Two identical uniform spheres of radius R=29 cm and mass M=188 kg are stuck to each other and rotate at the rate of 28 rev/min about an axis that goes through the point of their contact and is perpendicular to the line connecting the sphere centers. What is the kinetic energy of this rotational motion?
To find the kinetic energy of this rotational motion, we need to find the moment of inertia and the angular velocity of the rotating spheres.
The moment of inertia of a uniform sphere rotating about an axis passing through its center is given by the formula:
I = (2/5) * m * r^2
where m is the mass of the sphere and r is its radius.
In this case, since we have two identical spheres stuck to each other, we can consider it as a single system with the same mass and radius. So, the moment of inertia of the system is:
I = (2/5) * 2m * r^2 = (4/5) * m * r^2
Substituting the given values:
m = 188 kg
r = 29 cm = 0.29 m
I = (4/5) * 188 kg * (0.29 m)^2 = 29.422 kg*m^2
Next, we need to convert the given rotational speed from revolutions per minute to radians per second. We know that 1 revolution is equal to 2π radians.
So, the angular velocity (ω) in radians per second is:
ω = (28 rev/min) * (2π rad/rev) * (1 min/60 s) = 2.933 rad/s
Finally, we can calculate the kinetic energy (KE) of rotational motion using the formula:
KE = (1/2) * I * ω^2
KE = (1/2) * 29.422 kg*m^2 * (2.933 rad/s)^2 = 1279.88 J
Therefore, the kinetic energy of this rotational motion is approximately 1279.88 Joules.