3. a) Write down and explain the relationship between the angular velocity, w, and the angular...

  

Transcribed Image Text:

3. a) Write down and explain the relationship between the angular velocity, w, and the angular momentum, I, for a rotating rigid body. b) Given that the rate of change of I in an inertial frame S is related to the rate of change of I in a frame R, which is rotating relative to S with angular velocity, via [Ž]s = [Ż]R + W × L, show that Euler's equations for the motion of a rigid body subject to an external torque Gare G₁ = I₁₁+ (13- 12) W3W2 G₂ I202 + (11-13)w1W3 G3 I3W3+ (1211)w₂w1. and determine Ω. = c) In what frame of reference are Euler's equations valid? Is this an inertial frame? d) A space rocket is rotationally symmetric about its length, taken to be the 3 axis in the body frame, so that I₁ = I2 I3. The engines drive the rocket so that it rotates about its symmetry axis with constant angular velocity. However, the rocket thrust is slightly misaligned, so that there is a constant torque perpendicular to the symmetry axis. Assume that this torque is in the 2 direction in the body frame. Show that the equations of motion for the rocket in the body frame are النا ولنا ولنا = = Qwz G I₁ = 0 - - Ωωι [3] [6] [2] [5] 3. a) Write down and explain the relationship between the angular velocity, w, and the angular momentum, I, for a rotating rigid body. b) Given that the rate of change of I in an inertial frame S is related to the rate of change of I in a frame R, which is rotating relative to S with angular velocity, via [Ž]s = [Ż]R + W × L, show that Euler's equations for the motion of a rigid body subject to an external torque Gare G₁ = I₁₁+ (13- 12) W3W2 G₂ I202 + (11-13)w1W3 G3 I3W3+ (1211)w₂w1. and determine Ω. = c) In what frame of reference are Euler's equations valid? Is this an inertial frame? d) A space rocket is rotationally symmetric about its length, taken to be the 3 axis in the body frame, so that I₁ = I2 I3. The engines drive the rocket so that it rotates about its symmetry axis with constant angular velocity. However, the rocket thrust is slightly misaligned, so that there is a constant torque perpendicular to the symmetry axis. Assume that this torque is in the 2 direction in the body frame. Show that the equations of motion for the rocket in the body frame are النا ولنا ولنا = = Qwz G I₁ = 0 - - Ωωι [3] [6] [2] [5]

Answer rating: 100% (QA)

a The relationship between the angular velocity and the angular momentum L for a rotating rigid body is given by the equation L I where I is the moment of inertia of the body This equation states that ... View the full answer