A small coin is inside a bowl. The bowl is a surface of revolution of the curve y=100x^4 m^-3. This coin slides around the inside of the bowl at a constant height of y=0.01 m above the bottom of the bowl. What is its angular velocity in rad/s?

Accel due to gravity = 9.8 m/s^2

5.91

it is wrong

To find the angular velocity of the coin, we need to consider the forces acting on it. In this case, the coin is sliding around the inside of the bowl at a constant height above the bottom. The only force acting on it is the gravitational force.

The gravitational force can be written as F = m * g, where m is the mass of the coin and g is the acceleration due to gravity. Since the coin is sliding at a constant height, the gravitational force is balanced by the centrifugal force.

The centrifugal force can be written as F = m * ω^2 * r, where ω is the angular velocity and r is the radius of curvature of the bowl at the height of the coin.

To find the radius of curvature (r), we need to differentiate the equation of the curve with respect to x:

dy/dx = 400x^3

To find the radius of curvature at the height of the coin (y = 0.01 m), we can use the formula:

r = (1 + (dy/dx)^2)^(3/2) / |d^2y/dx^2|

Substituting the given values:

r = (1 + (400(0.01)^3)^2)^(3/2) / |d^2y/dx^2|

Now, we need to find the second derivative of y with respect to x:

d^2y/dx^2 = 1200x^2

Substituting the given values:

d^2y/dx^2 = 1200(0.01)^2 = 0.12 m^-2

Substituting the values of r and g into the equation for the centrifugal force:

m * g = m * ω^2 * r

Simplifying the equation:

g = ω^2 * r

Solving for ω:

ω = √(g / r)

Substituting the given values of g = 9.8 m/s^2 and r, which we previously calculated:

ω = √(9.8 / r)

Now, we can plug in the value of r into the equation to find the angular velocity (ω) in rad/s.