PHY 501 Fall 2013
Homework 1: Solutions
In each of the following problems:
(i) introduce convenient generalized coordinate(s) qj of the system.
(ii) write down the Lagrangian L as a function of q j , q j and (if appropriate) time,
(iii) write down the Lag
PHY 501 Fall 2013
Problems for initial self-check: Solutions
Problem 1.1. Find acceleration of a rocket due to working jet motor, and explore the resulting
equation of rockets motion.
Hint: For the sake of simplicity, you may consider the 1D motion.
Solut
PHY 501 Fall 2013
Optional problems Set 1: Solutions
Problem 2.5. For the system of two pendula which was the subject of Problem 1.3:
(i) introduce convenient generalized coordinate(s) qj of the system.
(ii) write down the Lagrangian L as a function of q
PHY 501 Fall 2013
Midterm Exam: Solutions
Problem M.1 (to be graded of 900 points). A pendulum of mass m is hung on
another point mass m that may slide, without friction, along a straight horizontal rail see Fig. on the right. Assuming that pendulums moti
PHY 501 Fall 2013
Homework 7: Solutions
Problem 6.3. A uniform, round disk of radius R can rotate, without
friction, in the vertical plane, about a horizontal axis A displaced by R/2 from
the disk center see Fig. on the right. Find the eigenfrequency of s
PHY 501 Fall 2013
Homework 6: Solutions
Problem 5.5 (to be graded of 40 points). Calculate and analyze the dispersion relation (k) for
longitudinal waves in an infinite 1D chain of coupled oscillators with alternating masses - see Fig. on
the right. In pa
PHY 501 Fall 2013
Homework 5: Solutions
Problem 5.1. For the system of two elastically coupled pendula,
confined to a vertical plane, with the parameters shown in Fig. on the right
(cf. Problem 1.3), find possible frequencies of small sinusoidal oscillati
PHY 501 Fall 2013
Homework 4: Solutions
Problem 4.5 (to be graded of 20 points). Find the fixed point of the following system of
equations:
q1 q1 4q 2 ,
q 2 q1 q 2 .
Analyze its stability. What type of the fixed point is it? (Node? saddle? focus? center?)
PHY 501 Fall 2013
Homework 3: Solutions
Problem 4.2. A square-wave pulse of force (see Fig.
on the right) is exerted on a linear oscillator with
eigenfrequency 0 (no damping), initially at rest. Calculate
the law of motion q(t), sketch it, and interpret t
PHY 501 Fall 2013
Homework 2: Solutions
Problem 3.1. Use Eq. (3.27) of the lecture notes to calculate the functional dependence of period
T of oscillations of a 1D particle of mass m in potential U(q) = q2n (where > 0, and n is a positive
integer) on ener
Classical Mechanics and Dynamics
PHY 501
Fall 2009
Homework 1 with solutions
In each problem of this assignment:
(i) introduce a set of convenient generalized coordinate(s) qj of the system,
(ii) write down the Lagrangian L as a function of q j , q j , an