If an airplane propeller rotates at 1570 rev/min while the airplane flies at a speed of 333 km/h relative to the ground, what is the linear speed of a point on the tip of the propeller, at radius 1.97 m, as seen by (a) the pilot and (b) an observer on the ground? The plane's velocity is parallel to the propeller's axis of rotation.

Question

If an airplane propeller rotates at 1570 rev/min while the airplane flies at a speed of 333 km/h relative to the ground, what is the linear speed of a point on the tip of the propeller, at radius 1.97 m, as seen by (a) the pilot and (b) an observer on the ground? The plane's velocity is parallel to the propeller's axis of rotation.

Answer 1

To find the linear speed of a point on the tip of the propeller, we can use the formula:

Linear speed = 2π * radius * rotational speed

(a) To find the linear speed as seen by the pilot, we can use the given rotational speed of the propeller, which is 1570 revolutions per minute (rev/min). However, we need to convert this to radians per second (rad/s) as follows:

Rotational speed in rad/s = rotational speed in rev/min * (2π rad/1 rev) * (1 min/60 s)

Rotational speed in rad/s = 1570 rev/min * (2π rad/1 rev) * (1 min/60 s)

Now, we can substitute the rotational speed into the formula and solve for the linear speed:

Linear speed = 2π * radius * (rotational speed in rad/s)

Linear speed as seen by the pilot = 2π * 1.97 m * [(1570 rev/min) * (2π rad/1 rev) * (1 min/60 s)]

(b) To find the linear speed as seen by an observer on the ground, we need to consider the velocity of the airplane relative to the ground. Given that the airplane flies at a speed of 333 km/h relative to the ground, and the plane's velocity is parallel to the propeller's axis of rotation, the linear speed of the point on the propeller as seen by the observer on the ground would be the sum of the linear speed seen by the pilot and the linear speed due to the forward motion of the airplane.

Linear speed as seen by the observer on the ground = Linear speed as seen by the pilot + Velocity of the airplane

Since the velocity of the airplane is given in km/h, we need to convert it to m/s:

Velocity of the airplane = 333 km/h * (1000 m/1 km) * (1 h/3600 s)

Now we can substitute the values into the equation and solve for the linear speed as seen by the observer on the ground:

Linear speed as seen by the observer on the ground = Linear speed as seen by the pilot + Velocity of the airplane

Note: The result will be in meters per second (m/s).