Question

A particle which can move along x-axis is released from rest at the position x=x0.The potential energy (U) of the block is described below:
\[U=\left\{ \begin{align}
  & -ax;x<0 \\
 & b{{x}^{2}};x\ge 0 \\
\end{align} \right\}\]
Which of the following statements are/is correct:-
A.The subsequent motion is simple harmonic.
B.The subsequent motion is periodic.
C.The speed is a continuous function of time.
D.The magnitude of acceleration is a continuous function of time.

Answer 1

Hint: Here Potential energy equation is given. We will study the r=equation clearly either theoretically or graphically and justify the motion of particles as either simple harmonic or not because the condition of SHM should be satisfied if not we will check the periodicity of a function. Then we apply force and potential energy equations to find the nature of acceleration and velocity function with time.

Complete answer:
The equation of potential energy given in the question is not a quadratic equation when the value of x is less than zero. So the motion of a particle is not simple harmonic because for simple harmonic function acceleration is proportional to negative of displacement.
\[acceleration\alpha -displacement\]
But here this condition is not obeyed. So Option A is wrong.
Whenever a particle is bounded by potential, then motion is periodic since this particle is bounded by the potential. For periodic value of function it should be repeated after a particular interval of time. Here valid functions get repeated. So option B is correct.
Here in this equation, the energy is constant and so the particle will move back and forth between a maximum negative position and the position x=x0​.Since here this potential energy is continuous and according to law of conservation of energy we know that Sum of Kinetic Energy and Potential Energy is constant which means \[KE+U=TE\].
This equation can be written as
\[KE=TE-U\]
since potential energy is continuous so from this equation Kinetic energy is also continuous. Therefore, speed is also continuous with time. So option C is correct.
Since, this derivative of the potential energy is not continuous at x = 0.
Since, force is equal to the negative derivative of the potential energy represented by
\[F=-\dfrac{dU}{dx}\], this force is not continuous at x=0. Therefore, acceleration is not continuous at x=0.So option D is incorrect.

Finally , Option B (Motion is Periodic )
Option C (The speed is a continuous function of time ) is Correct.

Note:
A motion of a particle is said to be simple harmonic motion when it follows the condition of acceleration is proportional to negative of displacement. Since we have to remember also all Simple harmonic motion are period but period motion is not simple harmonic. The motion of earth about its axis is periodic but not simple harmonic.