1000 Solved Problems in Modern Physics

1.2 Problems 27

1.56 Solve:

d^3 y

dx^3

− 3

d^2 y

dx^2

+ 4 y=0 (Osmania University)

1.57 Solve:

d^4 y

dx^4

− 4

d^3 y

dx^3

+ 10

d^2 y

dx^2

− 12

dy

dx

+ 5 y=0 (Osmania University)

1.58 Solve:

d^2 y

dx^2

+m^2 y=cosax

1.59 Solve:

d^2 y

dx^2

− 5

dy

dx

+ 6 y=x

1.60 Solve the equation of motion for the damped oscillator:

d^2 x

dt^2

+ 2

dx

dt

+ 5 x= 0

subject to the condition,x=5, dx/dt=−3att=0.

1.61 Two equal masses are connected by means of a spring and two other identical
springs fixed to rigid supports on either side (Fig. 1.3), permit the masses
to jointly undergo simple harmonic motion along a straight line, so that the
system corresponds to two coupled oscillations.
Assume thatm 1 =m 2 =mand the stiffness constant iskfor both the
oscillators.
(a) Form the differential equations for both the oscillators and solve the
coupled equations and obtain the frequencies of oscillations.
(b) Discuss the modes of oscillation and sketch the modes.

Fig. 1.3Coupled oscillator

1.62 A cylinder of massmis allowed to roll on a spring attached to it so that it

encounters simple harmonic motion about the equilibrium position. Use the

energy conservation to form the differential equation. Solve the equation and

find the time period of oscillation. Assumekto be the spring constant.