Physical Applications of Second-Order Differential Equations

Physical Applications of Second-Order Differential Equations
Simple Harmonic Motion
We consider the motion of an object with mass at the end of a spring that is either vertical or horizontal on a level surface as in a figure .

Hooke’s Law, says that if the spring is stretched units from its natural length, then it exerts a force that is proportional to : restoring force
where is a positive constant(called the spring constant). If we ignore any external resisting forces then, by Newton’s Second Law we have

Example 1: A frictionless spring with a 10kg mass can be held stretched 1 meters beyond its natural length by a force of 40 N. If the spring begins at its equilibrium position, but a push gives it an initial velocity of 2.5 m/sec, find the position of the mass after t seconds.
Solution: From Hooke’s Law, the force required to stretch the spring is








Damped Vibrations
We next consider the motion of a spring that is subject to a frictional force .An example is the damping force supplied by a shock absorber in a car or a bicycle. We assume that the damping force is proportional to the velocity of the mass and acts in the direction opposite to the motion. Thus:

where is a positive constant, called the damping constant. Thus, in this case, Newton’s Second Law gives :

Example 2: A spring with a mass of has natural length . A force of is required to maintain it stretched to a length of . If the spring is immersed in a fluid with damping constant . Find the position of the mass at any time if it starts from the equilibrium position and is given a push to start it with an initial velocity of
Solution:










Electric Circuits
The circuit shown in Figure contains an electromotive force , a resistor , an inductor and a capacitor in series. If the charge on the capacitor at time is , then the current is the rate of change of with respect to :

It is known from physics that the voltage drops across the resistor, inductor, and capacitor are

respectively. Kirchhoff’s voltage law says that the sum of these voltage drops is equal to
the supplied voltage

Since , this equation becomes

Example 3: A series circuit consists of a resistor with an inductor with a capacitor with and a 12-V battery. If the initial charge and current are both 0, find the charge at time
. Solution:














1. A spring with a mass of has damping constant and spring constant Find the position of the mass at time if it starts at the equilibrium position
with a velocity of
2. A series circuit consists of a resistor with an inductor with
a capacitor with and a 12-V battery. If ,
find the charge at time