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 (in radians per second), we need to find the linear velocity of the propeller tip first.
The Mach number (M) is defined as the speed of an object relative to the speed of sound (a). It is given by the formula:
M = v / a
where v is the velocity of the object.
In this case, the Mach number is given as 0.8, and the speed of sound at sea level is 340.26 m/s.
Using the formula, we can rearrange it to find the velocity v:
v = M * a
v = 0.8 * 340.26
v ≈ 272.21 m/s
Since the propeller is rotating in a circular motion, the linear velocity at the propeller tip can be calculated using the formula:
v = ω * r
where ω is the angular velocity (in radians per second) and r is the radius of the propeller.
The diameter of the propeller is given as 2.5m, so the radius (r) is half of that:
r = 2.5m / 2
r = 1.25m
Plugging in the values, we have:
272.21 = ω * 1.25
Solving for ω, we get:
ω = 272.21 / 1.25
ω ≈ 217.77 radians per second
Therefore, the rotational speed of the propeller is approximately 217.77 radians per second.
To determine the rotational speed of the propeller in radians per second, we need to find the linear speed of the propeller tip.
The Mach number (M) is defined as the ratio of the velocity of an object to the speed of sound (a). Mathematically, it can be expressed as:
M = V / a
Where:
M = Mach number
V = Velocity of the object
a = Speed of sound
In this case, the Mach number is given as 0.8, and the speed of sound (a) at sea level is given as 340.26 m/s.
Rearranging the equation, we can calculate the linear velocity (V) of the propeller tip at sea level:
V = M * a
Substituting the given values:
V = 0.8 * 340.26
V = 272.21 m/s
Now, the linear speed of the propeller tip is related to the rotational speed (ω) and the radius (r) of the propeller according to the formula:
V = ω * r
where:
V = Linear speed
ω = Rotational speed
r = Radius
The diameter of the propeller is given as 2.5m, so the radius (r) is half the diameter:
r = 2.5 / 2
r = 1.25m
Substituting the values, we can solve for ω:
272.21 = ω * 1.25
Dividing both sides by 1.25:
ω = 272.21 / 1.25
ω ≈ 217.77 rad/s
Therefore, the rotational speed of the propeller is approximately 217.77 radians per second.
The Mach number (M) is defined as the ratio of the speed of an object to the speed of sound in the surrounding medium. In this case, the propeller tip is moving at a speed of M = 0.8 times the speed of sound.
Given that the speed of sound at sea level is a = 340.26 m/s, we can find the speed of the propeller tip by multiplying the speed of sound by the Mach number:
v = M * a
v = 0.8 * 340.26
v = 272.21 m/s
The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. In this case, the radius is half the diameter of the propeller, so r = 2.5m/2 = 1.25m. Therefore, the circumference of the propeller tip is:
C = 2π * 1.25
C = 7.85 m
The rotational speed of the propeller can be determined by dividing the speed of the propeller tip by the circumference of the propeller tip:
ω = v / C
ω = 272.21 / 7.85
ω ≈ 34.67 rad/s
Therefore, the rotational speed of the propeller is approximately 34.67 radians per second.