An electric ceiling fan is rotating about a fixed axis with an initial angular velocity magnitude of 0.280 rev/s . The magnitude of the angular acceleration is 0.916 rev/s2 . Both the the angular velocity and angular accleration are directed counterclockwise. The electric ceiling fan blades form a circle of diameter 0.700 m . What is the tangential speed vt of a point on the tip of the blade at time t = 0.197 s ?

To find the tangential speed \(v_t\) of a point on the tip of the blade at time \(t = 0.197 \, \text{s}\), we need to use the relationship between angular speed and tangential speed.

The tangential speed \(v_t\) at a point on the blade is given by:
\[
v_t = \omega \cdot r
\]
where:
- \(\omega\) is the angular velocity in radians per second,
- \(r\) is the radius of the fan blade.

Let's break the solution into steps:

### Step 1: Convert angular quantities to standard units
We are given:
- Initial angular velocity: \(\omega_0 = 0.280 \, \text{rev/s}\),
- Angular acceleration: \(\alpha = 0.916 \, \text{rev/s}^2\).

Since we need angular velocity in radians per second, we can use the conversion:
\[
1 \, \text{rev} = 2\pi \, \text{radians}
\]
Thus:
\[
\omega_0 = 0.280 \times 2\pi \, \text{rad/s} = 1.7593 \, \text{rad/s}
\]
\[
\alpha = 0.916 \times 2\pi \, \text{rad/s}^2 = 5.7545 \, \text{rad/s}^2
\]

### Step 2: Find angular velocity at \(t = 0.197 \, \text{s}\)
The angular velocity \(\omega\) at time \(t\) can be found using the equation:
\[
\omega = \omega_0 + \alpha t
\]
Substituting the known values:
\[
\omega = 1.7593 + (5.7545 \times 0.197)
\]
\[
\omega = 1.7593 + 1.1342 = 2.8935 \, \text{rad/s}
\]

### Step 3: Find the radius of the blade
The radius \(r\) is half of the diameter. Given the diameter of the fan blade is \(0.700 \, \text{m}\), we have:
\[
r = \frac{0.700}{2} = 0.350 \, \text{m}
\]

### Step 4: Calculate the tangential speed
Now, using the formula \(v_t = \omega \cdot r\):
\[
v_t = 2.8935 \times 0.350 = 1.0127 \, \text{m/s}
\]

Thus, the tangential speed at the tip of the blade at time \(t = 0.197 \, \text{s}\) is approximately:
\[
v_t \approx 1.01 \, \text{m/s}
\]

Would you like more details on any part of this solution, or have any questions?

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Here are 5 related questions:
1. What would be the tangential speed at \(t = 1 \, \text{s}\)?
2. How would the tangential speed change if the angular acceleration were doubled?
3. What is the centripetal acceleration of the point on the blade tip at \(t = 0.197 \, \text{s}\)?
4. How long would it take for the fan to reach a tangential speed of \(2 \, \text{m/s}\)?
5. How many revolutions does the fan complete in \(0.197 \, \text{s}\)?

**Tip:** Always check units to ensure consistency when solving physics problems involving motion.

An electric ceiling fan is rotating about a fixed axis with an initial angular velocity magnitude of 0.280 rev/s . The magnitude of the angular acceleration is 0.916 rev/s2 . Both the the angular velocity and angular acceleration are directed counterclockwise. The electric ceiling fan blades form a circle of diameter 0.700 m . What is the tangential speed vt of a point on the tip of the blade at time t = 0.197 s?

Learn how to calculate the tangential speed of a point on the tip of a ceiling fan blade rotating with an angular velocity of 0.280 rev/s and an angular acceleration of 0.916 rev/s². This problem covers key physics concepts like angular velocity, circular motion, and tangential speed. Suitable for Grades 10-12.

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