Given Data:

• Radius of the grinding wheel, $r = 0.330 \, \text{m}$
• Tangential force, $F_t = 210 \, \text{N}$
• Angular acceleration, $\alpha = 0.854 \, \text{rad/s}^2$
• Time, $t = 6.00 \, \text{s}$

Relevant Equations:

1. Torque: $\tau = I \alpha$
2. Relationship between torque and force: $\tau = F_t r$
3. Moment of inertia for a solid cylinder: $I = \frac{1}{2} M r^2$
4. Angular velocity: $\omega = \omega_0 + \alpha t$, where $\omega_0$ is the initial angular velocity.

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(a) Moment of Inertia

From $\tau = I \alpha$ and $\tau = F_t r$, we can write: $I \alpha = F_t r \quad \implies \quad I = \frac{F_t r}{\alpha}$

Substitute the values: $I = \frac{210 \cdot 0.330}{0.854} = 81.15 \, \text{kg·m}^2$

So, the moment of inertia is: $I = 81.15 \, \text{kg·m}^2$

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(b) Mass of the Wheel

For a solid cylinder, $I = \frac{1}{2} M r^2$. Rearranging for $M$: $M = \frac{2I}{r^2}$

Substitute $I = 81.15 \, \text{kg·m}^2$ and $r = 0.330 \, \text{m}$: $M = \frac{2 \cdot 81.15}{(0.330)^2} = \frac{162.3}{0.1089} = 1490.58 \, \text{kg}$

So, the mass is: $M = 1490.58 \, \text{kg}$

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(c) Angular Velocity after 6.00 seconds

From the angular velocity equation: $\omega = \omega_0 + \alpha t$

Since the wheel starts from rest ($\omega_0 = 0$), the equation becomes: $\omega = \alpha t$

Substitute $\alpha = 0.854 \, \text{rad/s}^2$ and $t = 6.00 \, \text{s}$: $\omega = 0.854 \cdot 6.00 = 5.124 \, \text{rad/s}$

So, the angular velocity after 6 seconds is: $\omega = 5.124 \, \text{rad/s}$

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Final Answers:

(a) Moment of inertia: $81.15 \, \text{kg·m}^2$
(b) Mass: $1490.58 \, \text{kg}$
(c) Angular velocity after 6.00 seconds: $5.124 \, \text{rad/s}$

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Would you like a step-by-step breakdown or have any additional questions?
Here are some related questions for practice:

1. How would the moment of inertia change if the wheel's radius doubled?
2. What is the tangential velocity of a point on the edge after 6 seconds?
3. What would happen to the angular acceleration if the applied force is halved?
4. How much work is done by the tangential force over 6 seconds?
5. If the wheel were hollow, how would its mass and moment of inertia change?

Tip: Always confirm the units when calculating quantities like angular velocity and moment of inertia to avoid mistakes!