Tangential Acceleration Formula

For an object exhibiting a circular motion, there are always some parameters to describe its nature. 

If we talk about a particle’s velocity, which is an angular velocity, that remains constant throughout the motion; however, angular acceleration makes two types of components and they are tangential and radial acceleration.

Tangential acceleration acts tangentially to the direction of motion of a particle and remains perpendicular to the direction of the radial component. Now, we’ll discuss the tangential and radiation acceleration formula.

Tangential Acceleration and Centripetal Acceleration Formula

Tangential acceleration meaning is a measure of how the tangential velocity of a point at a given radius varies with time. Tangential acceleration is just like linear acceleration; however, it’s more inclined to the tangential direction, which is related to circular motion.

If we talk about the narrow gap between the centripetal acceleration, which is an acceleration that acts towards the center of the circle along which the body or a particle is creating a circular motion.

We can see there is a narrow line of difference between the two types of acceleration, and that the difference lies in the way the acceleration acts on the particle in a circular motion.

Now, let’s discuss the tangential acceleration equation followed by the centripetal acceleration.

Tangential Acceleration Formula

Let’s suppose that you and your friends are playing with a string. You are in the middle of the string and your friends have joined the string from hand to hand and are moving with high speed or changing speed in a circular motion.

Here, we are talking about angular velocity and we know that change in the velocity is called acceleration, which is angular acceleration. So, we can write the first derivative of angular velocity concerning time for angular acceleration. 

Acceleration Tangential Formula 

Here, we aim to describe the tangential acceleration formula, so we will focus more on it, as our article relies on the same. 

Now, writing the tangential acceleration equation in the following manner:

Tangential acceleration \[a_{t} = r \times \frac{\text{d}\omega}{\text{d}t}\]......(1)

So, we denote the tangential acceleration with a subscript ‘ct’ along with the English letter ‘a’. 

  Here, \[\frac{\text{d}\omega}{\text{d}t}\] = angular acceleration

r = is the radius of the circle

Since the motion talks about the position of a particular object that’s why we call ‘r’ as the radius vector.

We also know that the angular velocity can be written, so we can rewrite the above equation (1) to get the Tangential Acceleration Formula Circular Motion in a new form:

at = r𝛼….(2)

Centripetal Acceleration Formula

The centripetal acceleration of an object making a circular motion with a circle ‘r’ and having a speed ‘v’ in meter per second is given by the following centripetal acceleration equation:    

                      aC = v2/r

So, we denote the centripetal acceleration with a subscript ‘c’ along with the English letter ‘a’. 

Now, we will discuss the radial and tangential acceleration formula in detail.

Tangential and Radial Acceleration Formula 

We already discussed the acceleration tangential formula in the above context, while talking about the narrow difference between the centripetal and tangential acceleration, we also saw a minor difference between tangential acceleration and the centripetal acceleration formula.

Now, let’s discuss the radial acceleration:

Radial Acceleration

We define radial acceleration as the component that points along the radius vector, the position vector that points from a center, usually of rotation, and the position of the particle that is accelerating.

The formula for radial acceleration is given by: 

                   ar = v2/r …..(3)

Here, we can see the term ‘r’ or the radius vector has a difference in the tangential acceleration and the centripetal acceleration formula. Also, we notice that the centripetal acceleration and the radial acceleration have the same formula.

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Finding Tangential Acceleration

Now, we will look at one problem to find the tangential acceleration of an object. 

Example: 

If an object is experiencing a circular motion, then what will be its tangential acceleration? Also, determine the overall acceleration of the object.

Answer: 

The overall acceleration of an object is given by the following equation:

\[\vec{a_{(total)}} = \vec{a_{(r)}} + \vec{a_{(t)}}\]

Now, tangential acceleration can be determined by subtracting the radial component acceleration from the overall acceleration in the following manner:   

 \[\vec{a_{(t)}} = \vec{a_{(total)}} - \vec{a_{(r)}}\] 

 \[\vec{a_{(t)}} \theta cap = \vec{a_{(total)}} - \vec{a_{(r)}} r(cap)\]

If we wish to find out the total acceleration in the modulus function, we have the following equation:

\[\vec{a_{(total)}} = |\vec{a_{(total)}}| = \sqrt{a_r^2}+a_t^2\] 

So, the total acceleration is the square root of the sum of the squares of the radial and tangential acceleration.

What are the possibilities for the Value of Tangential Acceleration?

The result of tangential acceleration may have the following three possibilities:

1. The value of tangential acceleration can be greater than zero, that is, positive. This generally happens when the magnitude of the velocity vector increases with time, that is, the body has accelerated motion.

2. The value of tangential acceleration can be less than zero, that is, negative. This happens when the magnitude of the velocity vector decreases with time, that is when the object has slowed or decelerated motion.

3. The value of tangential acceleration can be equal to zero. This happens when the magnitude of the velocity vector remains constant, that is, the object is in uniform motion.

Summary of Tangential Acceleration

The object has a constant speed when it is executing circular motion around a circle of fixed radius in case of uniform circular motion. But if in case, the speed of the object is not constant, this is, changing with time then there will be an additional acceleration applied on the object, that is known as tangential acceleration, and is applied in the direction of the tangent of the circle. The tangential acceleration is calculated as the product of the radius of the circle and angular acceleration. The rate of change of angular velocity is denoted by angular acceleration. The centripetal acceleration is the radial component of the acceleration and is calculated as the square of velocity divided by the radius of the circle. Centripetal acceleration is directed towards the center of the circular path, that is, radially inwards. Both these accelerations, that is, centripetal acceleration and tangential acceleration are mutually perpendicular to each other. Therefore, the total acceleration acting on a particular object is the sum of both the tangential acceleration and the centripetal acceleration.

For example, a particle moving in a circular path of radius 2m has a linear speed equal to the square of time in seconds. The linear speed is given in meters per second. Calculate the radial acceleration and the tangential acceleration on the particle at t = 2s. Also, compute the net acceleration acting on a particle.

Solution: Since, the linear speed of the particle is the square of time in seconds, therefore, the linear speed of the particle at t = 2s will be the square of 2, that is, 4 meters per second. 

The radial acceleration of a particle is calculated as the square of velocity divided by the radius of the circle. Since the velocity as calculated above is 4 m/s, therefore the square of 4 will be 16m/s and the radius of the circle given in the question is 2m. Thus, radial acceleration will be 16 divided by 2, that is, 8 meters per second square.

The tangential acceleration of a particle is calculated by differentiation of velocity divided by time. Since the velocity is square of time, therefore on differentiation, we will get 2t. Putting the value of t as 2sec, the acceleration will be the product of 2 and 2, which is 4 meters per second square.

Now, for calculating the net acceleration, under the center root of the sum of squares of both the acceleration will be taken. On calculating, this will come out to be 8.944 meters per second square.