Question

A particle moves under force $F = - 5{\left( {x - 2} \right)^3}$. Motion of the particle is
(1) Translatory
(2) Oscillatory
(3) SHM
(4) All of these

Answer 1

The motion of a particle under the influence of force $F = - k{\left( {x - {x_0}} \right)^n}$ depends on the value of k and n. Translatory, oscillatory, and simple harmonic motion are three different kinds of motion possible in nature, depending upon k and n's distinct values.

Complete step by step solution:
For a given force $F = - k{\left( {x - {x_0}} \right)^n}$, we know that:
(a) If the value of k is greater than zero and n is unity, the particle's motion is simple harmonic motion.
For $k > 0$ and $n = 1$, motion is SHM
(b) If the value of k is greater than zero and n is an odd value, the particle's motion is oscillatory.
For $k > 0$ and $n = {\rm{odd}}$, motion is oscillatory
(c) If the value of k is greater than or less than zero and n is an even value, the particle's motion is translatory.
For $k > 0$ or $k < 0$ and $n = {\rm{even}}$, motion is translatory

The value of force is $F = - 5{\left( {x - 2} \right)^3}$.
On comparing this force with the standard expression $F = - k{\left( {x - {x_0}} \right)^n}$, we find:
$k = 5 > 0$ and $n = 3 = {\rm{odd}}$

Therefore, we can say that the particle's motion under the given force F is oscillatory, and option (2) is correct.

Additional information: Oscillatory motion is the to and fro motion of a particle from its mean position. Examples of oscillatory motion are the motion of a simple pendulum, a spring's movement, etc.

Note: Simple harmonic motion is an oscillatory motion of a particle moving in a straight path with some acceleration value. The translatory motion of a body is said to be done when the body moves in a straight path and all points are also moving uniformly with it.