Question

Which of the following relationships between the acceleration a and the displacement x of a particle involve SHM: (a) a=0.5x, (b) a=400${\mathbf{x}}^{\mathbf{2}}$, (c) a=20x, (d)a=-3${\mathbf{x}}^{\mathbf{2}}$?

Short answer

The relationship between acceleration and displacement is a=-20x .

Step-by-step solution

The given data 

The relationships between acceleration and position are given as:

1. a=0.5x
2. $\mathrm{a}=400{\mathrm{x}}^2$
   
3. $\mathrm{a}=-20\mathrm{x}$
   
4. $\mathrm{a}=-3{\mathrm{x}}^2$
   

Understanding the concept of Hooke’s law

We can use Hook’s law to determine the relation between acceleration and displacement of the particle. Comparing it with the given options, we can find the correct option showing this relationship.

Formulae:

The force on a spring due to Hooke’s law $\mathrm{F}=-\mathrm{kx}$, (i)

The force due to Newton’s second law of motion, $F=ma$ (ii)

Calculation of the correct acceleration and displacement relation

$$\begin{gathered} ma=-kx \\ a=-\frac{k}{m}x \end{gathered}$$

Comparing this result with the options given in the problem,

The option (c) matches with the acceleration we wrote from Hook’s law.

Therefore, the relation a=-20x involves the SHM.