A rotating uniform cylindrical platform of mass 240 kg and radius 5.7 m slows down from 3.9 rev/s to rest in 16 s when the driving motor is disconnected.

Incomplete.

To solve this problem, we can use the principles of rotational dynamics.

1. Find the initial angular velocity (ω₀) of the rotating platform:
- Given: ω₀ = 3.9 rev/s
- Convert revolutions to radians: ω₀ = 3.9 rev/s * (2π rad/rev) = 3.9 * 2π rad/s

2. Find the final angular velocity (ω) of the rotating platform:
- Given: ω = 0 (platform comes to rest)
- ω = ω₀ + α * t, where α is the angular acceleration and t is the time taken
- Rearranging the equation, α = (ω - ω₀) / t
- Substituting the values, α = (0 - 3.9 * 2π) rad/s / 16 s

3. Find the angular acceleration (α) of the rotating platform:
- Substituting the values, α ≈ -3.9 * 2π / 16 rad/s²

4. Find the torque (τ) acting on the platform:
- τ = I * α, where I is the moment of inertia of the platform
- The moment of inertia for a uniform cylindrical platform is I = 0.5 * m * r²
- Given: m = 240 kg (mass of the platform) and r = 5.7 m (radius of the platform)
- Substituting the values, I = 0.5 * 240 kg * (5.7 m)²
- τ = (0.5 * 240 kg * (5.7 m)²) * (-3.9 * 2π / 16 rad/s²)

5. Find the torque (τ) in Newton-meters (Nm):
- Since torque is defined as τ = F * r, where F is the force and r is the radius,
- We can find the force F by substituting τ / r: F = τ / r
- Given: r = 5.7 m
- Substituting the values, τ / r = (0.5 * 240 kg * (5.7 m)²) * (-3.9 * 2π / 16 rad/s²) / 5.7 m

6. Calculate the magnitude of the torque (τ):
- With the given values and calculations, you can determine the magnitude of the torque.

Note: The negative sign in the angular acceleration (α) indicates deceleration or slowing down of the rotating platform.

To find the torque acting on the rotating platform, we need to use the equation:

Torque = Moment of inertia × Angular acceleration

First, we need to find the moment of inertia (I) of the rotating platform. The moment of inertia for a uniform cylinder rotating around its central axis is given by the formula:

I = 0.5 × mass × radius^2

Given:
Mass (m) = 240 kg
Radius (r) = 5.7 m

Substituting the values:

I = 0.5 × 240 kg × (5.7 m)^2
I = 0.5 × 240 kg × 32.49 m^2
I = 3887.76 kg·m^2

Now, we need to find the angular acceleration (α) of the rotating platform. We can use the formula:

Angular acceleration = (Final angular velocity - Initial angular velocity) / Time

Given:
Initial angular velocity (ωi) = 3.9 rev/s (convert to rad/s)
Time (t) = 16 s
Final angular velocity (ωf) = 0 rev/s (as it comes to rest)

To convert the initial angular velocity from rev/s to rad/s, we multiply it by 2π radians (1 revolution) (ωi = 3.9 rev/s × 2π rad/rev).

Substituting the values:

Angular acceleration = (0 rad/s - 3.9 rev/s × 2π rad/rev) / 16 s
Angular acceleration = -3.9 × 2π / 16 rad/s^2

Now, we can calculate the torque:

Torque = Moment of inertia × Angular acceleration
Torque = 3887.76 kg·m^2 × (-3.9 × 2π / 16 rad/s^2)

Calculating this expression will give us the value of torque in units of newton-meters.