If the vector field given below section c describes the velocity of a fluid and you place a small cork in the plane at (2, 0), what path will it follow? Vector fields Sketch representative vectors of the following vector fields.a. F (x, y) = ⟨0, x⟩ = x j (a shear field)b. F (x, y) = ⟨1 - y2, 0⟩ = (1 - y2) i, for | y | ≤ 1 (channel flow)c. F (x, y) = ⟨ -y, x⟩ = -y i + x j (a rotation field)
If the vector field given below section c describes the velocity of a fluid and you place a small cork in the plane at (2, 0), what path will it follow?
Vector fields Sketch representative
a. F (x, y) = ⟨0, x⟩ = x j (a shear field)
b. F (x, y) = ⟨1 - y2, 0⟩ = (1 - y2) i, for | y | ≤ 1 (channel flow)
c. F (x, y) = ⟨ -y, x⟩ = -y i + x j (a rotation field)
As per the question, we will analyze the behavior of a small cork placed in a fluid with different velocity vector fields. We are given three distinct vector fields: a shear field, a channel flow field, and a rotation field.
Our task is to determine the path that the cork will follow in each vector field, starting from the initial position (2, 0).
Additionally, we will plot each of these vector fields, along with the cork's initial position. This will help us visualize the cork's movement in the context of each vector field.
First, let's determine the path the cork will follow in the vector field
F(x, y) = ⟨0, x⟩ = xj
Since the velocity is given by the vector field, the cork's position will change according to the velocity at its current position.
The given vector field has no component in the x-direction, and the y-component depends only on the x-coordinate.
Thus, the cork will move only in the y-direction, and its speed in the y-direction will be equal to the x-coordinate of its position.
Initially, the cork is placed at (2, 0).
Therefore, it will start moving in the positive y-direction with a velocity equal to its x-coordinate (2).
Since there is no force acting on the cork in the x-direction, its x-coordinate will remain constant at 2.
For the vector field F(x, y) = ⟨1 - y2, 0⟩ = (1 - y2)i with |y| ≤ 1, the cork's velocity depends only on the y-coordinate of its position.
The x-component of the velocity is affected by the y-coordinate, while the y-component is zero.
This means that the cork will move only in the x-direction, and its speed in the x-direction will be determined by the expression (1 - y2)
Initially, the cork is placed at (2, 0). The y-coordinate is 0, so the cork will start moving in the positive x-direction with a velocity of (1 - 02) = 1.
Since there is no force acting on the cork in the y-direction, its y-coordinate will remain constant at 0.
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