91Ó°ÊÓ

Torque is a measure of how much a force acting on an object causes it to rotate. Think of it like the rotational equivalent of linear force. When we apply force to twist something, like turning a doorknob or opening a jar, we are applying torque.
Torque depends on two things: the size of the force applied and how far the force is from the pivot point, which is where the object rotates. The formula for torque, \( \tau \), is given by:\[ \tau = r \cdot F \cdot \sin(\theta) \]Where:

• \( r \) is the distance from the pivot point to the point where the force is applied.
• \( F \) is the magnitude of the force.
• \( \theta \) is the angle between the force and the lever arm.

In our exercise, a net torque of \(1.8 \text{ N} \cdot \text{m}\) is applied to the ceiling fan's blades. This torque makes the fan blades spin about their axis.

The moment of inertia determines how difficult it is to change the rotational state of an object, either starting to spin or stopping. It is the rotational analog of mass in linear motion. The larger the moment of inertia, the harder it is to change the rotational state.

The formula for the moment of inertia, \( I \), varies depending on the shape of the object and the axis it rotates around. For simple geometric objects rotating about known axes, such as fans or wheels, standard formulas can be used. The general idea is:\[ I = \sum m_i r_i^2 \]Where:

• \( m_i \) is the mass of a point in the object.
• \( r_i \) is the distance of the point from the axis of rotation.

In this problem, we know the moment of inertia of the fan blades is given as \(0.22 \text{ kg} \cdot \text{m}^{2}\). This value shows the resistance of the blades against rotational motion.

Rotational motion refers to the motion of an object around a center point or axis. Examples include wheels turning, gears rotating, and even planets orbiting the sun. In our exercise, the ceiling fan's blades experience rotational motion as they spin around the central motor axis.

Key quantities that describe rotational motion include:

• Angular velocity: How fast an object spins. It's the rotational equivalent of linear velocity.
• Angular acceleration: The rate of change of angular velocity with respect to time. This is directly related to torque and moment of inertia.

To find the angular acceleration, \( \alpha \), we use the formula:\[ \tau = I \cdot \alpha \]Rearrange it to solve for \( \alpha \):\[ \alpha = \frac{\tau}{I} \]By substituting the known values from the exercise, \( \tau = 1.8 \text{ N} \cdot \text{m} \) and \( I = 0.22 \text{ kg} \cdot \text{m}^{2} \), we find:\[ \alpha = \frac{1.8}{0.22} \approx 8.18 \text{ rad/s}^2 \]This is the angular acceleration experienced by the fan blades.