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.