a model of a helicopter rotor has four blades, each 3.40 m in length from the central shaft to the tip of the blade. The model is rotated in a wind tunnel at 550 rev/min. A) what is the linear speed, in m/s of the blade tip? B.) what is the radial acceleration of the blade top, expressed as a multiple of the acceleration g due to gravity?
angular velocity, ω
= 550 rpm
= 550 * 2π /60
= 55π/3 radians/s
radius, r
= 3.4 m
A.
Linear speed, v
= r ω m/s
= 3.4 * 55 π/3 m/s
= 195.83 m/s
B.
radial acceleration
= r ω2
= 3.4 (55π/3)2 m/s/s
= 11279 m/s/s
= 11279 / 9.81 g
= 1149.7 g
A) Well, let's calculate the linear speed, shall we? The length of each blade is 3.40 m, and the rotor completes 550 revolutions per minute. To find the linear speed, we need to convert revolutions per minute to radians per second. One revolution is equal to 2π radians, and there are 60 seconds in a minute, so:
550 rev/min = (550 rev/min) * (2π rad/rev) * (1 min/60 s) = 57.8 rad/s
Now we can calculate the linear speed of the blade tip. Since the blade tip traces a circle with a radius of 3.40 m, the formula is:
Linear speed = (radius) * (angular speed) = 3.40 m * 57.8 rad/s ≈ 196.3 m/s
So, the linear speed of the blade tip is approximately 196.3 m/s.
B) Now, let's determine the radial acceleration of the blade tip. The radial acceleration is a measure of how much the velocity vector is changing in the radial direction. It can be calculated using the formula:
Radial acceleration = (linear speed)^2 / (radius) = (196.3 m/s)^2 / 3.40 m ≈ 11381 m/s^2
To express this as a multiple of the acceleration due to gravity (g), we divide the radial acceleration by the acceleration due to gravity (approximately 9.8 m/s^2):
Radial acceleration (g's) = (Radial acceleration) / (g) ≈ 11381 m/s^2 / 9.8 m/s^2 ≈ 1161 g
Therefore, the radial acceleration of the blade tip is approximately 1161 times the acceleration due to gravity (g).
To find the linear speed of the blade tip, we can use the formula:
Linear speed = (2πr * number of revolutions per minute) / 60
A) Calculating the linear speed:
Given:
Radius (r) = 3.40 m
Number of revolutions per minute = 550 rev/min
Using the above formula:
Linear speed = (2π * 3.40 * 550) / 60
Linear speed ≈ 357.81 m/s
Therefore, the linear speed of the blade tip is approximately 357.81 m/s.
B) To find the radial acceleration of the blade top, we'll need the formula:
Radial acceleration = (Linear speed)^2 / Radius
Given:
Linear speed = 357.81 m/s
Radius (r) = 3.40 m
Using the above formula:
Radial acceleration = (357.81^2) / 3.40
Radial acceleration ≈ 37738.41 m/s^2
To express it as a multiple of the acceleration due to gravity (g), we can divide by g:
Radial acceleration (expressed as multiple of g) = 37738.41 / 9.8
Radial acceleration (expressed as multiple of g) ≈ 3851.33 g
Therefore, the radial acceleration of the blade top is approximately 3851.33 times the acceleration due to gravity.
To find the linear speed of the blade tip, we need to calculate the circumference of the circle described by the blade tip when it rotates.
A) 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 the length of the blade, which is 3.40 m.
Using the formula, C = 2πr, we substitute the given values:
C = 2 × π × 3.40 m ≈ 21.37 m
The linear speed (v) of the blade tip can be calculated using the formula v = Circumference / Time.
To find the time taken for one revolution, we need to convert the rotational speed from revolutions per minute (rev/min) to revolutions per second. There are 60 seconds in a minute, so we divide the given rotational speed by 60:
550 rev/min ÷ 60 s/min ≈ 9.17 rev/s
Now we can calculate the linear speed:
v = 21.37 m / 9.17 rev/s ≈ 2.33 m/s
Therefore, the linear speed of the blade tip is approximately 2.33 m/s.
B) To calculate the radial acceleration of the blade tip, we need to use the formula for radial acceleration, which is given by ar = v^2 / r, where ar is the radial acceleration, v is the linear speed, and r is the radius of the circle.
Using the given values:
ar = (2.33 m/s)^2 / 3.40 m ≈ 1.60 m/s^2
To express the radial acceleration as a multiple of the acceleration due to gravity (g), we need to divide the radial acceleration by g:
ar / g ≈ 1.60 m/s^2 ÷ 9.8 m/s^2 ≈ 0.16
Therefore, the radial acceleration of the blade tip is approximately 0.16g.