Question 14.4: The functions f and g are given by f : x = a sin (ωt + ε) ...
The functions f and g are given by
f : x = a sin (ωt + ε) and g : x = a cos (ωt + ε)
where ε (epsilon) is a positive constant.
For each of these functions:
i) Differentiate it with respect to time to find v (or \dot{x}) and show that v² = a²(ω² – x²).
ii) Differentiate v with respect to time to find \ddot{x} and show that \ddot{x} = – ω²x .
i) For f:
x = a sin(ωt + ε)
⇒ v = \dot{x} = aω cos(ωt + ε) ①
Therefore v² = a²ω² cos²(ωt + ε)
= ω²[ a² – a² sin²(ωt + ε)]
⇒ v² = ω²(a² – x²)
Similarly for g:
x = a cos(ωt + ε)
⇒ v = \dot{x} = – aω sin(ωt + ε) ②
Therefore v² = a²ω² sin²(ωt + ε)
= ω²[ a² – a² cos²(ωt + ε)]
⇒ v² = ω²(a² – x²).
ii) Differentiating equation ① gives
\ddot{x} = – aω² sin(ωt + ε)
⇒ \ddot{x} = – ω²x. \ \ → \boxed{\begin{matrix}\text{These results show that } \\ \text{the functions f and g represent SHM}\end{matrix}}
Similarly differentiating ② gives
\ddot{x} = – aω² cos(ωt + ε)
⇒ \ddot{x}= – ω²x. \ \ → \boxed{\begin{matrix}\text{These results show that } \\ \text{the functions f and g represent SHM}\end{matrix}}