1: In this question we will study the damped harmonic oscillator. Consider the following spring-mass system. kr Figure 1: Spring-mass system with a friction force. That is, an object of mass m is attached at the end of a spring with spring-constant k. We will also consider the effect of friction. Friction works in the opposite direction of motion, as illustrated in the figure. A) Show that the differential equation governing the damped spring-mass system is: mä = -bi – kr (1) dr where i = and i = dt dt2 b) To solve the differential equation (1), we will take the following ansatz: r(t) = Ae. (2) If the above r(t) is a solution of Eq. (1) show that A has to satisfy the following equation: 1² + 27A +w = 0, (3) k Solve the above equation to find A. where y = and w = c) For w >, show that the general solution can be written as: %3D 2m r(a) = Ae- cos (wt + o) (4) where w? = w - 72. 2: Using Fermat's principle prove the law of reflection and the law of refraction of light.
1: In this question we will study the damped harmonic oscillator. Consider the following spring-mass system. kr Figure 1: Spring-mass system with a friction force. That is, an object of mass m is attached at the end of a spring with spring-constant k. We will also consider the effect of friction. Friction works in the opposite direction of motion, as illustrated in the figure. A) Show that the differential equation governing the damped spring-mass system is: mä = -bi – kr (1) dr where i = and i = dt dt2 b) To solve the differential equation (1), we will take the following ansatz: r(t) = Ae. (2) If the above r(t) is a solution of Eq. (1) show that A has to satisfy the following equation: 1² + 27A +w = 0, (3) k Solve the above equation to find A. where y = and w = c) For w >, show that the general solution can be written as: %3D 2m r(a) = Ae- cos (wt + o) (4) where w? = w - 72. 2: Using Fermat's principle prove the law of reflection and the law of refraction of light.
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Transcribed Image Text:1: In this question we will study the damped harmonic oscillator. Consider the following
spring-mass system.
kr
bà
Figure 1: Spring-mass system with a friction force.
That is, an object of mass m is attached at the end of a spring with spring-constant k. We
will also consider the effect of friction. Friction works in the opposite direction of motion, as
illustrated in the figure.
A) Show that the differential equation governing the damped spring-mass system is:
më = -bå – kx
(1)
dr
and i =
dt
where i =
dt2
b) To solve the differential equation (1), we will take the following ansatz:
r(t) = Ae.
(2)
If the above r(t) is a solution of Eq. (1) show that A has to satisfy the following equation:
1² +27d + wi = 0,
(3)
* and wi =
k
Solve the above equation to find A.
m2
b
where
y 3=
2m
c) For wi >?, show that the general solution can be written as:
x(a) = Ae-t cos (wt +0)
(4)
where w? = w - 7².
2: Using Fermat's principle prove the law of reflection and the law of refraction of light.
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