Final answer:
The angular acceleration of the grindstone is 0.78 rad/s². The stone will make approximately 80 turns before coming to rest.
Step-by-step explanation:
To calculate the angular acceleration of the grindstone, we can use the equation:
τ = Iα
Where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
The moment of inertia for a solid disk rotating about its central axis is given by:
I = (1/2)MR^2
Where M is the mass of the grindstone and R is the radius.
Substituting the values into the equations, we have:
τ = (1/2)MR^2α
Since the torque is equal to the force times the distance from the axis of rotation, we can write:
τ = FR
Where F is the force and R is the radius.
Substituting the values into the equation, we have:
(1/2)MR^2α = FR
Simplifying the equation, we get:
α = (2F)/(MR)
Now, let's calculate the angular acceleration.
α = (2F)/(MR) = (2 * 20.0 N) / (90.0 kg * 0.340 m) = 0.78 rad/s²
To find out how many turns the stone will make before coming to rest, we need to determine the time it takes for the stone to stop.
Using the equation:
ω = ω₀ + αt
Where ω is the angular velocity, ω₀ is the initial angular velocity, α is the angular acceleration, and t is the time.
Since we know that the final angular velocity is 0 (ω = 0), we can rearrange the equation to solve for the time:
t = - ω₀ / α
Substituting the values into the equation:
t = - (90.0 rpm * 2π rad/1 min) / (0.78 rad/s²) = - 18π s ≈ - 56.55 s
Since we can't have a negative time, we can take the absolute value of the time:
t = | - 18π s | ≈ 56.55 s
Lastly, we can calculate the number of turns by dividing the time by the period (T) of one turn:
T = 1 / (90.0 rpm) = (1 min / 90.0) * (60 s / 1 min) = 2π / (90.0 rpm * (2π rad/1 rev)) = 0.706 s/rev
Number of turns = t / T = 56.55 s / 0.706 s/rev ≈ 80 turns