GATE 2010 AEROSPACE ENGINEERING (Q 31-40)

  1. In a certain region a hill is described by the shape z(x,y) = (1/50)x4 + y2– xy – 3y, where the axes x and y are in the horizontal plane and axis z points vertically upward. If î, ĵ and k̂ are unit vectors along x, y and z, respectively, then at point x = 5, y = 10 the unit vector in the direction of the steepest slope of the hill be:
    1. î
    2. ĵ
    3. î + ĵ + k̂

    Answer:- ĵ
    z(x,y) = (1/50)x4 + y2 – xy – 3y
    ⇒ ∂z/∂x = (2/25)x3 – y & ∂z/∂y = 2y – x -3
    At point (5,10), ∂z/∂x = 0 & ∂z/∂y = 12
    Therefore, ∇z = (∂z/∂x)î + (∂z/∂y)ĵ = 12ĵ
    ⇒ unit vector in the direction of the steepest slope of the hill is ĵ
    The gradient of p, ∇p, at a given point in space is defined as a vector such that:

    1. Its magnitude is the maximum rate of change of p per unit length of the coordinate space at the given point
    2. Its direction is that of the maximum rate of change of p at the given point.
  2. An aircraft is cruising at an altitude of 9 km. The free-stream static pressure and density at this altitude are 3.08 × 104 N/m2 and 0.467 kg/m3 respectively. A pitot tube mounted on the wing senses a pressure of 3.31 × 104 N/m2. Ignoring compressiblity effects, the cruising speed of the aircraft is approximately
    1. 50 m/s
    2. 100 m/s
    3. 150 m/s
    4. 200 m/s

    Answer:- 100 m/s
    Pitot tube senses total pressure.
    p0 = p + 0.5ρv2
    ⇒3.31 × 104 = 3.08 × 104 + (0.5)(0.467)v2
    ⇒v = 99.25 m/s ~ 100 m/s

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s