Question

PLEASE HELP. I truly just want to understand how to do this problem.
A 1.20 kg mass on a horizontal spring oscillates on a frictionless surface with a displacement as a function of time given by x(t) = 0.075cos(4.16t – 2.45). (Units are standard units.)
a.) Find the time for one complete vibration.
b.) Find the force constant of the spring.
c.) Find the maximum force on the mass.
d.) Find the maximum speed of the mass.
e.) Find the position, speed and acceleration of the mass at t = 1.00 s.
f.) Find the total energy of the oscillating spring.

Answer 1

A sinusoidal oscillation takes the form
\[ y(t) = A\sin(\omega t - \phi)\]
where A is the amplitude of the oscillation, omega is the angular frequency, and phi is the phase offset. Do you know calculus?

Answer 2

If you do, the acceleration is given by
\[ a(t) = y''(t) = -A\omega^2\sin(\omega t - \phi)\]
which clearly has a maximum value of
\[a_{max} = A\omega^2\]
since F= ma, we have that the maximum force is
\[F_{max} = ma_{max} = mA\omega^2\]
This should correspond to the force exerted by the spring when x is at it's maximum, or
\[ kx_{max} = kA = mA\omega^2 \rightarrow k = m\omega^2\]
The total energy of the oscillator is equal to the potential energy when x = A, so
\[ E_{total} = \frac{1}{2} k A^2\]
The maximum velocity is achieved when the potential energy is zero, or
\[\frac{1}{2} mv^2 = E_{total} = \frac{1}{2} kA^2 \rightarrow v = A\sqrt{\frac{k}{m}}= A\omega \]
The time for a complete vibration is
\[T = \frac{2\pi}{\omega} \]
I think that covers just about everything...