Which one of the following equations of motion represents simple harmonic motion(A) Acceleration= - {k_0}x + {k_1}{x^2}(B) Acceleration= - k(x + a)(C) Acceleration=k(x + a)(D) Acceleration= kx where k,{k_0},{k_{1,}}a all are positive

We are tasked with identifying the equation of motion that represents simple harmonic motion (SHM).
Definitions and Analysis:
For a system to undergo simple harmonic motion, the restoring force (and thus the acceleration) must be directly proportional to the displacement from the equilibrium position and must act in the opposite direction to the displacement. Mathematically, the equation for SHM is:
a=−kxa = -kx
Where:
• aa is the acceleration.
• kk is the constant of proportionality (the spring constant in the case of a spring).
• xx is the displacement from the equilibrium position.
In SHM, the force acting on the particle is proportional to the negative of its displacement, which leads to the acceleration being proportional to −x-x.
Evaluating Each Option:
1. Option A:
Acceleration=−k0x+k1x2\text{Acceleration} = -k_0 x + k_1 x^2
This equation has both linear and quadratic terms in xx. The presence of the x2x^2 term indicates that the motion is not simple harmonic, as SHM requires the force to be linearly related to displacement. Thus, this is not SHM.
2. Option B:
Acceleration=−k(x+a)\text{Acceleration} = -k(x + a)
This equation involves a shift in the displacement by aa. While this equation could represent SHM, the negative sign and the direct proportionality to displacement suggest that the system undergoes harmonic motion, but the shifted equilibrium position doesn't change the nature of the SHM. This is possible SHM but with a shifted equilibrium.
3. Option C:
Acceleration=k(x+a)\text{Acceleration} = k(x + a)
Here, the acceleration is proportional to x+ax + a, which means the restoring force is not directed towards the equilibrium position but rather towards a shifted position. This is not SHM.
4. Option D:
Acceleration=kx\text{Acceleration} = kx
This equation represents acceleration being directly proportional to the displacement xx. However, in SHM, acceleration should be directed opposite to the displacement, which would be a=−kxa = -kx. This option is incorrect for SHM because the sign is incorrect.
Conclusion:
Option B: Acceleration = -k(x + a) correctly represents a shifted SHM, where the displacement is measured relative to a shifted equilibrium position.
Thus, the correct answer is: (B) Acceleration = -k(x + a)