A large grinding wheel in the shape of a solid cylinder of radius 0.450m is free to rotate on a frictionless, vertical axle. A constant tangential force of 180 N applied to its edge causes the wheel to have an angular acceleration of 0.870 rad/s2. (a) What is the moment of inertia of the wheel? (b) What is the mass of the wheel? (c) If the wheel starts from rest, what is its angular velocity after 8.00 s have elapsed, assuming the force is acting during that time?

Nnewton’s 2 Law for rotation

M = I•ε,
where angular acceleration ε=0.87 rad/s²,
the torque M = F•R ,
the moment of inertia I=mR²/2 ,
F•R= m•R²•ε/2,
m=2•F/R•ε=2•180/0.45•0.87=919.54 kg
I=mR²/2=919.54•0.45²/2=93.1 kg•m².
ω= ε•t=0.87•8=6.96 rad/s

To find the moment of inertia of the wheel, we can use the formula:

Moment of Inertia (I) = (Angular Acceleration (α)) / (Angular Velocity (ω))

Given values:
Radius (r) = 0.450 m
Tangential Force (F) = 180 N
Angular Acceleration (α) = 0.870 rad/s²

(a) To find the moment of inertia (I) of the wheel:
We need to find the angular velocity (ω).

First, we use Newton's second law, which states:
Tangential Force (F) = Moment of Inertia (I) * Angular Acceleration (α)

Plugging in the given values:
180 N = I * 0.870 rad/s²

Now, rearrange the equation to solve for I:
I = (180 N) / (0.870 rad/s²)
I ≈ 206.90 kg·m²

Therefore, the moment of inertia of the wheel is approximately 206.90 kg·m².

(b) To find the mass (m) of the wheel:
We can use the formula:

Mass (m) = Moment of Inertia (I) / (Radius (r))²

Plugging in the given values:
m = 206.90 kg·m² / (0.450 m)²
m ≈ 1035.58 kg

Therefore, the mass of the wheel is approximately 1035.58 kg.

(c) To find the angular velocity (ω) after 8.00 s:
We can use the formula:

Angular Velocity (ω) = Initial Angular Velocity (ω₀) + (Angular Acceleration (α)) * (Time (t))

Since the wheel starts from rest, the initial angular velocity is zero (ω₀ = 0).

Plugging in the given values:
ω = 0 + (0.870 rad/s²) * (8.00 s)
ω = 6.96 rad/s

Therefore, the angular velocity of the wheel after 8.00 s is 6.96 rad/s.