\(x(t) = A cos(\omega t + \phi_o)\)
where A is the amplitude of the motion, \(\omega\) is the (angular) frequency (in radians per second), and \(\phi_o\) is the phase (in radians). The phase plays a crucial role..., well, all the parameters do, but it is important to understand the role of the phase. The (angular) frequency is related to k and m via, \(\omega^2 = (k/m)\), ( \(\omega = \sqrt{k/m}\) ). When you graph the equation for x(t) (above) you see that the period is \( T = 2 \pi/\omega\), or, equivalently, \( \omega = 2 \pi/T\). There is a 2nd kind of frequency, f, which is pretty much redundant since f = 1/T = \(\omega/(2 \pi)\).
Velocity:
The equation for velocity the goes with the above equation for x(t) is:
\(v(t) = -A \omega sin(\omega t + \phi_o)\). (This comes from taking the derivative of x(t).)
Acceleration:
Acceleration is related to force, F(t). At any given time or place in the motion,
F(t) = -k x(t), since that is our starting ansatz (assumption) regarding the nature of the spring.
It is also true, always, that F(t) = m a(t) (Newton's 2nd law).
So you can combine those two relationships (equations) to get:
m a(t) = F(t) = -k x(t), or, equivalently,
a(t) = -(k/m) x(t).
(See video of problem 14.24 below).
May 2:
I would suggest making a better version of this table yourself. I think that might help you become more familiar with cos and sin. I would suggest maybe by increments of pi/12 and one for cos and one for sin. (You can use wolfram-alpha.)
Soln's:
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