Consider a point particle with position vector r = (x, y, z) in Cartesian coordinates, moving with a velocity v = (β, αz, −αy), where α and β are positive constants. (a) What are the physical dimensions of α and β? (b) Find the general form of r(t), the position of the particle, as a function of time t, assuming the initial position of the particle is r0 = (0, 2, 0) (hint: write v = (β, αz, −αy) as a system of first order ODEs and note that the equation for x is decoupled from the others). Describe in words the motion of the particle and sketch its trajectory in R3 (you can use software packages for the plot). (c) Show that the speed of the particle is constant, but the acceleration vector a(t) is nonzero. Justify. Assuming the particle has a constant mass m, use Newton’s second law to show that the force acting on the particle is (as a function of the position r = (x, y, z)) F(x, y, z) = mα2(0, −y, −z).
Consider a point particle with position vector r = (x, y, z) in Cartesian coordinates, moving with a velocity v = (β, αz, −αy), where α and β are positive constants.
(a) What are the physical dimensions of α and β?
(b) Find the general form of r(t), the position of the particle, as a function of time t, assuming the initial position of the particle is r0 = (0, 2, 0) (hint: write v = (β, αz, −αy) as a system of first order ODEs and note that the equation for x is decoupled from the others). Describe in words the motion of the particle and sketch its trajectory in R3 (you can use software packages for the plot).
(c) Show that the speed of the particle is constant, but the acceleration vector a(t) is nonzero. Justify.
Assuming the particle has a constant mass m, use Newton’s second law to show that the force acting on the particle is (as a function of the position r = (x, y, z))
F(x, y, z) = mα2(0, −y, −z).
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