To determine which equation can be used to model simple harmonic motion, let's review the properties and general form of the equation representing simple harmonic motion.
Simple harmonic motion (SHM) occurs when an object oscillates back and forth about an equilibrium position in such a way that the restoring force is proportional to the displacement from that position and acts in the direction opposite to that displacement. The mathematical expression commonly used to describe SHM is:
[tex]\[ d = a \sin (\omega t) \][/tex]
where:
- [tex]\( d \)[/tex] is the displacement of the object at time [tex]\( t \)[/tex],
- [tex]\( a \)[/tex] is the amplitude of the motion (the maximum displacement from the equilibrium position),
- [tex]\( \omega \)[/tex] (omega) is the angular frequency, and
- [tex]\( t \)[/tex] is the time.
Given these properties, let's examine each of the provided options:
A. [tex]\( d = a \sin (\omega t) \)[/tex]
- This equation fits perfectly with the standard form of the SHM equation, where [tex]\( a \)[/tex] is the amplitude and [tex]\( \omega \)[/tex] is the angular frequency.
B. [tex]\( d = \sin (2 \omega t) \)[/tex]
- This equation does not include the amplitude [tex]\( a \)[/tex] and the argument of the sine function is different (i.e., [tex]\( 2 \omega t \)[/tex]), which complicates modeling SHM directly.
C. [tex]\( d = a \sin (\omega t) + k \)[/tex]
- This equation introduces an additional constant [tex]\( k \)[/tex], which shifts the entire graph vertically by [tex]\( k \)[/tex]. This shift implies that the equilibrium position is no longer at [tex]\( d = 0 \)[/tex], making it not a pure simple harmonic motion.
D. [tex]\( d = a \cos [\omega(t + k)] \)[/tex]
- This equation represents a cosine function with a phase shift [tex]\( k \)[/tex]. Although cosine functions can describe harmonic motion, SHM is typically described by a sine function without phase shifts in its simplest form.
Based on the examination above, the correct equation that can be used to model simple harmonic motion is:
Option A: [tex]\( d = a \sin (\omega t) \)[/tex].
