Question

You have a grindstone (a disk) that is 80.0 kg, has a 0.490-m radius, and is turning at 63.0 rpm, and you press a steel axe against it with a radial force of 15.0 N. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.40, calculate the angular acceleration (in rad/s2) of the grindstone. (Indicate the direction with the sign of your answer.) rad/s2 (b) How many turns (in rev) will the stone make before coming to rest? rev +

Answer 1

(a) The angular acceleration of the grindstone is 1.246 rad/s². (b) The stone will make approximately 1875.381 turns before coming to rest.

Step-by-step explanation:

(a) 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 torque can be calculated as the product of the radial force and the radius:

τ = Fr

Substituting the given values:

τ = (15 N)(0.490 m) = 7.35 N·m

The moment of inertia of a disk can be calculated as:

I = (1/2)MR²

Substituting the given values:

I = (1/2)(80.0 kg)(0.490 m)² = 5.892 kg·m²

Now we can rearrange the equation for torque to solve for angular acceleration:

α = τ / I

Substituting the values we calculated:

α = 7.35 N·m / 5.892 kg·m² = 1.246 rad/s²

The angular acceleration of the grindstone is 1.246 rad/s². Since the torque and force are in opposite directions, the angular acceleration is negative.

(b) To calculate the number of turns the stone will make before coming to rest, we can use the equation:

ω² = ω₀² + 2αθ

Where ω is the final angular velocity, ω₀ is the initial angular velocity (given as 63.0 rpm), α is the angular acceleration, and θ is the angle of rotation.

At rest, the final angular velocity is 0. Rearranging the equation, we get:

θ = (ω² - ω₀²) / (2α)

Substituting the given values:

θ = (0 - (63.0 rpm)²) / (2(1.246 rad/s²)) = -1875.381 rad

The stone will make approximately 1875.381 turns before coming to rest.

Answer 2

The angular acceleration of the grindstone is 1.153 rad/s². The stone will make approximately 8 turns before coming to rest.

Step-by-step explanation:

We can use the following formula to determine the grindstone's angular acceleration: τ = Iα. where α is the angular acceleration, I is the moment of inertia, and τ is the torque.

The torque can be calculated using the equation:

τ = rF

Where r is the radius and F is the force.

Substituting the given values, we have:

τ = (0.340m)(20.0N) = 6.8 Nm

The moment of inertia of a disk is given by:

I = (1/2)mr²

Substituting the given values, we have:

I = (1/2)(90.0kg)(0.340m)² = 5.899 kgm²

Now, we can rearrange the equation τ = Iα and solve for α:

α = τ/I

Substituting the values, we have:

α = (6.8 Nm)/(5.899 kgm²) = 1.153 rad/s²

Therefore, the angular acceleration of the grindstone is 1.153 rad/s². (b) To calculate how many turns the stone will make before coming to rest, divide the initial angular velocity by the angular acceleration:

Number of turns = initial angular velocity / angular acceleration

Given that the initial angular velocity is 90.0 rpm, we convert it to rad/s by multiplying by (2π/60):

Initial angular velocity = 90.0 rpm × (2π/60) = 9.425 rad/s

Substituting the values, we have:

Number of turns = 9.425 rad/s / 1.153 rad/s² = 8.17 turns

Therefore, the stone will make approximately 8 turns before coming to rest.