You have a grindstone (a disk) that is 92.0 kg, has a 0.490-m radius, and is turning at 83.0 rpm, and you press a steel axe against it with a radial force of 25.0 N.
(a)
Assuming the kinetic coefficient of friction between steel and stone is 0.50, calculate the angular acceleration (in rad/s2) of the grindstone. (Indicate the direction with the sign of your answer.)
(b)
How many turns (in rev) will the stone make before coming to rest?
To answer these questions, we can use Newton's second law of motion and the equation for angular acceleration.
(a) Calculate the angular acceleration (in rad/s^2):
The angular acceleration can be calculated using the equation:
τ = Iα
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
The torque can be calculated using the formula:
τ = rF
where r is the radius and F is the force applied.
In this case, the torque is equal to the applied force multiplied by the radius:
τ = (25.0 N) * (0.490 m) = 12.25 N*m
The moment of inertia for a solid disk can be calculated using the formula:
I = (1/2) * m * r^2
where m is the mass and r is the radius.
Substituting the given values, we have:
I = (1/2) * (92.0 kg) * (0.490 m)^2 = 5.6124 kg*m^2
Now, we can find the angular acceleration:
12.25 N*m = (5.6124 kg*m^2) * α
α = 12.25 N*m / (5.6124 kg*m^2) ≈ 2.183 rad/s^2
The angular acceleration of the grindstone is approximately 2.183 rad/s^2, and the direction of the acceleration depends on the sign of the torque. Since torque is a vector quantity and we are given no information about the direction, we cannot determine the direction of the angular acceleration.
(b) Calculate the number of turns before the stone comes to rest:
For a body undergoing constant angular acceleration, the final angular velocity (ωf) can be calculated using the equation:
ωf^2 = ωi^2 + 2αθ
where ωi is the initial angular velocity, α is the angular acceleration, and θ is the displacement.
In this case, the initial angular velocity is given as 83.0 rpm, which needs to be converted to rad/s:
ωi = (83.0 rpm) * (2π rad/1 min) * (1 min/60 s) = 8.68 rad/s
The final angular velocity is zero because the grindstone comes to rest.
Using the equation:
ωf^2 = ωi^2 + 2αθ
0^2 = (8.68 rad/s)^2 + 2 * (2.183 rad/s^2) * θ
Rearranging the equation:
-8.68^2 = 4.366 * θ
θ = -8.68^2 / 4.366 ≈ -17.34 rad
The negative sign indicates the opposite direction of displacement. To convert this to revolutions, divide by 2π:
θ_revolutions = -17.34 rad / (2π rad/rev) ≈ -2.75 rev
Therefore, the grindstone will make approximately -2.75 revolutions before coming to rest.