Here are the detailed notes on the time period in terms of [math]k[/math] and [math]m[/math], incorporating the latest formatting preference:
Time Period in Terms of [math]k[/math] and [math]m[/math]
The time period [math]T[/math] of an oscillating system can be derived by analyzing the relationship between force, angular frequency, and the spring constant.
1. Force in SHM
From Newton’s second law:
[math]F = m a[/math]
Using the equation for acceleration in SHM:
[math]a(t) = -\omega^2 x[/math]
we substitute:
[math]F = -m \omega^2 x[/math]
From Hooke’s law, the restoring force is:
[math]F = -k x[/math]
Equating the two expressions for force:
[math]-m \omega^2 x = -k x[/math]
Canceling [math]x[/math] (as [math]x \neq 0[/math]):
[math]m \omega^2 = k[/math]
Thus,
[math]\omega = \sqrt{\frac{k}{m}}[/math]
2. Time Period in SHM
The angular frequency [math]\omega[/math] is related to the time period [math]T[/math] by:
[math]\omega = \frac{2\pi}{T}[/math]
Substituting [math]\omega = \sqrt{\frac{k}{m}}[/math]:
[math]\frac{2\pi}{T} = \sqrt{\frac{k}{m}}[/math]
Rearranging for [math]T[/math]:
[math]T = 2\pi \sqrt{\frac{m}{k}}[/math]
Final Equations
- Force:
[math]F = -k x[/math]
[math]F = -m \omega^2 x[/math]
Angular Frequency:
[math]\omega = \sqrt{\frac{k}{m}}[/math]
Time Period:
[math]T = 2\pi \sqrt{\frac{m}{k}}[/math]
Let me know if you need more explanations or further assistance!
Here is the LaTeX code for the equation written between the required tags:[math] \textstyle R = \frac{u^2 \sin 2\theta}{g} [/math]
