91Ó°ÊÓ

To determine the linear speed of the helicopter rotor's blade tip, we first need to understand the relationship between the angular velocity and the linear speed. In rotational motion, the linear speed \(v\) is connected to the angular velocity \(\omega\) through the formula \(v = r \omega\), where \(r\) represents the radius, or here, the length of the blade.
In our example, each blade measures 3.40 meters, and that becomes our radius. The rotor spins at an angular velocity of 57.6 radians per second, calculated from converting revolutions per minute to radians per second.
You can compute linear speed with the formula:

• \(v = 3.40 \, \text{m} \times 57.6 \, \text{rad/s}\)

By multiplying these, the answer is 196.0 meters per second.

This linear speed is the speed at which the tip of the blade moves along its circular path. A higher angular velocity or a longer radius increases the linear speed, meaning the blades slice through the air quicker.

Centripetal acceleration is crucial for any object that is moving in a circular path. It's the force that "pulls" an object towards the center of its circular path, allowing it to continue revolving rather than moving off in a straight line.

For the helicopter rotor, centripetal acceleration \(a_r\) is calculated through the formula \(a_r = \frac{v^2}{r}\), where \(v\) is the linear speed and \(r\) is the radius of rotation. The faster the blade tip moves, the higher the centripetal force needed to keep it on its circular path.
Replace the known values with:

• \(v = 196.0 \, \text{m/s} \)
• \(r = 3.40 \, \text{m} \)

You get:

• \(a_r = \frac{(196.0)^2}{3.40}\)

The calculation results in approximately 11288.2 m/s². To express this as a multiple of gravitational acceleration \(g\) (approximately 9.81 m/s²), simply divide:

• \(\frac{11288.2}{9.81} \)

This equals around 1151 times the force of gravity. This high acceleration shows how strong the forces acting on the rotor blades are.

Angular velocity conversion is pivotal when shifting units from rotational to linear motion calculations. Angular velocity \(\omega\) often starts in units of revolutions per minute (rev/min) but calculations in physics favor radians per second (rad/s) for precise measurements.

These two units describe the same motion but require conversion factors. To convert from rev/min to rad/s, multiply by \(2\pi\) (due to a full circle being \(2\pi\) radians) and divide by 60 (since there are 60 seconds in a minute). The formula looks like this:

• \(\omega = \text{rev/min} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}}\)

For the rotor model:

• \(\omega = 550 \times \frac{2\pi}{60}\)

After calculation, you find \(\omega \, \approx \, 57.6 \, \text{rad/s}\).
Remember that radians provide a straightforward measure of angular distance, making this conversion essential for solving problems concerning rotational dynamics, ensuring accuracy and understanding in calculations.