Question
Use this to write down the equation of motion of a particle under a force F = cu3 , where c is a constant. Solve the equation of motion. You will need to discuss three cases c > h2 , c = h2 and c < h2 . Discuss, in which of these cases, is a closed orbit possible.
Answer :
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Skip in 10
Let's solve the equation of motion for a particle under the force \( F = c u^3 \), where \( c \) is a constant. To start, we apply Newton's second law: \[ F = ma = m \frac{d^2 u}{dt^2} \] This gives: \[ m \frac{d^2 u}{dt^2} = c u^3 \] Now, rearranging the equation: \[ \frac{d^2 u}{dt^2} = \frac{c}{m} u^3 \] To make the problem more solvable, we introduce the parameter \( \omega = \sqrt{\frac{c}{m}}
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