Answer :
A solid ball of mass 2.40 kg and diameter 13.0 cm rotating about its center of mass at 45.0 rev/min, the kinetic energy is 0.17926 J. None of the given option correct.
To find the kinetic energy of the rotating solid ball, we first need to calculate its moment of inertia I and then use the formula for kinetic energy of rotating objects:
KE = (1/2) I ω²
Where:
- KE is the kinetic energy
- I is the moment of inertia
- ω is the angular velocity in radians per second
Given that the ball is rotating about its center of mass, we'll use the formula for the moment of inertia of a solid sphere rotating about its diameter:
I = (2/5) m r²
Where:
- m is the mass of the ball
- r is the radius of the ball (half of the diameter)
First, let's convert the given diameter to radius:
r = 13.0 cm / 2 = 0.065 m
Given:
- m = 2.40 kg
- r = 0.065 m
- ω = 45.0 rev/min
Let's calculate:
I = (2/5) * (2.40 kg) * (0.065 m)²
I = (2/5) * 2.40 * 0.004225
I = (2/5) * 0.01008
I = 0.004032 kg·m²
Now, let's convert the given angular velocity to radians per second:
ω = 45.0 rev/min = (45.0 * 2π) / 60 rad/s
ω = (45.0 * π) / 30 rad/s
ω = 3π rad/s
Now, let's calculate the kinetic energy:
KE = (1/2) * (0.004032 kg·m²) * (3π rad/s)²
KE = (1/2) * 0.004032 * (9π²) J
KE ≈ 0.006048 * 29.608 J
KE ≈ 0.17926 J
So the kinetic energy is 0.17926 J.
Therefore, the correct answer is approximately 0.17926 J. None of the given option correct.
