Helicopter blades withstand tremendous stresses. in addition to supporting the weight of a helicopter, they are spun at rapid rates and experience large centripetal accelerations, especially at the tip. calculate the centripetal acceleration at the tip of a 4.20 m long helicopter blade that rotates at 270 rev/min.
compare the linear speed of the tip with the speed of sound (taken to be 340 m/s), i.e., calculate the ratio of the linear speed over speed of sound.

A.) 
(Centripetal Acceleration) = (radius)*(angular velocity)^2 
325 rev/min = 34.0339204 radians/second 
(Centripetal Acceleration) = (4.00 m)*(34.0339204 radians/second)^2 
(Centripetal Acceleration) = 4 633.23095 meters/second^2 

b.) 
(tangential velocity) = sqrt((centripetal acceleration)*(radius)) 
just plug them in and get a velocity, then divide that by 340 m/s to get your percentage of the speed of sound, so if you get for example 680, that would be twice the speed of sound or Mach 2

R = ( 24.68 i^ + 16j^ + 25.68k^) lb

Explanation:

The way to work the vectors is to add each component independently of the others and when you have the resulting components the vector is constructed.

Let's write the competency of each vector

X axis

     Ax = 10 lb

     Bx = -3 lb

The vector C is given in the form of magnitude (C = 25 lb) and angle 45º so we will use trigonometry to find its components

      Cos θ = Cx / C

      Cx = C cos θ

      Cx = 25 cos 45

      Cx = 17.68 lb

Axis y

       Ay = 16 lb

       By = 0

       Cy = 0

Z axis

      Az = 6 lb

      Bz = 2 lb

      sin θ = Cz / C

      Cz = C sin θ

      Cz = 25 sin 45

      Cz = 17.68 lb

 

We calculate in the components of the resulting vector

      Rx = Ax + bx + Cx

      Rx = 10 -3 + 17.68

      Rx = 10 -3 + 17.68

      Rx = 24.68 lb

      Ry = Ay + By + Cy

      Ry = 16 + 0 + 0

      Ry = 16 lb

      Rz = Az + Bz + Cz

      Rz = 6 +2 + 17.68

      Rz = 25.68 lb

We build the resulting vector

      R = ( 24.68 i^ + 16j^ + 25.68k^) lb

      R = √ (Rrx² + Ry² + Rz²)

      R = √ (24.68² + 16² + 25.68²)

      R = 39.05 lb

Note that this is a three-dimensional system so we have angles between xy, xz and yz

Let us calculate each angle separately, for this we will use the concept of cosine directors

       Cos α = x / R

       Cos β = y / R

       Cos γ = z / R

       cos α = 24.68 / 39

       α = cos⁻¹ 0.632

       α = 50.8º

       Cos β = 16/39

       β = cos⁻¹ (04097)

       β = 68.8º

      cos γ = 25.68/39

      γ= cos⁻¹ (0.658)