 Solution

ID: past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/section-b/8/d/solution

The starting radius is

r_{0} = 2 c

. In uniform circular motion, the required centripetal acceleration has magnitude

ω^{2} r_{0} = k r_{0} / m

. The initial velocity must be tangent to the starting radius, with the orientation chosen anticlockwise:

v (0) = ω z \times (c, c, 0) = (- ω c, ω c) .

The resulting trajectory is

x = c (cos ω t - sin ω t), y = c (cos ω t + sin ω t) .

It has constant

x^{2} + y^{2} = 2 c^{2}

, so

U (t) = k c^{2} for all t .

This is the horizontal line in the right panel of the figure. The kinetic energy also equals

k c^{2}

2 k c^{2}

, twice that for release from rest at the same point.

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