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Answer :
The kinetic energy of the rotating ball is 0.0394 J.
To find the kinetic energy of the rotating ball, we first need to calculate its angular velocity. We know that the ball is rotating about its diameter, so the distance traveled by any point on the ball in one revolution is equal to the diameter of the ball (10 cm). The circumference of the ball is 2*pi*r = 2*pi*(10/2) = 31.4 cm, which is the distance traveled by any point on the ball in one complete revolution.
At 67 revolutions per minute, the angular velocity of the ball can be calculated as follows:
Angular velocity = (67 rev/min) x (2*pi radians/rev) x (1 min/60 sec) = 7.02 radians/sec
Next, we can use the formula for rotational kinetic energy:
Rotational kinetic energy = (1/2) x I x w^2
where I is the moment of inertia of the ball and w is its angular velocity.
The moment of inertia of a solid sphere rotating about its diameter can be calculated as:
I = (2/5) x m x r^2
where m is the mass of the ball and r is its radius (5 cm).
Substituting the given values, we get:
I = (2/5) x 1.6 kg x (5/100)^2 = 0.0004 kg m^2
Now we can calculate the rotational kinetic energy:
Rotational kinetic energy = (1/2) x 0.0004 kg m^2 x (7.02 rad/sec)^2 = 0.0394 J
Therefore, the kinetic energy of the rotating ball is 0.0394 J.
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