Math 371-04: Fall 2010
Sect 3.8 (Linear Models: Initial Value Problems)
• Basics:
W
.
s
Where s is the elongation of the mass hanging on the spring at equilibrium
(measured in f t, m, etc), and W is the the Weight of the mass on the end
of the spring.
– Spring Constant: Denoted k. Is calculated using: k =
– You can convert back and forth from Mass ↔ Weight with w = m · accg .
Where w is weight, m is mass, and accg (acceleration due to gravity) is
9.8m/s2 or 32f t/s2 .
– Note, the english unit of mass is a slug. 1 slug =
1lb · sec2
ft
• Spring/Mass: Free, Undamped Motion
• The DE for a spring (with spring constant k) and a mass (with mass m) that is
released (at time t = 0) from position x0 with initial velocity x1 is:
m
d2 x
+ kx = 0
dt2
d2 x
+ ω2x = 0
dt2
or
subject to x(0) = x0 , x′ (0) = x1
where
ω2 =
k
m
x0 = +2f t, mean the mass was released from 2 ft below equilibrium position
x1 = −1f t/s, means when the mass was released it had a velocity of 1f t/sec in
the upward direction.
• We can easily solve this DE with section 3.3:
x(t) = c1 cos(ωt) + c2 sin(ωt) (we call this the equation of free motion).
• Period of free vibration: Denoted T is T =
2π
.
ω
• Frequency of free vibration: Denoted f is f =
1
T
• Free Damped Motion
• The DE for a spring (with spring constant k) and a mass (with mass m) that
is released (at time t = 0) from position x0 with initial velocity x1 in a medium
with dampening constant β is:
d2 x
dx
m 2 +β
+ kx = 0
dt
dt
or
d2 x
dx
+ 2λ + ω 2 x = 0
2
dt
dt
subject to x(0) = x0 , x′ (0) = x1
where 2λ =
k
β
and ω 2 =
m
m
• Note that we can solve this DE using 3.3.
• Driven Motion:
• Rather than having a closed system, there could be attached to something that
is imparting force into the system (e.g. moving the ’fixed’ point of the spring).
• The DE for a spring (with spring constant k) and a mass (with mass m) that
is released (at time t = 0) from position x0 with initial velocity x1 in a medium
with dampening constant β is with the mass is driven by an external force of
f (t) is:
m
dx
d2 x
+β
+ kx = f (t)
2
dt
dt
or
d2 x
dx
+ 2λ + ω 2 x = F (t)
2
dt
dt
subject to x(0) = x0 , x′ (0) = x1
where F (t) =
f (t)
m
• Note that we can solve this DE using 3.5 (or maybe 3.4, depending on F (t)).
• Series Circuit Analogue:
• Charge on the capacitor in a LRC:
L
dq
1
d2 q
+ R + q = E(t)
2
dt
dt C
where
q(t) is the charge (on the capacitor)
L is the inductance (across the inductor)
R is the resistance (across the resistor)
C is the capacitance (across the capacitor)
E(t) is the voltage (impressed on the the circuit)
i(t) is the current (in the circuit)
dq
and i(t) =
dt
measured
measured
measured
measured
measured
measured
in
in
in
in
in
in
C (coulombs)
h (henry’s)
Ω (ohms)
f (farahs)
V (volts)
A (amperes)
• Note: this is a DE we can solve with 3.5 (or maybe 3.4, depending on E(t))
• Examples
1. A mass weighing 1 lbs stretches a string 2 feet. At time t = 0 the mass is
released from a point 6 inches below equilibrium position with an upward
velocity of 18 inches per second. (Ignore any dampening forces like friction). Determine the equation of free motion. What is the period of simple
harmonic motion?
2. A 5 kg mass stretches a spring .7 m. The whole system is suspended in
molasses, which has a dampening coefficient of 40. The mass is started from
the equilibrium position with an initial upward velocity of 2 m/s. (Note: I
made up the dampening coefficient of molasses.)
(a)
(b)
(c)
(d)
Find the equation of motion.
What is the position of the mass after 1 second?
Is this motion overdamped, critically damped, or under damped?
Without doing any math, where (approximately) do you expect the
mass to be after 200 seconds?
3. A mass of 51 slug is attached to a spring with spring constant = 2lb/f t. The
mass is released from rest 1/2 foot below equilibrium position. The system
is in a medium with dampening coefficient β = 1.2. The system is being
driven by an external force given by f (t) = 5 cos(4t). Find the equation of
motion.
4. Find the charge on the capacitor and the current in the LRC circuit with
L = 1h, R = 100Ω, C = .00004f, E(t) = 30V, q(0) = 0C and i(0) = 2A.
Find the maximum charge on the capacitor.
5. A 5 kg mass stretches a spring 0.7m. What is the period of simple harmonic
motion for this spring/mass system (ignoring dampening forces).
6. A 16 foot weight is attached to a 5 foot long spring. At equilibrium, the
spring measures 8.2 feet. The weight is pushed up and released from rest
at a point 2 feet above equilibrium position, and it is known that the surrounding medium offers a resistance numerically equal to its instantaneous
velocity. Find the displacement x(t).