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Posted By: arunkumar Member Level: Gold Posted Date: 11 Jun 2008
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2004 Classical Mechanics I model paper 2 Question paper
Classical Mechanics I
Mid-Semester Examination, Semester I
marks:25 September, 2004
• Use the formual for cross product of 2 vectors in terms of their lengths
1.
to show that
a b c
= =
sin(A) sin(B) sin(C)
where A, B, C are the angles of the triangles and a, b, c are the sides
opposite A, B, C respectively.
• Prove that in a coordinate-independent way (i.e) not using compo-
nents:
x ? (y ? z) = (x · z)y - (x · y)z
2. Given vectors x, y, z, not coplanar, define
y?z z?x x?y
x = ,y = ,z =
S(x, y, z) S(x, y, z) S(x, y, z)
where S(x, y, z) = x · (y ? z). Show that:
•
1
S(x, y, z) =
S(x , y , z )
•
y ?z
x=
S(x , y , z )
• A particle of unit mass is shot into a medium offering a resistive
3.
force F = kv 3 , v = speed. (There are no other forces). Solve the
equation of motion and find the distance travelled by the particle
when it comes to rest.
• Do the same problem for F = kv 2 and comment on the difference in
answer.
• Show that the centre of mass X of 2 points of mass m1 and m2 at
4.
positions x1 and x2 lie on the line joining x1 and x2 and divides it
in the ratio
X - x1 m2
=
m1
X - x2
1
• The centre of mass of 3 particles of mass m1 , m2 , m3 at positions
x1 , x2 , x3 is defined as
m1 x1 + m2 x2 + m3 x3
X=
m1 + m2 + m3
Show that the centre of mass can be equivalently defined as the centre
of mass of 3 particles and the equivalent one-particle of particles 1 and
2, independantly of the order in which we ennumerate the particle.
If you didn’t understand what he tried to say here, he is saying that
if u take 2 particles, find their CM, consider this as a particle, then
find the CM with the other particle, it is equivalent to finding the CM
of the 3 using the formula, and does not matter as to what order of
particles u choose.
5.In ?ABC, there are points A , B , C on BC, CA, AB respectively
such that
BA ? CB a AC ß
= , = , =
AC ß BA ? CB a
for some a, ß, ?. Show that the lines AA , BB , CC meet at one
point.
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