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

Hint: 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 $$n=\mathrm{o}\mathrm{d}\mathrm{d}$$, 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 $$n=\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}$$, 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 $$n=3=\mathrm{o}\mathrm{d}\mathrm{d}$$

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.