A small fan has blades that have a radius of 0.0600 m. When the fan is turned on, the tips of the blades have a tangential acceleration of 11.4 m/s² as the fan comes up to speed. What is the angular acceleration a of the blades?
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![### Problem Statement: Angular Acceleration of Fan Blades
A small fan has blades that have a radius of 0.0600 m. When the fan is turned on, the tips of the blades have a tangential acceleration of 11.4 m/s² as the fan comes up to speed. What is the angular acceleration \(\alpha\) of the blades?
#### Given:
- Radius of the blades, \( r = 0.0600 \, \text{m} \)
- Tangential acceleration, \( a_t = 11.4 \, \text{m/s}^2 \)
#### Required:
- Angular acceleration, \( \alpha \)
#### Solution:
To find the angular acceleration, we use the relationship between tangential acceleration (\( a_t \)) and angular acceleration (\( \alpha \)):
\[ a_t = r \alpha \]
Rearranging for \( \alpha \):
\[ \alpha = \frac{a_t}{r} \]
Substitute the given values:
\[ \alpha = \frac{11.4 \, \text{m/s}^2}{0.0600 \, \text{m}} \]
\[ \alpha = 190 \, \text{rad/s}^2 \]
#### Conclusion:
The angular acceleration of the fan blades is \( \alpha = 190 \, \text{rad/s}^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F445dc1ba-0059-4453-a67d-7747e0259e94%2Fb1139c06-cbd4-4015-9dbb-1cf46b1ab096%2F56gf2v_processed.png&w=3840&q=75)
Step by step
Solved in 3 steps with 10 images
