The angular speed of the ball is approximately 13.1 radians per second.
To determine the angular speed of the ball, we'll utilize the principle of conservation of energy.
The work done by the player is converted into rotational kinetic energy of the ball.
The formula for rotational kinetic energy is:
KE = (1/2) * I * ω^2
where:
KE is the rotational kinetic energy (in Joules)
I is the moment of inertia (in kg·m²)
ω is the angular speed (in radians per second)
Given that the work done is 1.97 J, we can set up the equation:
1.97 J = (1/2) * I * ω^2
We need to determine the value of I, the moment of inertia of the basketball.
For a solid sphere, the moment of inertia is given by:
I = (2/5) * MR²
where:
M is the mass of the basketball (in kilograms)
R is the radius of the basketball (in meters)
Assuming the basketball is a standard men's basketball, the mass is approximately 0.624 kg and the radius is approximately 0.121 m. Plugging these values into the equation for I, we get:
I = (2/5) * 0.624 kg * (0.121 m)² ≈ 0.026 kg·m²
Now we can substitute the values of I and KE into the original equation and solve for ω:
1.97 J = (1/2) * 0.026 kg·m² * ω^2
Solving for ω, we get:
ω ≈ 13.1 rad/s
Therefore, the angular speed of the ball is approximately 13.1 radians per second.