The tip of a 2.5m diameter propeller turns with a Mach number of M = 0.8 at sea level, under ISA conditions. Determine the rotational speed of the propeller (in radians per second).
The speed of sound at sea level is: a = 340.26 m/s.
To determine the rotational speed of the propeller, we need to find the linear speed of the propeller tip first.
The Mach number (M) is defined as the ratio of the speed of an object to the speed of sound in the surrounding medium.
Given:
Diameter of the propeller (D) = 2.5m
Mach number (M) = 0.8
Speed of sound at sea level (a) = 340.26 m/s
The linear speed (v) of the propeller tip can be calculated using the formula:
v = M * a
Substituting the given values, we have:
v = 0.8 * 340.26
v ≈ 272.21 m/s
The linear speed of the propeller tip is 272.21 m/s.
To convert this linear speed to rotational speed, we need to find the circumference of the propeller and divide it by the linear speed.
The circumference (C) of the propeller can be calculated using the formula:
C = π * D
Substituting the given value for the diameter, we have:
C = π * 2.5
C ≈ 7.85 m
The rotational speed (ω) of the propeller can be calculated using the formula:
ω = v / C
Substituting the values, we have:
ω = 272.21 / 7.85
ω ≈ 34.69 rad/s
Therefore, the rotational speed of the propeller is approximately 34.69 radians per second.
To determine the rotational speed of the propeller in radians per second, we need to calculate the linear speed of the propeller tip first.
Given:
Diameter of the propeller = 2.5m
Mach number = 0.8
Speed of sound at sea level = 340.26 m/s
First, we need to calculate the linear speed of the propeller tip. The linear speed is calculated by multiplying the speed of sound with the Mach number.
Linear speed of the propeller tip = Speed of sound * Mach number
Substituting the given values:
Linear speed of the propeller tip = 340.26 m/s * 0.8
Next, we need to calculate the circumference of the propeller, which is the distance traveled in one rotation.
Circumference of the propeller = π * Diameter
Substituting the given value:
Circumference of the propeller = π * 2.5m
Now, we can calculate the time taken to complete one rotation by dividing the circumference by the linear speed.
Time taken to complete one rotation = Circumference / Linear speed
Substituting the calculated values:
Time taken to complete one rotation = (π * 2.5m) / (340.26 m/s * 0.8)
Finally, we can calculate the rotational speed of the propeller in radians per second by taking the reciprocal of the time taken to complete one rotation.
Rotational speed of the propeller = 1 / Time taken to complete one rotation
Simply plug in the calculated value to get your answer.
To determine the rotational speed of the propeller, we need to find the tangential velocity of the propeller tip.
Given:
Diameter of the propeller, d = 2.5m
Mach number, M = 0.8
Speed of sound, a = 340.26 m/s
The tangential velocity of the propeller tip, V, can be calculated using the formula:
V = M * a
Substituting the values, we get:
V = 0.8 * 340.26
V = 272.208 m/s
To find the rotational speed, we need to determine the circumference of the propeller and divide the tangential velocity by the circumference.
The circumference of the propeller, C, can be calculated using the formula:
C = π * d
Substituting the values, we get:
C = π * 2.5
C ≈ 7.85 m
The rotational speed of the propeller, ω, can be calculated using the formula:
ω = V / C
Substituting the values, we get:
ω = 272.208 / 7.85
ω ≈ 34.69 rad/s
Therefore, the rotational speed of the propeller is approximately 34.69 radians per second.