We reviewed springs; in particular, that they have a restoring force that is proportional to the amount
they are displaced from their equilibrium position, i.e.,
F = - k x
We then considered the oscillatory motion of a mass on a spring, and determined that the function
x(t) = A cos ( ω t + φ )
satisfies the equation
F = m a = m
d2x / dt2 = - k x
provided
ω2 = k/m
The amplitude A corresponds to the maximum displacement of the mass from its equilibrium
position. Next, we related ω to the period T, the time required for the behavior to start repeating itself.
ω = 2 π / T
Finally, the phase constant φ was associated with the "initial conditions" – the part of the cycle in which
the motion started.
Wednesday:
We determined useful formulas for the maximum velocity and maximum acceleration of an oscillating
mass.
vmax = ω A
amax = ω2 A
We looked at an energy analysis of a mass vibrating on a spring. It would have kinetic energy
( m v2 / 2 ) and spring potential energy ( k x2 / 2 ) and if it is oscillating
in the vertical direction gravitational potential energy (mgh). We verified that the solution
x(t) = A cos ( ω t + φ ) satisfies conversation of energy for a horizontal spring with
no friction or other external force acting on the system by using the result ω2 = k/m
and the trigonometric identity cos2 θ + sin2 θ = 1. We also determined
two useful expressions for the energy:
E = k A2 /2
when we think of the energy as being all spring potential (when the mass is at the
amplitude position); and
E = m ω2 A2 /2
when we think of the energy as being all kinetic potential (when the mass is at the equilibirum position).
We solved problems 15-33 and 15-36 from Halliday, Resnick and Walker, solving for a number of the quantities
in two ways, for example taking a kinematic approach versus an energy approach or by using the two different
expressions above for the energy.
