Radius of curvature. Hello friends, today it’s all about the radius of curvature of any curve. Have a look!!

Radius of curvature_compressed

**Radius of curvature**

Well, it’s an old topic from high school. So I’ll not go into much detail.

Suppose is the equation of any curve.

Now the equation of the radius of curvature at any point is

(1)

Next I will give you an example.

**Example of the radius of curvature of any curve**

Disclaimer: This example does not belong to me. I have chosen it from some book. I have also given the due reference at the end of the post.

So here is the example.

**Example **

According to Stroud and Booth (2013)* “If with prove that the radius of curvature at the point is ”

**Solution**

Now here the equation of the curve is And I have to prove that the radius of curvature at the point is

So I’ll start with the differentiation.

You can also check out the **rules for differentiation**

**Step 1 **

First of all, I’ll differentiate with respect to .

Thus it will be

\end{equation*}

As I can see, here is a constant with a value greater than .

On the right hand side, is a product of two functions: and .

So, for that, I’ll use the product rule of differentiation.

You can also check out the * product rule of differentiation*.

Thus it will be

And that gives

Therefore I can say

(2)

Next, I’ll differentiate equation (2) with respect to to get the value of

**Step 2**

Now I’ll differentiate equation (2) with respect to .

So it will look like

Again I’ll use product rule of differentiation on both sides to get

Then I’ll simplify it. So this gives

Therefore I can say

(3)

Next, I’ll get the radius of curvature at the point

And for that, I’ll start with the value of at the point

**Step 3**

Now I’ll put and in equation (2).

Hence it will be

Next, I’ll simplify it to get

So I can say that the value of at the point is

(4)

Now I’ll get the value of at the point

**Step 4**

Therefore I’ll substitute and in equation (3).

And that gives

Next, I’ll simplify it to get

Now I’ll divide both sides with . So it gives

Therefore will be

Thus the value of at the point as

(5)

Now comes the last step of this problem.

**Step 5**

Next, I’ll substitute the values of and at the point from equations (4) and (5) to equation (1).

So the radius of curvature of the given curve at the point will be

Now I’ll simplify it to get

So this means

Hence I have proved that the radius of curvature of this curve at the point is

This is the answer to the given example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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