18.01 Section, October 7, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Applications of differential equations: springs and pendulums
1. A spring is attached to the wall, and a weight is attached to the other end of the spring. The
weight is pulled 5 cm beyond the equilibrium position, and then let go. Write an equation
for the motion of the spring, given that the period is 0.5 seconds.
2. A pendulum consists of a (heavy) weight on the end of a (light) stick, which swings back
and forth under the action of gravity. Suppose the weight is 5 kg, the stick is 2 m. At the
start, the pendulum is raised to an angle of 1 radian and then let go.
(a) The only force in the picture is gravity. If the pendulum is at an angle of θ from
vertical, what is the force acting in the direction of motion? What is the acceleration
of the weight? What is d2 θ/dt2 ?
(b) Write an equation for the angle θ in terms of time t. (You may use the simplification
sin θ ≈ θ.)
1
3. Use the same setup as in the previous problem, and, as in the previous problem, assume
that pendulums are modelled by simple harmonic motion. This time, I raise the pendulum
to an angle of 3 radians, and then give it a little push, so its initial velocity is 0.1 m/s
towards the center. Write an equation for its motion (θ in terms of time).
4. Bonus question: If you plot (cos t, sin t) for a lot of t’s, you get a circle. (Why?) What
happens if you plot (cosh t, sinh t)?
Review
• Table of differential equations discussed in lecture
Diff. eq.
dy 2
=c
dt2
d2 t
c
=− 2
2
dt
y
Solution
Comments
y(t) = 2c t2 + At + B
A = initial vel., B = initial position
constant acceleration ⇐⇒ constant force
y(t) = At2/3
e.g. gravitational force F = −GmM
√
√
d2 y
= −cy, c > 0 y(t) = A cos( c · t) + B sin( c · t)
2
dt
√
simple harmonic motion (springs, etc.)
√
y(t) = Ae c·t + Be− c·t
√
√
⇐⇒ y(t) = C sinh( c · t) + D cosh( c · t)
√
√
2π
• Period of y(t) = A cos( c · t) + B sin( c · t) is √
c
d2 y
= cy, c > 0
dt2
• Hyperbolic sin and cosine
cosh(x) =
ex + e−x
2
sinh(x) =
d
cosh x = sinh x
dx
ex − e−x
2
d
sinh x = cosh x
dx
2
1
y2