00:01
In this problem, we are as to find the curvature and the radius of curvature at the stated point.
00:07
We are given that r of t is equal to 5 cost t i vector plus 11 sine t j vector plus 4 t g vector at t is equal to 5 by 2.
00:26
It is known that the curvature kappa can be given by the ratio r -dash cross r -double -dash divided by r -dash the whole cube.
00:40
So we need to find the value of r -dash and r -double -dash at t is equal to 5 by 2.
00:49
Let's do that.
00:50
If we differentiate r of t with respect to t, we will get the value of r -dash and it will be equal to minus 5 sine t i vector plus 11 cost t j vector plus 4 k vector let's consider this equation as equation number 1 and similarly let's find the value of r double dash that is we need to differentiate r dash with respect to t if we do that we will get the value of r double dash as minus 5 cost t i vector minus 11 sine t j vector plus 0 k vector let's consider this equation as equation number 2 now we need to find the value of r dash and r double dash at t is equal to pi by 2 so the equation number 1 becomes r dash of pi by 2 and it will be equal to minus 5 i plus 0 j plus 4 k this will be our equation number 3 and the second equation will become r double dash of 5 by 2 and it will be equal to 0 i vector minus 11 j vector plus 0 k vector now let's find the value of r dash cross or double dash that is we have to find the value of determinant of the matrix i j k minus 5 0 4 the coefficients of equation number 3 and this third row will be 0 minus 11 0 which is the coefficient of equation number 4.
02:48
So the determinant of this matrix will be equal to 44 i vector minus 0 j vector plus 55 .5.
03:01
K vector.
03:03
Now to find the value of kappa we need to find the value of modulus of r -dash cross or double -dash and it will be equal to root of 44 square plus 55 square and it will be equal to 11 root 41.
03:25
So we have found the value of this.
03:28
We need to find the value of modulars of r dash and it will be equal to root of minus 5 the whole square which is equal to 25 plus 0 square which is 0 plus 4 square that is 16 so it will be equal to root of 41 now we found the required values to find the value of curvature so using these two informations we can find the value of kappa and it will be equal to 11 root 41 divided by root 41 the whole cube...