A large, solid metal sphere is suspended from a the ceiling by a cable, and is free to rotate about the wire. If the mass of the sphere is 176 kg, its radius is 0.7 m, and it is initially rotating at 28 rpm (revolutions per minute), what work is required to increase its rotational speed to 114 rpm
To find the work required to increase the rotational speed of the sphere, we need to calculate the change in kinetic energy.
The formula for rotational kinetic energy is:
KE_rot = 1/2 * I * ω^2
Where:
KE_rot is the rotational kinetic energy
I is the moment of inertia of the sphere
ω is the angular velocity
The moment of inertia for a solid sphere rotating about its diameter is given by:
I = (2/5) * m * r^2
Where:
m is the mass of the sphere
r is the radius of the sphere
First, let's calculate the initial rotational kinetic energy:
KE_rot_initial = 1/2 * I_initial * ω_initial^2
We are given:
m = 176 kg
r = 0.7 m
ω_initial = 28 rpm = 28 rev/min = (2π * 28) rad/min = (2π/60 * 28) rad/s
Substituting these values into the equation, we can calculate the initial rotational kinetic energy.
Next, let's calculate the final rotational kinetic energy:
KE_rot_final = 1/2 * I_final * ω_final^2
We are given:
ω_final = 114 rpm = 114 rev/min = (2π * 114) rad/min = (2π/60 * 114) rad/s
Substituting these values into the equation, we can calculate the final rotational kinetic energy.
Finally, the work required to increase the rotational speed is the difference between the final and initial rotational kinetic energies:
Work = KE_rot_final - KE_rot_initial
By solving this equation, we can find the work required to increase the rotational speed of the sphere.