Answer to Question #247122 in Economics of Enterprise for Sekhar

Question

 Describe the motion of an object when displacement of the particle is zero at t=0s and its displacement is given by the equation  

Answer 1

complete question; Describe the motion of an object when displacement of the particle is zero at t=0s and its displacement is given by the equation  $x(t)=A\space cos(ωt+ϕ)$

solution

Here the equation for displacement with respect to time is given and the initial displacement conditions. Thus, we will look at the equation to determine the kind of motion it describes and, we will also use the initial conditions to determine the constants for this equation.

The given equation of motion is:

$x(t)=A\space cos(ωt+ϕ)$

On double differentiating w.r.t. time,

$x''=-Aω^2cos(ωt+ϕ)\\x''=-(ω^2)x\\x''∝-x$

Thus, it can be seen that the acceleration on the body is opposite in direction but proportional to the displacement. This is characteristic of a simple harmonic motion.

Using the initial condition,

$x(0)=0\\A\space (cos\space 0+ϕ)=0\\cosϕ=0\\ϕ=90°$

Thus, it is concluded that the equation $x(t)=A\space cos(ωt+ϕ)$  along with the initial condition x(0)=0 characterizes a simple harmonic motion with amplitude A, angular frequency ω

ω, and phase difference of ϕ=90°