The maximum speed at which you can spin an object on Earth without it flying off due to centrifugal force depends on various factors, including the mass and size of the object, as well as the specific gravitational force and the material strength of the object.
The centrifugal force experienced by an object rotating on Earth is given by the equation:
F = m * ω^2 * r
Where: F is the centrifugal force, m is the mass of the object, ω (omega) is the angular velocity (rotational speed) in radians per second, and r is the distance from the axis of rotation to the center of the object.
To prevent the object from flying off, the centrifugal force should not exceed the gravitational force acting on it. The gravitational force is given by:
F_gravity = m * g
Where: F_gravity is the gravitational force, m is the mass of the object, and g is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth's surface).
Setting the centrifugal force equal to the gravitational force, we have:
m * ω^2 * r = m * g
Canceling out the mass on both sides of the equation, we get:
ω^2 * r = g
Solving for ω (angular velocity), we have:
ω = √(g / r)
The speed of an object at a specific radius can be calculated by multiplying the angular velocity (ω) by the radius (r). So, the maximum speed without the object flying off can be given by:
v_max = ω * r = √(g * r)
Therefore, the maximum speed at which you can spin an object on Earth without it flying off due to centrifugal force depends on the radius of rotation (r) and the acceleration due to gravity (g).
Keep in mind that this calculation assumes a uniform and rigid object. Objects with different shapes or compositions may have different tolerances for centrifugal forces.