105A Discussion
Week 7
February 24 2014
1. A ring of radius r and mass m rolls without slipping inside the parabola y = ax2 . Find
the equation of constraint. Using Lagrange multipliers, find the equations of motion and
the generalized forces of constrain
105A Discussion
Week 4
February 3 2014
1. M. 10.12. Show that the small angular deviation of a plumb line from the true vertical
(i.e. toward the center or Earth) at a point on Earths surface at a latitude is
=
R 2 sin cos
g0 R 2 cos2
(1)
where R is the
Problem 1
Let's consider a two dimensional isotropic harmonic oscillator.
1
1
L m( x 2 y 2 z 2 ) m 2 ( x 2 y 2 )
2
2
(a) Which of the following are symmetries of the Lagrangian?
(i) translation in the x direction (ii) translation in the y direction (iii)
Problem 1
(a) (iii) (vi)
(b) p z mz Lz m( xy yx )
(c) L
1
1
m(r 2 r 2 2 z 2 ) m 2 r 2
2
2
p
L
mr 2 L z
Problem 2
(a) x g
(b) h
1 2
mx mgx . Yes, it's the total mechanical energy.
2
(c) L
1
m( x v) 2 mg ( x vt)
2
E.O.M x g which is equivalent to x g .
Problem 1
Textbook page 138 problem 3.6
Suppose the free length of the string is L and the two masses never collide with each other.
Problem 2
Suppose the equation of motion of a particle is x ax bx 0 . a and b are two constants.
(a) For a 0 b 0 , solve t
Problem 1
In the old coordinate system:
0
A B 1 A B 1
1
Transformation matrix:
0
1
1
0
2
0 1
2
0
1
2
1
2
In the new coordinate system:
1
0
1
A 0 B 2 A B 1 A B 0
0
2
0
Problem 2
F U mgy , where y means the unit vector in y direct
105A Discussion
Midterm 1 Review
January 30 2014
1. M. 3.43. A point mass m slides without friction on a horizontal table at one end of a
massless spring of natural length a and spring constant k as shown in the Figure. The
spring is attached to the table
Problem 1
A bead slides without friction around a rotating ring, as shown in the figure below.
g
R
(a) Write down the equation of motion of the bead as a differential equation of
the angular coordinate .
(b) What's condition (in terms of , g, R ) for the
Problem 1
(a) mx kx mg x 2 x g
x(t )
g
V
(1 cost ) sin t
2
When x (t ) 0 , t
V
and x ( 2 1)
4
(b) mx kx (mg qxB) x
Critical damping (
qB
m
qB
m
x 2 x g
) 2 4 2 0
(c) mx kx mg 3mg cos 2t x 2 x g 3g cos 2t
x(t )
g g
V
2 cos 2t sin t
2
When x (t ) 0 , t
Problem 1
Equations of motion for each mass
m1 x1 k ( x 2 x1 L)
m2 x2 k ( x1 x 2 L)
Do the variable substitution y x1 x2 L , we get y k (
oscillator with angular frequency
k(
1
1
) y 0 . It's a simple harmonic
m1 m2
1
1
) . The frequency is f
.
m1 m2
2
105A Discussion
Week 10
March 10 2014
k
1. (a) A particle of mass m moves in a repulsive potential of the form V (r) = , where k is
r
a positive constant. Write down the Lagrangian and Hamiltonian. Show that is a cyclic
coordinate, and that H is constant.
HW 1
Solve problems 1-10, 2-3, 2-9, 2-25 from the textbook, and the following problem: A particle is moving on a plane, where we identify its position using polar coordinates r(t), (t). Find the expression for the radial and angular components of the
HW 7
Due 2/27/2008 Solve the following problems: 7-21, 8-3, 8-5. A pendulum of mass m is suspended by a massless spring, with equilibrium length L and spring constant k. The point of support moves vertically with constant acceleration. Write the Lagr
HW 3
Due 1/30/2008 Solve the following problems: 2-47, 2-53, 2-55, 3-17. A particle of mass M moving along the x-axis is subject to an elastic force kx. The particle is also subject to a friction force !2M " x and a driving force F = F0 cos(3 k / M t
Problem 1
g
R
(a) mR mg sin m 2 R sin cos or sin 2 sin cos
(b) Set 0 we get cos
g
2R
g
2R
. In order to have a solution for we must have
1 because the magnitude of cosine of any angle cannot be greater than 1.
arccos
g
2R
.
Problem 2
(a) x B cost
(b)
Problem 1
A block of mass m is moving one-dimensionally on a horizontal table. The kinetic friction
coefficient between the block and the table is . One end of a spring of spring constant k is
attached to the block, with the other end attached to a wall.