A metal disk lies in the xy-plane, centered about the origin, and rotates with a constant angular velocity about the z-axis. There is a uniform 0.0314 T magnetic field parallel to the z-axis. The radius of the disk is 1.56 m. At what rate is the disk turning if an E of 3.86 V develops between the center of the disk and a point on its rim

 w = -101 rad / s

Explanation:

For this exercise we will use Faraday's law

          E = - dФ / dt

where the magnetic flux is

          Ф = B. A = B A cos θ

In this case, the angle between the magnetic field and the normal to the disk is zero, cos 0 = 1, they indicate that the field is constant, let's find the area

The area rotated by the disk is

         A = ½ r s

if we express the angles in radians

         θ = s / r

         s = r θ

where is the arc supported

         A = ½ r (r θ)

let us substitute in the Faraday equation

          E = - d (B ½ r² θ) / dt

          E = - ½ B r² dθ/dt

the definition of angular velocity is

          w = dθ/dt

          E = - ½ B r² w

           w = - 2E / B r²

let's calculate

         w = - 2 3.86 / (0.0314 1.56²)

         w = -101 rad / s

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .