Question 4 (23 marks)
Two blocks M_(1) and M_(2), of the same mass m, located on the horizon, are connected by a spring S_(1) of natural
length l_(0) and stiffness k as shown in Figure 3 .
Let x_(j)=x_(j)(t) be the rightward displacement of block M_(j) measured from their corresponding equilibrium point
O_(j) at time t respectively, where j=1,2. Take +i and +j be the unit vectors pointing rightward and upward
respectively if necessary. Assume that there is no air resistance and damping force applied to the system, and
all the contacts are smooth.
(a) (5 marks) Draw the force diagrams showing all the forces acting on each block. Please indicate the
direction of each force by an arrow and briefly define all the newly introduced symbols used in the force
diagrams.
(b) (2 marks) By applying Newton's second law, show that the equations of motion of blocks M_(1) and M_(2) are
given by
x_(1)^(¨)=(k)/(m)(-x_(1)+x_(2)),
x_(2)^(¨)=(k)/(m)(x_(1)-x_(2)).
(c) (1 mark) Show that the equations of motion can be written as a matrix form as x^(¨)=A_(1)x, where
x=([x_(1)],[x_(2)]),A_(1)=(k)/(m)([-1,1],[1,-1]).
In this case, A_(1) is called the dynamic matrix.
(d) (4 marks) Find the eigenvalues and the corresponding eigenvectors of the dynamic matrix A_(1).
(e) (7 marks) Find the normal mode angular frequencies and their corresponding periods of oscillation (if
there is no period of oscillation, you may answer "no period of oscillation".). State whether the motion
of each normal mode is in-phase or phase-opposed (Please refer to Example 1.5 and Exercise 1.6 in Unit
10 (p.20-21) for the answering skills of this part. ). Write down the general solutions to x_(j) for jin{1,2}
based on what you found in the previous parts. (Hint: please pay attention to all the answers related to
the zero mode. )
(f) (2 marks) If the initial conditions are
x_(1)(0)=0
x_(2)(0)=0
show that the particular solutions of x_(1) and x_(2) are
x(t)=([x_(1)(t)],[x_(2)(t)])=(ut)/(2)([1],[1])+(u)/(2)sqrt((m)/(2k))([1],[-1])sin(sqrt((2k)/(m))t).
(g) (2 marks) If the other initial conditions are
x_(1)(0)=L
x_(2)(0)=0
show that the particular solutions of x_(1) and x_(2) are
x(t)=([x_(1)(t)],[x_(2)(t)])=(L)/(2)([1],[1])+(L)/(2)([1],[-1])cos(sqrt((2k)/(m))t).
Question 4 23 marks
Two blocks Mi and M2of the same mass m,located on the horizon, are connected by a spring Si of natural length fo and stiffness k as shown in Figure 3.
Let ; = (t) be the rightward displacement of block M; measured from their corresponding equilibrium point O; at time t respectively, where j = 1,2. Take +i and +j be the unit vectors pointing rightward and upward respectively if necessary. Assume that there is no air resistance and damping force applied to the system, and all the contacts are smooth.
direction of each force by an arrow and briefly define all the newly introduced symbols used in the force diagrams.
(b) (2 marks) By applying Newton's second law, show that the equations of motion of blocks M and M2 are given by
= -1+2) m
(20)
i2= m
c) (1 mark) Show that the equations of motion can be written as a matrix form as = A, where
A=m1 (21) In this case, A, is called the dynamic matrix. d) (4 marks) Find the eigenvalues and the corresponding eigenvectors of the dynamic matrix A. e) (7 marks) Find the normal mode angular frequencies and their corresponding periods of oscillation (if there is no period of oscillation, you may answer no period of oscillation. ). State whether the motion of each normal mode is in-phase or phase-opposed (Please refer to Example 1.5 and Exercise 1.6 in Unit 10 (p.20-21) for the answering skills of this part.). Write down the general solutions to for j {1,2} based on what you found in the previous parts. (Hint: please pay attention to all the answers related to the zero mode.)
f)(2 marks) If the initial conditions are
x10=0 x0=0 show that the particular solutions of 1 and x2 are
x10= 20=0
(22)
=8=1+ (g) (2 marks) If the other initial conditions are x10=L x0=0 x20=0 x20=0 show that the particular solutions of and 2 are
(23)
(24)
xt= x1t
(25)
