To find the angular acceleration of the rod about the pivot point, we can use the following steps:

Step 1: Moment of inertia about the pivot

The given moment of inertia is about the center of mass. However, we need the moment of inertia about the pivot point. To do this, we'll use the parallel axis theorem:

$I_{\text{pivot}} = I_{\text{cm}} + m d^2$

where:

• $I_{\text{cm}} = \frac{m \ell^2}{12}$ is the moment of inertia about the center of mass,
• $m = 2.8 \, \text{kg}$ is the mass of the rod,
• $\ell = 1.3 \, \text{m}$ is the length of the rod,
• $d = \frac{\ell}{2} = 0.65 \, \text{m}$ is the distance from the center of mass to the pivot point.

First, calculate the moment of inertia about the center of mass: $I_{\text{cm}} = \frac{2.8 \times (1.3)^2}{12} = 0.395 \, \text{kg} \cdot \text{m}^2$

Now, using the parallel axis theorem: $I_{\text{pivot}} = 0.395 + 2.8 \times (0.65)^2$ $I_{\text{pivot}} = 0.395 + 2.8 \times 0.4225 = 0.395 + 1.183 = 1.578 \, \text{kg} \cdot \text{m}^2$

Step 2: Torque exerted by the force

The torque $\tau$ due to a force $F$ applied at a distance $r$ from the pivot point is given by: $\tau = r F \sin(\theta)$ where:

• $F = 5.1 \, \text{N}$,
• $r = 1.3 \, \text{m}$ (the length of the rod),
• $\theta = 106^\circ$ is the angle between the force and the horizontal.

Now, calculate the torque: $\tau = 1.3 \times 5.1 \times \sin(106^\circ)$ Using $\sin(106^\circ) \approx 0.961$: $\tau = 1.3 \times 5.1 \times 0.961 = 6.381 \, \text{N} \cdot \text{m}$

Step 3: Angular acceleration

Using Newton's second law for rotation, the angular acceleration $\alpha$ is related to the torque by the equation: $\tau = I_{\text{pivot}} \alpha$

Solving for $\alpha$: $\alpha = \frac{\tau}{I_{\text{pivot}}} = \frac{6.381}{1.578} \approx 4.044 \, \text{rad/s}^2$

Final Answer:

The angular acceleration of the rod is approximately $\boxed{4.04 \, \text{rad/s}^2}$.

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Would you like more details or have any questions?

Here are some related questions:

1. How does the parallel axis theorem apply to other shapes, like disks or spheres?
2. What would happen if the force were applied at a different angle?
3. How is torque related to angular momentum?
4. What is the role of angular acceleration in rotational kinetic energy?
5. How does the distribution of mass affect the moment of inertia?

Tip: When calculating torques, always check the perpendicular distance and angle for accuracy!