To solve this problem, let's analyze the information provided and calculate the magnitude of $F_2$.

Given Data:

1. Radius of disk ($R$) = 1.6 cm = 0.016 m
2. Mass of disk ($m$) = 20 grams = 0.02 kg
3. Initial angular velocity ($\omega_0$) = 0 (since it starts from rest)
4. Final angular velocity ($\omega$) = 280 rad/s (counterclockwise)
5. Time interval ($t$) = 1.2 s
6. Force $F_1$ = 0.096 N

The question requires us to find the magnitude of the force $F_2$.

Approach:

1. Calculate the moment of inertia (I) of the disk: For a uniform disk rotating around its center, the moment of inertia is: $I = \frac{1}{2} m R^2$

2. Calculate the angular acceleration ($\alpha$): Using the formula for angular acceleration: $\alpha = \frac{\omega - \omega_0}{t}$

3. Calculate the net torque ($\tau$): Torque ($\tau$) is related to angular acceleration by: $\tau = I \cdot \alpha$

4. Set up the equation for net torque with $F_1$ and $F_2$: Since the forces are applied tangentially, the torque due to each force is: $\tau = F_1 \cdot R - F_2 \cdot R$ Since we know $\tau$, we can solve for $F_2$.

Let's go through the calculations:

1. Moment of inertia: $I = \frac{1}{2} \times 0.02 \, \text{kg} \times (0.016 \, \text{m})^2 = 2.56 \times 10^{-6} \, \text{kg} \cdot \text{m}^2$

2. Angular acceleration: $\alpha = \frac{280 \, \text{rad/s} - 0}{1.2 \, \text{s}} = 233.33 \, \text{rad/s}^2$

3. Net torque: $\tau = I \cdot \alpha = (2.56 \times 10^{-6} \, \text{kg} \cdot \text{m}^2) \times 233.33 \, \text{rad/s}^2 = 0.000597 \, \text{N} \cdot \text{m}$

4. Torque equation with $F_1$ and $F_2$: $\tau = F_1 \cdot R - F_2 \cdot R$ Substituting $\tau = 0.000597 \, \text{N} \cdot \text{m}$, $F_1 = 0.096 \, \text{N}$, and $R = 0.016 \, \text{m}$: $0.000597 = (0.096) \cdot (0.016) - F_2 \cdot (0.016)$

5. Solve for $F_2$: $F_2 = \frac{(0.096 \cdot 0.016) - 0.000597}{0.016}$ $F_2 = \frac{0.001536 - 0.000597}{0.016} = \frac{0.000939}{0.016} = 0.0587 \, \text{N}$

Answer:

The magnitude of $F_2$ is approximately $0.0587 \, \text{N}$.

Would you like a further breakdown of any part of this solution?

Here are some related questions you might consider:

1. How would the solution change if the disk had a different mass?
2. What if the disk’s radius were larger? How would $F_2$ be affected?
3. How does the direction of the forces impact the angular velocity?
4. Can this approach be applied to non-uniform objects, like a ring?
5. What role does the moment of inertia play in rotational dynamics?

Tip: Always double-check unit conversions, especially when switching from grams to kilograms or centimeters to meters.