The paraboloid is formed be revolving the shaded area are th

The paraboloid is formed be revolving the shaded area are the x axis. Determine the required
applied moment to revolve the object about the x axis with an angular acceleration ? =5 rad/s2. The
density of the material is ? = 25 kg/m3.

Solution

The equation posted right next to the parabola, that is, y^2 = 13x doesn\'t actually satisfy the values for the right most end of the parabola. That is for x = 82 mm, you cannot possibly get 1255 mm for the y axis.

However, I will solve the problem in terms of variables and use h, as the maximum y and a as the rightmost x coordinate for the parabola. Also, I will take the equation of the parabola as y² = h²x/a, which is a standard equation for a parabola. Once we find the expression, we can find the numerical value by simply substituting the correct values for h and a.

Now let us consider a disc of thickness dx at a distance x from the origin. The radius of this disc is given as the y coordinate on the parabola.

Hence, radius = r = hx/a

hence the moment of inertia of the disc would be dI = [p(h²x/a)dx](h²x/a)/2 [Since MI for disc is MR²/2. Here M = density x volume]

We can integrate the expression above to determine the inertia of the whole body as:

I = dI = [p(h²x/a)dx](h²x/a)/2 = p(h⁴/a²)(a³/6) = p(h⁴a/6)

Therefore the moment required to rotate the given body with angular acceleration of is given as I

NOTE: As I have mentioned above the correct values for h and a are needed to determine the moment. h here is the y coordinate for the rightmost point of the parabola while a is the x-coordinate.