What is the angular momentum of a 2.43 kg uniform cylindrical

grinding wheel of radius 12.5 cm when rotating at 1600 rpm? (b) How
much torque is required to stop it in 8.00 s?

To determine the angular momentum of the grinding wheel, we can use the formula:

Angular momentum (L) = Moment of inertia (I) × Angular velocity (ω)

First, we need to calculate the moment of inertia of the cylindrical grinding wheel. The moment of inertia for a uniform cylindrical object is given by the formula:

Moment of inertia (I) = (1/2) × mass (m) × radius (r)^2

Given:
Mass (m) = 2.43 kg
Radius (r) = 12.5 cm = 0.125 m

Calculating the moment of inertia:
I = (1/2) × 2.43 kg × (0.125 m)^2

Now, let's calculate the angular velocity (ω) in radians per second (rad/s) using the formula:

Angular velocity (ω) = 2π × frequency (f)

Given:
Frequency (f) = 1600 rpm

Calculating the angular velocity:
ω = 2π × 1600 rpm / 60 s

Now, we can calculate the angular momentum (L) using the formula mentioned earlier:

L = I × ω

Finally, to calculate the torque required to stop the grinding wheel in 8.00 seconds, we can use the formula:

Torque (τ) = change in angular momentum (ΔL) / time (t)

Since the grinding wheel comes to rest, the change in angular momentum is equal to the initial angular momentum. Therefore, ΔL = L.

Using the given time, t = 8.00 s, we can calculate the torque.

Now, let's calculate all the values step-by-step.

To determine the angular momentum of the grinding wheel, we can use the formula:

Angular momentum (L) = Moment of inertia (I) * Angular velocity (ω)

The moment of inertia for a cylinder can be calculated using the formula:

Moment of inertia (I) = 0.5 * mass (m) * radius^2

First, let's calculate the moment of inertia (I) of the cylindrical grinding wheel:

Given:
Mass (m) = 2.43 kg
Radius (r) = 12.5 cm = 0.125 m

Using the formula, we have:
I = 0.5 * m * r^2
I = 0.5 * 2.43 kg * (0.125 m)^2

Now, let's calculate the angular velocity (ω) in radians per second. We know that 1 revolution is equal to 2π radians, and 1 minute is equal to 60 seconds. So, we can convert the given angular velocity from rpm to rad/s:

Given:
Angular velocity (ω) = 1600 rpm

Using the conversion factor, we have:
ω = 1600 rpm * (2π rad/1 revolution) * (1 min/60 s)
ω = 1600 rpm * (2π rad/1 revolution) * (1/60 s)

Now that we have the moment of inertia (I) and the angular velocity (ω), we can calculate the angular momentum (L):

L = I * ω

Substitute the values into the equation and calculate L.

Once we have the angular momentum (L), we can calculate the torque required to stop the wheel in 8.00 seconds. The formula for torque is:

Torque (τ) = Change in angular momentum (ΔL) / Time interval (Δt)

In this case, the initial angular momentum (L_initial) is the angular momentum calculated in part (a), and the final angular momentum (L_final) is zero because we want to stop the wheel.

Given:
Time interval (Δt) = 8.00 s

Substitute the values into the equation and calculate τ.