PHYSICS WORD PROBLEM-TORQUE

Question

A tradesman sharpens a knife by pushing it against the rim of a grindstone. The 50 cm diameter stone is spinning at 200 rpm and has a mass of 29 kg. The coefficient of kinetic friction between the knife and the stone is 0.20. If the stone loses 10% of its speed in 10 s of grinding, what is the force with which the man presses the knife against the stone?

1. What are the initial and final angular velocities of the grindstone in radians/second?
2. What was the angular acceleration that causes this change in angular velocity?
3. What is the moment of inertial of the grindstone?
4. What torque must be applied to the grindstone to produce this angular acceleration?
5. What force must be applied to give the kinetic friction that causes this torque?

Answer 1

Let's solve this problem step by step:

**Step 1:** Calculate the initial and final angular velocities of the grindstone in radians per second.

Given:

- Diameter of the stone (D) = 50 cm = 0.5 m

- Radius of the stone (R) = D/2 = 0.5 m

- Initial angular velocity (ω_initial) = 200 rpm

- Time (t) = 10 s

- Percentage speed loss (Δv) = 10%

First, convert the initial angular velocity from rpm to radians per second:

ω_initial = (200 rpm) * (2π rad/1 min) * (1 min/60 s) ≈ 20.94 rad/s

Now, calculate the final angular velocity:

The percentage speed loss is 10%, which means the final speed is 90% of the initial speed:

Final angular velocity (ω_final) = 0.90 * ω_initial

**Step 2:** Calculate the angular acceleration (α) that causes this change in angular velocity.

Use the equation for angular acceleration:

α = (ω_final - ω_initial) / t

Plug in the values:

α = (ω_final - 20.94 rad/s) / 10 s

**Step 3:** Calculate the moment of inertia (I) of the grindstone.

The moment of inertia for a solid disk rotating about its axis is given by:

I = (1/2) * m * R^2

Where:

- m = Mass of the grindstone = 29 kg

- R = Radius of the grindstone = 0.5 m

**Step 4:** Calculate the torque (τ) that must be applied to the grindstone to produce this angular acceleration.

Use the rotational analog of Newton's second law:

τ = I * α

Substitute the values:

τ = (1/2) * 29 kg * (0.5 m)^2 * α

**Step 5:** Calculate the force (F_friction) that must be applied to give the kinetic friction that causes this torque.

The torque is also related to the force applied at a distance from the center of rotation (the radius R):

τ = F_friction * R

Solve for F_friction:

F_friction = τ / R

Now, we have all the necessary values to calculate F_friction.

Once you have calculated F_friction, that is the force with which the man presses the knife against the stone, as it must equal the frictional force to keep the stone grinding.