A particle moves along the x-axis while acted on by a conservative force describ
ID: 1534758 • Letter: A
Question
A particle moves along the x-axis while acted on by a conservative force described by the potential energy function shown in the figure below. At which labeled points does the force act to the left? To the right? At which labeled points is the force zero? What are the positions corresponding to points of stable equilibrium? Of unstable equilibrium? If the particle is released from rest at point B, describe the motion of the particle. At what point does the particle have the most kinetic energy? Describe the motion of the particle if it is instead released from rest at Point L? At what point does the particle have the most kinetic energy?Explanation / Answer
(a)
We know that, force, F(x) = -dU(x) / dx
At point A, dU(x) / dx is negative, therefore, the force acts to the right. Similarly at points B, F and G, the force acts to the right. ie., In the region AC and EH, the slope is negative (dU(x) / dx < 0) and force is positive.
At point D, dU(x) / dx is positive, therefore the force acts to the left. Similarly at points J, K and L, the force acts to the left.
At point C, dU(x) / dx = 0, therefore the force is also zero here. Similarly at points E and H, the force is zero. These points are the equilibrium points.
(b)
Points C and H are stable equilibrium points. A local minimum is a point of stable equilibrium, since an object placed at a local minimum will return to its equilibrium position after a slight displacement.
Point E is unstable equilibrium point. A local maximum is known as a point of unstable equilibrium, because an object placed at such a point will not return to its equilibrium position after being displaced slightly.
(c)
At point B, the particle is at rest, therefore the potential energy is maximum. When the particle is released, it reaches the stable equilibrium point C at which the kinetic energy is maximum. The particle move towards the CE region and stops at a point where the potential energy is equal to the potential energy at the point B.
At point C, the kinetic energy will be maximum.
(d)
When the particle is released from point L, it will pass through the points K and J, and it will reaches the equilibrium point H at which the kinetic energy is maximum. And it will move in the HE direction and reaches a point where the potential energy is equal to the point L.
Kinetic energy will be maximum at point H.
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