Show that the relation x= A cos (ωt + Φ) satisfies Newtons 2nd Law...?
Show that the relation x= A cos (ωt + Φ) satisfies Newtons 2nd Law of Motion applied to the oscillatory motin of a spring, ie
F = ma = m(d^2x/dt^2) = -Kx
by deriving that ω = root(K/m).
Please describe in detail how to come to the solution as I wish to understand the method not just get the answer.
Thank you in advance for your help.
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OK Well, this is just an excercise in differentiating trig functions.
We are given that
F = ma = m (d^2x/dt^2) = - Kx
and that
x = A cos (ωt + Φ)
so the obvious first step is to differentiate x twice with respect to t.
dx/dt = - A ω sin (ωt + Φ)
remember, this is the 'function of a function' rule for differentiating.
e.g. if
y = A cos f(x)
dy/dx = - A sin f(x) * d(fx)/dx
Continuing on, we have,
d^2x/dt^2 = - A ω^2 cos (ωt + Φ) = - ω^2 x
We now plug this into Newton's law to see what we get...
F = ma = - m ω^2 x = - Kx
m ω^2 = K
ω^2 = K/m
ω = ± √(K/m)
when taking square roots, you should always consider both the positive and negative roots unless one or other of them is physically meaningless.