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
To find the angular velocity of the coin as it slides around the inside of the bowl, we need to consider the vertical forces acting on the coin.
The upward force acting on the coin is the normal force exerted by the curved surface of the bowl. This force is equal to the weight of the coin which is given by:
Normal force = Weight = mass * acceleration due to gravity
To find the mass of the coin, we need to determine its volume. Since the coin is a small object, we can approximate its volume using the equation for the volume of a cylindrical object:
Volume = π * r^2 * h
In this case, "h" represents the constant height of the coin above the bottom of the bowl, which is 0.01 m.
Now, we need to find the radius of the coin at the position where it is sliding. We can do this by considering the equation of the curve that forms the surface of the bowl, which is given by y = 100x^4 m^-3.
Setting y = 0.01 m (the height of the coin), we can solve for x:
0.01 = 100x^4
Simplifying the equation, we get:
x^4 = 0.01/100 = 0.0001
Taking the fourth root of both sides, we find:
x ≈ 0.1
So the radius (r) of the coin at this position is approximately 0.1 m.
Now, we can calculate the volume of the coin:
Volume = π * r^2 * h
= π * (0.1)^2 * 0.01
≈ 0.0000314 m^3
Next, we can find the mass of the coin using the volume and the density:
Density = mass / volume
The density of the coin is not given in the question, so we need more information. If we assume a typical density for a metal coin of around 8000 kg/m^3, we can calculate the mass:
mass = density * volume
= 8000 * 0.0000314
≈ 0.251 kg
Finally, we can find the angular velocity:
Angular velocity = (Normal force) / (radius * mass * acceleration due to gravity)
The normal force is equal to the weight of the coin, which is the same as the mass multiplied by the acceleration due to gravity:
Normal force = mass * acceleration due to gravity
= 0.251 * 9.8
≈ 2.4638 N
Now we can calculate the angular velocity:
Angular velocity = (2.4638) / (0.1 * 0.251 * 9.8)
≈ 10.021 rad/s
Therefore, the angular velocity of the coin as it slides around the inside of the bowl is approximately 10.021 rad/s.