A model of a helicopter rotor has four blades, each 3.40 m long from the central shaft to the blade tip. The model is rotated in a wind tunnel at 550 rev/min. (b) What is the radial acceleration of the blade tip expressed as a multiple of g?

Verified step by step guidance

1. Calculate the angular velocity, \(\omega\), of the rotor blades in radians per second. Use the conversion factor that 1 revolution per minute (rev/min) is equivalent to \(\frac{2\pi}{60}\) radians per second. Multiply the given rotational speed by this factor: \(\omega = 550 \times \frac{2\pi}{60}\).
2. Determine the radius, \(r\), of the circle described by the blade tips. This is given directly as the length of each blade, 3.40 m.
3. Calculate the radial (centripetal) acceleration, \(a_r\), using the formula for centripetal acceleration: \(a_r = r\omega^2\), where \(r\) is the radius and \(\omega\) is the angular velocity.
4. Convert the radial acceleration from m/s^2 to multiples of the acceleration due to gravity, \(g\), where \(g\approx 9.81 \, \text{m/s}^2\). This is done by dividing the calculated radial acceleration by \(g\): \(\text{Multiple of } g = \frac{a_r}{g}\).
5. Interpret the result as how many times the acceleration due to gravity, \(g\), the radial acceleration of the blade tip is.

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of the circle. It is calculated using the formula a_c = v^2 / r, where v is the tangential velocity and r is the radius of the circular path. In the context of the helicopter rotor, the blade tips experience centripetal acceleration as they rotate around the central shaft.

Tangential Velocity

Tangential velocity is the linear speed of an object moving along a circular path, calculated as v = ωr, where ω is the angular velocity in radians per second and r is the radius. For the helicopter rotor, the tangential velocity of the blade tips can be determined from the rotor's rotational speed, which is given in revolutions per minute (rev/min).

Gravitational Acceleration (g)

Gravitational acceleration, denoted as g, is the acceleration due to Earth's gravity, approximately 9.81 m/s². In this problem, the radial acceleration of the blade tip is expressed as a multiple of g to provide a comparative measure of the acceleration experienced by the blades relative to the force of gravity. This helps in understanding the magnitude of the forces acting on the rotor blades.