A 6-kg bowling ball with a radius of 14 cm is spinning about a vertical axis (with its center of mass motionless) at an initial rate of 3 turns per second. Friction with the floor, however, causes the bowling ball to rotate more and more slowly until it stops after about 22s. What was the average torque that the friction interaction applied to the bowling ball?
To find the average torque applied by friction to the bowling ball, we can use the equation:
τ = Iα
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
Step 1: Calculate the initial angular velocity.
The initial angular velocity (ω₁) in radians per second can be calculated using the formula:
ω₁ = 2πf
where f is the initial rate of turns per second.
Plugging in the given value:
ω₁ = 2π(3) = 6π rad/s
Step 2: Calculate the final angular velocity.
The final angular velocity (ω₂) is zero, as the bowling ball comes to rest.
ω₂ = 0 rad/s
Step 3: Calculate the angular acceleration.
The angular acceleration (α) can be calculated using the formula:
α = (ω₂ - ω₁) / t
where t is the time taken for the ball to come to rest.
Plugging in the given values:
α = (0 - 6π) / 22 = -6π / 22 rad/s²
Step 4: Calculate the moment of inertia.
The moment of inertia (I) for a solid sphere rotating about its central axis is given by the formula:
I = (2/5) * m * r²
where m is the mass of the sphere and r is the radius.
Plugging in the given values:
I = (2/5) * 6 kg * (0.14 m)² = 0.03744 kg·m²
Step 5: Calculate the average torque.
Using the equation τ = Iα, plug in the calculated values:
τ = (0.03744 kg·m²) * (-6π / 22 rad/s²)
Evaluating this expression will give us the average torque applied by friction to the bowling ball.
To find the average torque applied by friction to the bowling ball, we can use the equation:
Δω = (Δt * τ) / I
Where:
Δω is the change in angular velocity (in radians per second),
Δt is the change in time (in seconds),
τ is the torque (in Newton-meters), and
I is the moment of inertia (in kg·m^2).
Since the bowling ball stops rotating after 22 seconds, the change in angular velocity, Δω, is simply the initial angular velocity, ω_i, which is given as 3 turns per second, converted to radians per second:
ω_i = 3 turns/s = 3 * 2π rad/1 turn = 6π rad/s
Also, the moment of inertia of a solid sphere can be calculated using the formula:
I = (2/5) * m * r^2
Where:
m is the mass of the ball,
r is the radius of the ball.
Plugging in the values:
m = 6 kg
r = 0.14 m (converted from 14 cm to meters)
I = (2/5) * 6 kg * (0.14 m)^2 = 0.09408 kg·m^2 (approximately)
Since the bowling ball comes to rest (Δω = 0), we can rearrange the equation to solve for torque:
τ = (Δω * I) / Δt
τ = (6π rad/s * 0.09408 kg·m^2) / 22 s
Calculating this expression gives us the average torque applied by friction to the bowling ball.