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A problem set from MIT's 16.07 Dynamics course, focusing on iceboat aerodynamics and vector transformations. Students are asked to explore the relationship between an iceboat's speed, the true wind's magnitude and direction, and the lift-to-drag ratio. They are also required to derive transformation matrices for rotations in three dimensions. The problem set includes figures, diagrams, and a photograph for reference.
Typology: Exercises
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Turn in each problem on separate sheets so that grading can be done in parallel
Problem 1 (20 points) Consider an iceboat traveling in the x direction at constant speed Vboat in a true wind of speed Vwind making an angle of θwind with the direction of travel. The forces between the boat and the ice are such that the friction force from the runners is effectively zero while still supporting necessary side forces. In addition, the design of the boat is such that it has a lift to drag ratio, L/D. The helmsman has the freedom to choose the lift to drag ratio within the capabilities of the rig as well as choosing the course relative to the wind direction by choosing θwind. In this problem, we will explore how best to set these quantities for maximum speed. Figure 1: Iceboat traveling on ice. Figure 2: Top view of boat. a) Solve for the magnitude of the apparent wind, Vapparent, seen by the helmsman and its angle relative to the direction of travel, θapparent. b) Show that if the boat is traveling at constant speed, the resultant aerodynamic force Ftotal acts in a direction normal to the direction of travel.
g) In sailing, the term “Velocity Made Good” (VMG) refers to the velocity component directed towards a desired waypoint or destination. VMG is a measure of how well progress towards the waypoint is being made. In the case of our race course, the VMG would be the component of velocity in the direction of the wind. Using the value of L/D from Part (e), calculate the maximum upwind VMG. What angle, θwind, should be chosen to achieve this? Repeat the calculations for the downwind leg.
Problem 2 (10 points) In two-dimensions, a coordinate transformation of the components of a general vector V from an x,y system to an X’, Y’ system by a rotation through an angle q is given by (in matrix notation) Derive the equivalent transformation matrix A in three dimensions for the following different coordinate transformations 1, 2, and 3 shown below. a) 1. A single rotation about the z axis through an angle θ b) 2. A single rotation about the x axis through an angle φ c) 3. A single rotation about the y axis through an angle ψ
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