Radius of Curvature
The radius of curvature, R, is the inverse of the curvature of differential geometry.
It is proportional to the radius of the circular arc that best approximates the curve at that point with a curve. The radius of curvature of a surface is the radius of a circle that best describes a normal section or combinations of normal sections.
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The radius of curvature is the approximate circle's radius at a given point. The radius of curvature of a space curve is equal to the length of the curvature vector. The curvature vector length defines the radius of curvature. When the curve shifts, the radius increases. The radius of curvature, denoted by R, is calculated using the radius of curvature formula is,
below.
R =\[\left | \frac{ds}{d\varphi } \right |=\frac{1}{k}\]
As a result, for plane curves with explicit equations,
The radius of curvature at a point is y =f(x).
R= \[\frac{(1+(\frac{dy}{dx})^{2})^{\frac{3}{2}}}{\left | \frac{d^{2}y}{dx^{2}} \right |}\]
What is radius of curvature?
The radius of curvature is the diameter of the circle created by the curved part of the lens.
We use the curved part of the lens to draw a circle to find its centre we then measure the radius of the circle from the centre to get the radius of curvature.
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The Radius of Curvature Formula Physics
The radius of curvature of a spherical mirror is equal to the radius of the sphere that the reflection surface of the spherical mirror is a part of, which is represented by r'
R = \[\frac{(Velocity)^{2}}{Normal acceleration}\]
How to Find Radius of Curvature?
Square the first derivative,\[\frac{dy}{dx}\], and plug the result into the formula for finding the radius of curvature. Put the result into the formula at\[(\frac{dx}{dy})^{2}\] . In the formula for calculating the radius of curvature, plug in the second derivative of the curve equation. Put the second derivative into the formula at\[(\frac{d^{2}y}{dx^{2}})\] .
Replace the vector "x" with a numerical value to solve the equation for a point "x" in your curve. Use a calculator to speed your calculations to find a curvature.
Examples of the Radius of Curvature
Find the radius of curvature for y = 7x3-x+6 at x=2.
Solution:
y = 7x3-x+6
\[\frac{dy}{dx}\] = 21x2-1
\[(\frac{d^{2}y}{dx^{2}})\] = 42x
R = \[\frac{(1+(\frac{dy}{dx})^{2})^{\frac{3}{2}}}{\left | \frac{d^{2}y}{dx^{2}} \right |}\]
R = \[\frac{(1+(21x^{2}-1)^{2})^{\frac{3}{2}}}{\left | 42x \right |}\]
R = \[\frac{(1+(441x^{4}-42x^{2}+1))^{\frac{3}{2}}}{42x}\]
R = \[\frac{(1+441x^{4}-42x^{2}+1)^{\frac{3}{2}}}{42x}\]
R = \[\frac{(441x^{4}-42x^{2}+2)^{\frac{3}{2}}}{42x}\]
Now, substitute the value of x=2
R = \[\frac{(7056-168+2)^{\frac{3}{2}}}{84}\]
R = \[\frac{(6890)^{\frac{3}{2}}}{84}\]
R = \[\frac{1.6354}{84}\]
R = 1.9469
Hence the value of the radius of curvature is 1.9469
FAQs on Radius of Curvature Formula
1. What is curvature?
The curvature of a curve or the curvature of a surface is the sum from which it deviates from being a straight line or a plane. The circle, which has a curvature equal to the reciprocal of its radius, is the canonical example of a curve.
2. What is the radius of curvature for a parabola?
The radius of curvature of a parabola is given by R =\[\frac{(p+2x)^{\frac{3}{2}}}{p^{\frac{1}{2}}}\], which can be found by differentiation and the general formula for the curvature of a plane curve.
3. Types of curvature
Extrinsic and inherent curvature are the two types of curvature that exist in general.
Extrinsic Curvature
It's a curvature that's a submanifold of a manifold and is dependent on how it's placed.
In addition, torsion of three-dimensional curves and curvature are examples. It also takes into consideration the mean curvature of surfaces in 3 directions.
Inherent Curvature
It's also a curvature, like Gaussian curvature, that can be detected by 2-D (two-dimensional) surface "populations" rather than only outside observers. Furthermore, they are unable to study in three dimensions (three dimensional).
