A student claims that he applied a force F = -k x on a particle, resulting in the particle executing simple harmonic motion. He declines to reveal whether k is a constant or not. Assume that he has worked exclusively with positive x and that no other force acted on the particle.

Step-by-step solution

For a particle to execute simple harmonic motion, the restoring force must be of the form F = -c(x - x₀) , where c > 0 is a constant and x₀ is the equilibrium position. Given that the applied force is F = -k x , we can equate the two expressions: -k x = -c(x - x₀) k = c(x - x₀) x = c ( x - x₀ x ) To determine the variation of k with respect to x , we differentiate k with respect to x : dk dx = c 2 x + cx₀ 2x x Since the particle executes SHM exclusively in the region x > 0 , its equilibrium position x₀ must also be