Download Tutorial 2 (Application Second Order Linear

Tutorial 2 (Application Second Order Linear Differential Equation)
1. Assuming that there is no external force acting on the spring system, the Newton’s second
law with the net or resultant force and weight, W, is given
d 2x
m 2  kx
dt
and Hooke’s law state that a restoring force F opposite to the direction of elongation and
proportional to the amount of elongation, s, simply stated as F  ks where k is the spring
constant and m is a mass attached to the spring. By defining, W  mg , find the equation
of the motion if a mass weighing 2 pounds stretches a spring 6 inches. At t = 0, the mass
is released from a point 8 inches below the equilibrium position with an upward velocity
of 3/4 ft/sec given g  32 ft / s 2 . Then, determine the period of free vibrations and its
frequency.
2. A mass weighing 10 N stretches a spring 2 cm. At t = 0, the mass is released from a point
10 cm below the equilibrium position with an upward velocity of 7 cm/s with a given
g  980 cm / s 2 . Find the equation of free motion. Next, determine the period of free
vibrations and its frequency.
3. A mass weighing 14 N stretches a spring 0.7 m. Initially, the mass is released from a
point 1 m below the equilibrium position with an upward velocity of 2/3 cm/s with a
given g  9.8 m / s 2 . Determine the equation of free motion, the period of free vibrations
and its frequency. Assume that there is no external force acting on the spring system. The
Newton’s second law with the net or resultant force and weight, W, is given
m
d 2x
 kx
dt 2
where k is the spring constant and m is a mass attached to the spring.
4. If a mass weighing 20 N stretches a spring 0.2 m, find the equation of the motion. At
t = 0, the mass is released from a point 0.8 m below the equilibrium position with an
upward velocity of 56 cm/s given g  980 cm / s 2 . Then, determine the period of free
vibrations and its frequency. According to the Newton’s second law with the net or
resultant force and weight, W and define W  mg , is given
m
d 2x
 kx
dt 2
where k is the spring constant and m is a mass attached to the spring.
Answer
1. xt  
F
2 2 
2
3
cos8t   sin 8t  , T 

 .
3
32

8
4
1
  4
1    .
T
4 


2. xt   10 cos 7 10t 
F
1

T
1
 2 


 7 10 
3. xt   cos7t  
F
1

T
1


7 10
.
2
2
2 2
sin 7t  , T 

.
21

7
 2 


 7 

7
.
2
4. xt   80 cos7t   8sin 7t  , T 
F
1

T
1

1
2
2
sin 7 10t , T 

.
 7 10
10
 2 


 7 

7
.
2
2


2
.
7