How much torque is required to accelerate a uniform sphere of mass 10,000.0 kg and radius 2.5 meters from rest to an angular velocity of 60 rpm in a time of 90 seconds?
To calculate the torque required to accelerate a uniform sphere, we need to use the formula:
Torque = Moment of Inertia * Angular Acceleration
First, let's calculate the moment of inertia of a uniform sphere. The moment of inertia for a solid sphere rotating around an axis through its center is given by the formula:
I = (2/5) * m * r^2
Where:
I is the moment of inertia
m is the mass of the sphere
r is the radius of the sphere
Plugging in the given values:
m = 10,000.0 kg
r = 2.5 meters
I = (2/5) * 10,000.0 kg * (2.5 meters)^2
I = 2,000 kg * 6.25 m^2
I = 12,500 kg * m^2
Next, let's calculate the angular acceleration. The angular acceleration is given by:
Angular Acceleration = (Final Angular Velocity - Initial Angular Velocity) / Time
Plugging in the given values:
Initial Angular Velocity = 0 (since the sphere starts from rest)
Final Angular Velocity = 60 rpm = 60 revolutions per minute = 60 * 2π radians per minute (since 1 revolution = 2π radians)
Final Angular Velocity = 60 * 2π * (1/60) rps = 2π rad/s
Time = 90 seconds
Angular Acceleration = (2π rad/s - 0 rad/s) / 90 s
Angular Acceleration = (2π rad/s) / 90 s
Angular Acceleration = π / 45 rad/s^2
Now, we can calculate the torque:
Torque = Moment of Inertia * Angular Acceleration
Torque = 12,500 kg * m^2 * (π / 45 rad/s^2)
Torque = 277.78 kg * m^2 * rad / s^2
Therefore, the torque required to accelerate the uniform sphere as described is 277.78 kg * m^2 * rad / s^2.