# Tangential & Normal acceleration in Circular Motion

• Shreya
In summary, the notes on curvature describe how the acceleration of a moving object is split up into the tangential and normal components. The normal acceleration is the component of the acceleration that is normal to the velocity, and the tangential acceleration is the component of the acceleration that is parallel to the velocity.
Shreya
Homework Statement
Can I get an intuition (or a derivation) of the equations in the coloured boxes. Please refer the image.
Please be kind to help
Relevant Equations
Please Refer image

The tangential acceleration is the component of the acceleration that is parallel to the velocity.
For any two vectors ##\frac{\vec x.\vec y}{|\vec y|}## is the magnitude of the component of ##\vec x## parallel to ##\vec y## , and ##\frac{\vec y}{|\vec y|}## is the unit vector parallel to ##\vec y##, so the product of the two ##\frac{\vec x.\vec y}{|\vec y|}\frac{\vec y}{|\vec y|}## is the component of ##\vec x## parallel to ##\vec y##. That yields the first part of the first circled equation.

The normal (or centripetal) acceleration is the component normal to the velocity, the tangential and normal accelerations add vectorially to give the whole acceleration. I'll have to think more on how to view the vector equation for that.

Shreya
Given any vector ##\vec A## and any vector ##\vec V##,
one can write ##\vec A## as the sum of two vectors:
one parallel to ##\vec V## and the other (the rest of ##\vec A##) perpendicular to ##\vec V##.

\begin{align*}
\vec A
&=\vec A_{||V} + (\vec A - \vec A_{||V})
\end{align*}Now,
\begin{align*}
\vec A_{||V}
&= (\vec A \cdot \hat V) \hat V\\
&= (\vec A \cdot \frac{\vec V}{V}) \frac{\vec V}{V}
\end{align*}
[As a special case, let $\hat V = \hat x$.]
(Check that $\vec A_{||V}\cdot \hat V=(\vec A \cdot \hat V)$
and that $(\vec A - \vec A_{||V})\cdot \hat V=0$.)

For a trajectory, $\hat V$ is tangent to the trajectory and (for a planar curve)
$\hat V_{\bot}$ is normal to the trajectory.

So, if $\vec A$ is the acceleration vector ($\vec A =\frac{d}{dt}\vec V$),
then $\vec A_{||V}$ is the tangential-part of the acceleration vector (responsible for changing the magnitude of $\vec V$ [i.e. speeding up or slowing down]),
and the other part is the normal-part of the acceleration vector (responsible for changing the direction of $\vec V$ [turning the velocity vector] ).

Next, one has to express these in terms of the angular pseudovectors $\vec \alpha$ and $\vec \omega$ [probably should use a different "arrowhead"] (which when crossed with ordinary [polar] vectors result in ordinary [polar] vectors).

You'll probably [implicitly] use the "BAC-CAB" rule https://en.wikipedia.org/wiki/Triple_product#Vector_triple_product
https://mathworld.wolfram.com/BAC-CABIdentity.html

Last edited:
Shreya and PeroK
haruspex said:
The tangential acceleration is the component of the acceleration that is parallel to the velocity.
That makes sense now. Thanks haruspex. About the second part of 1st equation: I think they are multiplying the perpendicular component of angular acceleration with radius vector giving dv/dt.

robphy said:
pseudovectors
What is meant by pseudovectors?

PeroK said:
Everything is online these days:
True, thanks a lot for the link though!

I think I get both the equations now! Thanks a lot PeroK, robphy and Haruspex! I really appreciate your help.

Shreya

## 1. What is tangential acceleration in circular motion?

Tangential acceleration is the rate of change of tangential velocity in circular motion. It is the component of acceleration that is parallel to the direction of motion and is responsible for the change in speed of an object moving in a circular path.

## 2. How is tangential acceleration calculated?

Tangential acceleration can be calculated using the formula at = rα, where a is the tangential acceleration, r is the radius of the circular path, and α is the angular acceleration.

## 3. What is normal acceleration in circular motion?

Normal acceleration is the component of acceleration that is perpendicular to the direction of motion in circular motion. It is responsible for the change in direction of an object moving in a circular path.

## 4. How is normal acceleration related to tangential acceleration?

Normal acceleration and tangential acceleration are always perpendicular to each other, and together they make up the total acceleration of an object in circular motion. As tangential acceleration increases, normal acceleration decreases, and vice versa.

## 5. Can an object have a constant speed and still have tangential and normal acceleration?

Yes, an object can have a constant speed in circular motion while still experiencing tangential and normal acceleration. This is because acceleration is a vector quantity and can change in direction even if the magnitude remains constant.

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