Askiitians Tutor Team
Last Activity: 4 Months ago
Let's break down the problem of simple harmonic motion (SHM) and explore how the choice of equation affects the time period. You’ve already identified that the shortest distance traveled by a particle in SHM from its mean position in 2 seconds is \(\frac{\sqrt{3}}{2}\) of its amplitude. This is a great starting point for understanding the relationship between the equations of motion and the time period.
Understanding the Equations of Motion in SHM
In SHM, the position of a particle can be described using two common equations:
- x = a sin(ωt)
- x = a cos(ωt)
Here, a is the amplitude, ω is the angular frequency, and t is time. Both equations describe the same motion but start from different points in the cycle. The sine function starts at zero, while the cosine function starts at the maximum amplitude.
Analyzing the Problem with the Sine Function
Using the equation x = a sin(ωt), you can find the position of the particle at any time t. Given that the distance traveled in 2 seconds is \(\frac{\sqrt{3}}{2} a\), you can set up the equation:
At \(t = 2\) seconds:
x = a sin(ω * 2) = \(\frac{\sqrt{3}}{2} a\)
This implies:
sin(ω * 2) = \(\frac{\sqrt{3}}{2}\)
The angle for which sin(θ) = \(\frac{\sqrt{3}}{2}\) is θ = \(\frac{\pi}{3}\) or θ = \(\frac{2\pi}{3}\). Therefore, we can write:
ω * 2 = \(\frac{\pi}{3}\)
Solving for ω gives:
ω = \(\frac{\pi}{6}\)
The time period (T) is related to ω by the formula:
T = \(\frac{2\pi}{ω}\) = \(\frac{2\pi}{\frac{\pi}{6}}\) = 12 seconds.
Exploring the Cosine Function
Now, let’s consider the equation x = a cos(ωt). If we apply the same logic, we need to find the time at which the particle has moved the same distance:
At \(t = 2\) seconds:
x = a cos(ω * 2) = \(\frac{\sqrt{3}}{2} a\)
cos(ω * 2) = \(\frac{\sqrt{3}}{2}\)
The angles for which cos(θ) = \(\frac{\sqrt{3}}{2}\) are θ = \(\frac{\pi}{6}\) and θ = \(\frac{11\pi}{6}\). Thus, we can write:
ω * 2 = \(\frac{\pi}{6}\)
or
ω * 2 = \(\frac{11\pi}{6}\)
From the first equation:
ω = \(\frac{\pi}{12}\)
Calculating the time period gives:
T = \(\frac{2\pi}{ω}\) = \(\frac{2\pi}{\frac{\pi}{12}}\) = 24 seconds.
Why the Difference in Time Period?
The difference in time periods arises from the initial conditions set by the choice of sine or cosine. The sine function starts at zero displacement, while the cosine function starts at maximum displacement. This means that the particle's motion is effectively "shifted" in time, leading to different values for ω and consequently different time periods.
In summary, while both equations describe the same physical motion, the choice of starting point (sine vs. cosine) affects the angular frequency and thus the time period. This is a fundamental aspect of SHM that highlights the importance of initial conditions in oscillatory systems.
