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. Apendnlun with a masslcss string of length ) and mass mis hanging fron aniong
platfor1 which^ has^ a^ constant^ accclcration cí. starting^ from^ an^ initial^ clocitr^ (Gt itational acccloration ()^ acts^ along^ -j.^ The^ string^ makcs^ an^ anglc^ )^ with^ thc^ rertical.
a) Obtain the kinctic cncrgy of this srsten in the ground framc at rest (not the moNing
frame of the platforn) in tems of the gcncralizcd rariablc 0.
b) Use virt ual work to obtain gcncrali cd ms of 0and its timc dorivatives usi.g (3 + 2 =5 marks)
- Show that Lagrangcs cquation ()- =F. where 7 is the kinctic cncrgy and F is the gencraliscd force. is cquivalcnt to the cquation - 2=F (2.5 mnarks).
force F and obtain the cquation of and Fn.
- Ile have lcarnt the invariancc propcrty that the Largrangian cquatiol C Nl.
Luxder the transformation :: L. ’ '= l.+t0) Hore /. = Li9, q, !) where 4s te
gencralizcd variablc and / docs not depcnd on
The Ligrangian of a charge y in magnrtir ficld /: is given hr 7= 7 4
where. is the magnctic vector potcnt ial and o is the scalar potcntial. Explot tie abore invariance propcrtr. to show that a gange transformation giren br
kecps the Lagrangian cquation invariant.
(Although you will not nced this here, for complcteness sakr. 1: = -D
(2.5 marks)
k
Fig.
k
M
Fig.
(4 + 2 +1= 7 marks)
2a (^60)
1. A chain of two masses m and M are connected by horizontal identical springs (with
spring constant k and unstretched length l) to a wall as shown in Fig.1. Ignore gravity
and consider 1-D motion along horizontal axis only. When the system is disturbed slightly
from equilibrium, small oscillations result. (a) Find the normal mode frequencies.
(b) Now set m = M = k= 1 and rewrite the normal mode frequencies. Then find the
eigen vector corresponding to any one of the frequencies. c) Draw the qualitative nature
of the displacements of the masses corresponding to this frequency.
Fig.
b) the total angular momentum of the system LT. c) Calculate dLr/dt. (6 marks)
Fig.
- Fig-2: Two masses my and m2, connected by a rigid rod of length l, are in motion in a frictionless horizontal plane. Tension (T) in the connector is the only operating force on the masses. Using the vectors r), I2, i] and r2 (w.r.t. the origin) write down
a) The Lagrangian of the system including the constraint term A[(r1 r2)- }, where
Ais the lagrange multiplier.
- Fig3: A mass m is connected to two identical springs (k) of rest length 2a whose other ends are connected to the ground. At equilibrium the springs make 60° angle with the vertical. Compute the frequencies and eigen modes for small 2D oscillations of the mass in the vertical plane and ignore gravity. (5+1=6 marks)
- Fig.4: Obtain an equation for the shortest line, on a cylindrical surface of radius R, passing from (R, 01, z1) to (R,o,, 22). Infinitesimal path length ds on a cylinderical surface is given by ds² = Rd0+ dz. Hint: lst write the functional that has to be minimized and then use Euler-Lagrange's equation. (4 marks)
- Two masses of m each are in motion in a horizontal 2D plane. They interact via an attractive central force potential V(r) = kr vhere r is the dist ance between them. Ignore gravity. (a) Obtain Lagrangian of thesysten in terms of the effective mass uand r, using plane polar coordinates.
(b) Get the EOM and identify the constants of motion (conserved quantities).
(c) Show that there can be a cireular orbit,obtain its radius ro and using energy argunents (i.e., through Vefr(r)), show that this circular orbit is stable. (d) Displace the particle slightly from its circular orbit (i.e., r(t) =ro + Rt), where
R(t) << ro) and show that R(t) willexhibit oscillation. Obtain its frequency. (1 +2+
2 + 2 =7 marks)
- Give brief answers:
a) Arigid body is made of 3 equal mass points located at , a, Y= (a,0,0), (0, a, 20), (0, 20, a).
Compute the moment of inertia matrix Iod = L;m,[(r)ag -(r:)a(r)sl
b) Fig. 1: Consider an elliptic central force orbit for a mass m. The force center is at 'O
and the closest and the farthest points are B and A, respectively, at distances R and r,
with R+ r= 2a (the major axis). Compute the ratio of the velocities va/vg and using
the energy cxpression E = +^ in^ terms^ of^ the^ constants ,^ k,^ m.
A
c) In the cartesian coordinate systen, write down the matrix for a rotation about the z
axis by angle and obtain its eigen values.
d) If Ais a rotation matrix for rotation angle180° about any axis, then show P+ =(1+4)
have the property P = P. (3+3+242=10 points)
2. Fig-2: Consider two masses mË and my connected by a massess rigid rod of length 20.
The midpoint of the rod is constrained to move on a circle of fixed radius R.
Fig-
2 2)2r
a) Obtain the kinetic cnergy of this system in terms of the generaised variables and a. b) Considering the presence of gravity (g) use virtual work to obtain generalized forces Fo and Fa: (2 + 3 =5 marks)
3. The EOM of 0 (with range [0, n/2] for a heavy symmetric top reads: = (1 - )(a
Bu) - (b- au)?, where u = cos and ß = 2Mgl/ I1, while a, a, b are positire dynamical
constants. If gravitaion was absent and a>b', dctermine how many turning points in 8
are expected. Recall that for nonzero g the answer was two. (5 points)
R
compute #
I B
Fig-
R
(^2) a m
Fig-
A,
b
B
H
- Fig.3: Arectangular solid fixcd at it center 0', is set into motion with an angular angular
velocity @, about one of its diagonals (ED as shown). The subsequent motion is torque
frce. Given, I, = ,= a' +6°) and I, =a'.
G
(a) Obtain the EOM fr the W(t) in the principal axis body frame and solve then.
(b) Which diagonal is the next one to assume the role of rotational axis?
(c) How long does it take for this to happen? (4+2+1=7 points)
5. Spherical pendum : A particle of mass m is constrained to move on the surface of a
sphere of (^) radius R. (^) Ignore (^) gravity.
(a) Write the Lagrangian of the system using spherical polar coordinates (r, 0, ). In
general i=j2+2 +7 sin
(b) Obtain the equations of motion, identify the conserved quantities.
(c) Obtain the total energy E as a function of only.
(d) Obtain Ver(0) and show graphically that when E > Eo (certain threshold) the motion
of (t) willbe confined between two bounding values of 8, i.e., 0, < et) < B,. Obtain
the threshold energy Eo. This type of motion is called nutation. e) Show that there can be a horizontal, circular orbit and obtain its radius Ro and
inclination o. (1+2+1+2+2= 8 marks)
6. The Lagrangian of a point charge e in an EM field is given by L = -eo+eA.r, where
Ais the magnetic vector potential and ¢ is the electric scalar potential. (a) Obtain the canonical momentunnp and the Hamiltonian H(p,), where G=r. (b) We know that the Lagrangian equations remain invariant under the following gauge transformation:
t Under this transformation L changes to I'. Obtain the new canonical momentum P as a function of p, r. (c) Compute the new Hamiltonian Has a function P, r,A,.
(1)
(d) Consider the changes p P= P(p, r) and ’ Q= f to be coordinate transfor
mations. Show that this is a canonical transformation by computing the fundamental
Poisson brackets {Q.,Q},(P, P}, {Qi, P}}.
(2+2+2+2+2 =10 marks)
(e) Determine the generating function F.( , F,t), using P=p =t^ O2 and Q, = , Remenmber
7. Determine the generating function F(g, P) for the identity transformation: Q=9, P=p
? Obtain the corresponding^ F(g.^ Q)^ from^ this^ F^ and^ show^ that^ indeed^ this^ Fi^ generates
an identity transformation. Use^ p=^ 2,Q= aF2^ and^ p=dq^ aF^ (5^ points)
OP; that g=Er here.
JP
