

In this section, let’s us try to answer a few questions like, “Is centripetal acceleration constant?”

The formula for centripetal acceleration is given as: ac = v2/r

Where,
ac = centripetal acceleration.
v = velocity of the object.
r = radius of the circle.

Centripetal acceleration is a vector quantity, and thus, to be a constant, its magnitude and direction should also be constants. For a given uniform circular motion, the magnitude of centripetal acceleration will be persistent as the velocity of the object and radius of the trajectory will be unwavering. But, the direction will be continuously changing, and therefore, centripetal acceleration will not be a constant.

## Is Centripetal Acceleration Always Constant?

The centripetal acceleration is never constant.

The centripetal acceleration is never constant, but if the radius of the orbit that the object is moving in is very large and the speed of the object is relatively less than for a fraction of a second or so, the centripetal acceleration might be considered as a constant value.

If the mentioned situation is not there, then the centripetal acceleration is never constant.

## When Is Centripetal Acceleration Constant?

When the radius of the circle is too large, the centripetal acceleration can be constant.

The radius of the circle is already a constant value. Considering the velocity also to be a constant, then for a distance equal to the tangent of a circle, the centripetal acceleration might be constant.

As the magnitude and direction both should be constant for a vector quantity to be a constant, the centripetal acceleration for that tangent is considered to be constant as the direction for that particular distance is not changing. The small yellow line at the top of the circle represents the tangent to a circle.

Read more about How To Find Centripetal acceleration.

## When Is Centripetal acceleration Not Constant?

The centripetal acceleration is generally never constant, as the direction of an object is continuously changing in a circular motion.

Centripetal acceleration, also known as radial acceleration, is a vector quantity consisting of direction along with magnitude. The magnitude in the uniform circular motion is always constant, but as the trajectory is circular, the direction will continuously change, resulting in an inconstant value of acceleration.

Thus, the centripetal acceleration will not be persistent in the presence a uniform circular motion.

## Is Centripetal Acceleration Constant In Magnitude?

When the object having centripetal acceleration is under uniform circular motion, then the magnitude of centripetal acceleration is constant.

If the object is under the influence of uniform circular motion then the centripetal acceleration will have a steady magnitude.

But, if the object is not moving in uniform circular motion, there will be varying outputs for the magnitude of centripetal acceleration.

## Is Centripetal Acceleration Constant In Uniform Circular Motion?

The uniform circular motion does not guarantee stability to the centripetal acceleration.

The centripetal acceleration is a vector quantity, and for a vector quantity to be constant, it should have constant direction as well as constant magnitude. But, as the motion of an object is circular, its direction will be changing continuously. Therefore, uniform circular motion can guarantee constant magnitude, but it does not ensure constant direction.

Only one exception can help the centripetal acceleration to be constant in a uniform circular motion, which is the orbit having a large radius. Large radius results in a large circumference and a large circumference for a fraction of distance looks like a straight line. So, the direction for a small unit of time might not change, and for that particular distance, the centripetal acceleration can be considered constant.

## Centripetal Acceleration Is A Constant Vector?

For a vector quantity to be a constant, its magnitude and direction, both ought to be constants.

The centripetal acceleration cannot be considered a constant vector, as the object in motion will follow a circular trajectory due to which the direction of the object will be continuously changing, which restricts the centripetal acceleration from become a constant vector.

Therefore, the centripetal acceleration is not a constant vector,

Read more about Centripetal acceleration Vs acceleration.

## What Happens To Centripetal Acceleration When Speed Is Constant?

There is no such impact of constant speed on the centripetal acceleration.

The only outcome will be the constant magnitude. Constant speed does not ascertain constant centripetal acceleration.

There are two formulas for the centripetal acceleration; one involves velocity (v), and another involves the angular velocity ω. Both the formulas are given as: ac = v2/r

Where,
ac = centripetal acceleration.
v = velocity of the object.
r = radius of the circle.

ac = r/ω2

Where,
ac = centripetal acceleration.
ω = angular velocity of the object.
r = radius of the circle.

From both the formulas, it is apparent that the magnitude of the centripetal acceleration will alter in accordance with change in the speed of the object, as the centripetal acceleration is directly proportional to the speed of the object. Thus, if there is an increase in the velocity, then the centripetal acceleration will also increase. Similarly, if there is a decline in the velocity, then the centripetal acceleration will also decline in the same format.

## Centripetal Acceleration Derivation

The centripetal acceleration can be derived from several different methods and formulas. One such easy way to derive the centripetal acceleration is by using the formula for centripetal force. The formula for centripetal force is given as: F = mv2/r

Where,
F = centripetal force.
m = mass of the object.
v = velocity of the object.
r = radius of the circle.

According to Newton’s second law of motion, force on an object is directly proportional to its acceleration. To remove the proportionality sign, a constant is added. The constant in this case is the mass (m). The formula for Newton’s second law of motion is given as: F = ma

Where,
F = force.
m = mass of the object.
a = acceleration of the object.
Equate both the equations of force to obtain the formula for centripetal acceleration.

ma = mv2/r

Therefore,

a = v2/r

Here, acceleration (a) is equal to centripetal acceleration (ac). Therefore, ac = v2/r

## Que: A car is traveling at a speed of 77 m/s on a circular track of radius 205 m. What is the centripetal acceleration of the car?

Ans: The formula to calculate the centripetal acceleration is given as: ac = v2/r

Substitute 77 m/s for v and 205 m for r into the formula to calculate the centripetal acceleration.

Therefore, the acceleration of the car is 28.92 m/s2 or around 29 m/s2.

## Que: The angular velocity of a boat is 75 km/hr, which is making circles in a large pond for an annual show. The radius of the circle is about 15 m. Calculate the centripetal acceleration of the boat.

Ans: The formula used to calculate the centripetal acceleration of the boat is: ac = rω2

The speed of the boat is given in km/hr. The first one needs to convert the speed of the boat from km/hr into m/s. To convert the speed from km/hr into m/s, the given speed needs to be multiplied by 1000 metres as 1 kilometre = 1000 m and divide the given speed by 3600 s as 1 hour = 3600 seconds. Therefore,

Substitute 20.83 m/s for ω and 15 m for r into the formula to calculate the centripetal acceleration.

Therefore, the centripetal acceleration of the boat is 6508.33 m/s2.

Durva Dave

I am Durva Dave, completed my post-graduation in Physics. Physics fascinates me a lot and I like in knowing the ‘Why’ and ‘How’ of everything that unfolds in our universe. I try to write my blogs in simple yet effective language so that it is easier for the reader to understand as well remember. I hope with my curiosity I am able to provide the readers for what their looking for through my blogs. Let’s connect through LinkedIn.