Question

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

Answer 1

Hint:In a mechanics and physics, simple harmonic motion is defined as a special type of periodic motion where the restoring force on a moving object is directly proportional to the object displacement magnitude and acts towards the object equilibrium position. a simple harmonic motion of a constant amplitude in which the acceleration is directly proportional and oppositely directed to the displacement of a body from a position of equilibrium

Complete step by step solution:
As we discussed we know the equation for simple harmonic motion
$a = - \dfrac{{{d^2}x}}{{d{t^2}}}$
Now we can write the equation as
$a = - {\omega ^2}x$
We know that the value of $\omega = \sqrt {\dfrac{k}{m}} $
Whereas now
$x = A\sin (\omega t + \delta )$
Now the acceleration due to gravity can be written as
$a = - Kx$
We have the value of $x$ that is
$x = x + a$
Now substitute the value of $x$ in the above equation
Hence $a = - K(x + a)$
In Simple harmonic motion, acceleration is directly proportional to the displacement from mean position and also the acceleration is in the opposite direction of the displacement.

Hence, the correct answer is option B

Note: In simple harmonic motion acceleration is directly proportional to the displacement from the mean position. Also the acceleration is in the opposite direction of displacement. The best example for simple harmonic motion is motion of a pendulum and the motion of a spring
Acceleration: it is defined as the rate of change of the velocity with respect to time. Usually, the acceleration means speed is changing, but not always. When an object moves in a circular path at the constant speed, it is still accelerating, because the direction of its velocity is changing.