Classical mechanics problems - MIT course

Massachusetts Institute of Technology Physics 809 Fall 2006

Homework 1 due Wednesday September 13 2006 before 1230 PM

Suggested Reading

Refresh your mechanics knowledge by reading from your favorite Mechanics book For example the book by Marion amp Thornton is very helpful Goldstein is a bit too dry for this initial ldquoNewtonian mechanics refresherrdquo

Problems

Problem 11 (20 pts)

A block of mass m1 rests on top of a block of mass m2 The static and kinetic coefficients of friction between the two blocks and the bottom block and the table are micros and microk A fixed force F is applied to the bottom block

m

m 1

2

friction

friction F

Describe the motion and find the accelerations of the two blocks a1 and a2 for all possible values of F from F = 0 rarr infin

Problem 12 (20 pts)

Consider a thin circular hoop of radius R and mass m attached to the wall with a frictionless screw The hoop can swing freely but it cannot slip Parametrize the position of the hoop in terms of the angular displacement from the vertical θ Write the equations of motion in terms of θ and its derivatives Find all solutions in the limit of small amplitude motion of the hoop What is the period of oscillation of the hoop What is the length of a simple mathematical pendulum of length L and mass m with the same period of oscillation Gravitational acceleration is g

Problem 13 (20 pts)

Mass m slides down a frictionless incline as shown in Figure 2 The mass is released at a height h above the bottom of the loop

a What is the magnitude and direction of the velocity and the force on the mass at the bottom of the incline (point A)

b What is the magnitude and direction of the velocity and the force on the mass at point B right before the mass leaves the track

c At what speed does the mass leave the track

d How far away from point A does the block land on the level ground

e Sketch the potential energy of the mass as a function of position U(x)

1

m

R

B 45

h

xA

Problem 14 (20 pts)

A particle moves in a two dimensional orbit defined by

x(t) = A(2αt minus sin αt)

y(t) = A(1 minus cos αt)

a Find the tangential acceleration at and normal acceleration an as a function of time where the tangential and normal components are taken with respect to the velocity

b Determine at what times in the orbit an has a maximum

Problem 15 (20 pts)

A particle of mass m slides down an inclined plane with inclination angle θ under the influence of gravity If the motion is resisted by a force f = kmv2 show that the time required to move a distance d after starting from rest is

coshminus1(ekd)t = radic

kg sin θ

2


Homework 2 due Wednesday September 20 2006 at 1230 PM

Suggested Reading

Goldstein Sections 11-12 and 21-25

Problems

Problem 21 (20 pts)

Goldstein Problem 6 Chapter 2 Page 64

Problem 22 (20 pts)


Problem 23 (20 pts)


Problem 24 (20 pts)

Consider a region of space divided by a plane The potential energy of a particle in region 1 is U1 and in region 2 is U2 If a particle of mass m and with speed v1 in region 1 passes from region 1 to region 2 such that its path in region 1 makes an angle θ1 with the normal to the plane of separation and an angle θ2 with the normal to the plane when in region 2 show that

sin θ1 U1 minus U2 = 1 +sin θ2 T1

where T1 = 12 mv12 What is the optical analog of this problem

Problem 25 (20 pts)

A pendulum consists of a mass m suspended by a massless spring with unextended length b and spring constant k Find Lagrangersquos equations of motion

1


Homework 3due Wednesday September 27 2006 at 1230 PM

Suggested Reading

Goldstein Chapter 13 14 24 26 27

Problems

Problem 31 (20 pts)


Problem 32 (20 pts)


Problem 33 (20 pts)


Problem 34 (20 pts)


Problem 35 (20 pts)


1


Homework 4due Wednesday October 11 2005 at 1230 PM

Suggested Reading

Goldstein Sections 81 and 82

Problems

Problem 41 (25 pts)


Problem 42 (25 pts)


Problem 43 (25 pts)


Problem 44 (25 pts)


1


Homework 5due Monday October 18 2006 at 1230 PM

Suggested Reading

Goldstein ChapterSection Canonical Transformations 91-939697

Problems

Problem 51 (20 pts)

Goldstein Chapter 9 Problem 4 page 422

Problem 52 (20 pts)


Problem 53 (20 pts)


Problem 54 (20 pts)


Problem 55 (20 pts)


1



Suggested Reading

Goldstein Kepler Problem 31-33

Problems

Problem 61 (25 pts)


Problem 62 (25 pts)


Problem 63 (25 pts)

Goldstein Chapter 3 Exercise 19 page 129

Problem 64 (25 pts)


1


Homework 7due Monday November 6 2006 at 1230 PM (Note unusual date)

Suggested Reading

Goldstein Collisions 310-311 lecture material Marion and Thornton 96-910 (Goldstein doesnrsquot contain a lot about scattering)

Problems

Problem 71 (20 pts)

Marion amp Thornton 9-26

Problem 72 (20 pts)


Problem 73 (20 pts)


Problem 74 (20 pts)


Problem 75 (20 pts)


1


Homework 8due Monday November 8 2006 at 1230 PM

Suggested Reading

Lecture Material Goldstein Sections 49 410 53 and 54 Also Chapter 10 Marion and Thronton

Problems

Problem 81 (25 pts)

Goldstein Exercise 22 page 183

Problem 82 (25 pts)

Marion and Thornton 10-12

Problem 83 (25 pts)


Problem 84 (25 pts)


1


Homework 9due Wednesday November 29 2006 at 1230 PM

Suggested Reading

Goldstein Chapter 5

Problems

Problem 91 (25 pts)

Goldstein Derivation 6 page 233 Ignore the last sentence of question c) (Poinsot construction)

Problem 92 (25 pts)


Problem 93 (25 pts)


Problem 94 (25 pts)


1


Homework 10due Wednesday December 6 2006 at 1230 PM

Suggested Reading

Goldstein Sections 15 85 and Chapter 13

Problems

Problem 101 (25 pts)








1

Massachusetts Institute of Technology Department of Physics

Course 809 Classical Mechanics Term Fall 2006

Quiz 1 October 4 2006

Instructions

bull Do not start until you are told to do so

bull Solve all problems

bull Put your name on the covers of all notebooks you are using

bull Show all work neatly in the blue book label the problem you are working on

bull Mark the final answers

bull Books and notes are not to be used Calculators are unnecessary

2

Useful Formulae

Newton and Basic Kinematics

F = p = ma for v c t=t

v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0

Lagrangian and Hamiltonian

L(q q) = T minus U H(p q) = T + U = pq minus L

Euler-Lagrange (without and with constraints)

partL d partL partL d partL partgaminus = 0 minus + λa = 0 partx dt partx partx dt partx partxa

Polar Coordinates

x = r sin θ cos φ y = r sin θ sin φ z = r cos θ

Cylindrical Coordinates

x = r cos φ y = r sin φ z = z

Possibly useful integrals

dx dx = arctan(x) = arctanh(x)

1 + x2 1 minus x2

3

Problem 1 Variable Length Pendulum (35 points)

Consider a simple pendulum consisting of mass m attached to a string of length L After the pendulum is set into motion (at t = 0) the length of the string is lenghtened at a constant rate

dL = α = constant

dt The suspension point remains fixed

a) How many degrees of freedom does the system have

b) Write the Lagrangian for the pendulum Calculate partL is it equal to zero partt

c) Write the equations of motion for the pendulum do not solve Show that the equations of motion become the equations of the fixed length pendulum for α = 0

d) Write the Hamiltonian for the system

e) Calculate the total mechanical energy of the system Compare to the Hamiltoshynian

f) The energy of the system is not conserved What is the rate of change Give physical interpretation of the sources and magnitude of power flowing in or out of the system

Problem 2 Particle on a rotating wire (35 points)

A particle of mass m is constrained to move along a straight frictionless wire The wire is rotating in a vertical plane at a constant angular velocity Ω as shown in the Figure 1 The system is in the gravitational field with gravitational acceleration g pointing downwards At t = 0 the mass was stationary with respect to the wire and it was at a distance R0 from the axis of rotation

a) Write the Lagrangian for a free mass constrained to the vertical plane in gravishytational field using polar coordinate system

b) Write explicitly the constraint equations that force the mass to remain on the rotating wire

c) Write equations of motion for all the coordinates introducing Lagrange Multiplishyers

4

m

g

R0 Ω

Figure 1 Particle on a rotating wire

d) Obtain the expressions for the forces of constraints Give the physical interpreshytation of these forces

e) For sufficiently large Ω the mass will be moving away from the rotation axis for all wire positions What is the minimum value of the angular velocity such that this is guaranteed

Problem 3 Double Pendulum (30 points)

Consider a system consisting of two masses m1 and m2 and connected with massless rigid rods of length L1 and L2 (see Figure 2) As the pendulum moves both masses remain in the vertical plane

L1

g

m 1

L 2

m 2

Figure 2 Double Pendulum


b) Write the Lagrangian for the system

c) Write equations of motion for the system Do not solve



Quiz 2 November 15 2006

Instructions




bull Show all work neatly in the white book label the problem you are working on


bull Books and notes are not to be used You may use your calculator

2

Useful Formulae


F = p = ma for v c t=t v = v0 + dtprimea

t=0 t=t t=t

r = r0 + v0t + dtprime dtprimeprimeF (tprimeprime)m t=0 t =0

Gravitational Law

F = minus Gm1m2

r122r12


partqL(q q) = T minus U H(p q) = T + U = p minus L

partt

Hamilton Equation of Motion

partH partH = minusp = q

partq partp

Generating function

partF (Q q) partF (Q q) = p = minusP

partq partQ

Poisson Brackets

partg partf partg partf [g f ] = minus

partq partp partp partq


partL d partL partL d partL partgminus = 0 minus + λ = 0 partx dt partx partx dt partx partx

Polar Coordinates


Orbit Equation

micro 1 u primeprime + u = minus F (u) with u =

2u2 r

ε =

3

Effective Potential

2

V (r) = U(r) + 2micror2

Keplerian Orbits

k α U(r) = minus = ε cos θ + 1

r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3

microk microk2 k α α

rmin = a(1 minus ε) = rmax = a(1 + ε) =1 + ε 1 minus ε

Spherical Coordinates


Inelastic Scattering (coefficient of restitution)

|v2 minus v1||u2 minus u1|

Scattering

dσ dσ b db σ(θ) = = =

dΩ dφ sin θdθ sin θ dθ sin θ π θ

tan ψ = ζ = minus cos θ + (m1m2) 2 2

Vector in fixed frame expressed in terms of vector in rotating frame and rotation velocity

dX dX( )inertial = ( )rotating + (ω times X )dt dt

Acceleration in accelerated and rotating frame

a = g minus V minus (ω times r) minus 2(ω times v) minus (ω times (ω times r))

Useful Trigonometrical Formulas

sin(A + B) = sin A cos B + cos A sin B

cos(A + B) = cos A cos B minus sin A sin B

4

Problem 1 Canonical Transformation (35 points)

Consider a harmonic oscillator with Hamiltonian

p2 mω2q2

H = + 2m 2

Introduce transformation of variables from q(t) and p(t) to Q(t) = q(t + τ) and P (t) = p(t + τ) where τ is constant The goal of this problem is to find generating function F (Q q) of this canonical transformation of variables

a) Solve Hamilton equations of motion and find q(t) and p(t) for arbitrary initial conditions

b) Express P and p as functions of only q Q m ω and τ You may want first to express Q and P as the functions of q and p and then rearrange equations Note that there should be no time dependence in these expressions

c) Integrate p as a function of q to obtain q-dependence of F Remember to include the undetermined integrating constant(q) C(Q)

d) Use equation for P (Q q) to find C(Q) and determine F (Q q)

e) Verify that F (Q q) is the correct generating function

5

Problem 2 Scattering from a rotationally symmetric surface (30 points)

Consider small particles moving with velocity parallel to the z-axis Particles are scattering from a perfectly elastic and frictionless surface The surface is rotationally symmetric about the z-axis and the shape of the surface is ρ(z) = z3 for z gt 0 We want to find differential scattering cross section σ(θ)

a) Find the relationship between the scattering angle θ and the impact parameter Note that the surface at the impact point can be considered as a perfectly elastic wall tangential to the curve at that point

b) Extract b(θ) and determine the differential cross section

c) Make a sketch of the scattering angle at some representative point ρ(z) for the case of the surface that is frictionless but totally inelastic

d) Make a sketch of the scattering angle at some representative point ρ(z) for the case of surface that is elastic but it has very high friction

6

Problem 3 Motion in rotating frame (35 points)

Consider a cannon located at geographical lattitude of exactly 45o North Cannon can fire identical projectiles in East North West and South directions All projectiles have identical initial speed of v0 = 200 ms and the cannon always points up at exactly 45o to the horizontal In your calculations of the projectile trajectory you can ignore the curvature of the Earth Assume that there is no air resistance

Assume that the radius of the Earth is R = 6 middot106 m and that the angular velocity of Earthrsquos rotation is Ω = 7 middot 10minus5 sminus1 Gravitational acceleration is g = 10 ms2 Do your calculations in the coordinate system where the z-axis is vertical pointing up with 0 at the center of the Earth the x-axis points North and the y-axis points West

a) Assume that the Earth is not rotating Obtain the algebraic formulas and the numericals value for the total range in meters maximum height above the ground in meters and the projectile flight time in seconds

b) Assume now that the Earth is rotating with its usual Ω Write down an expresshy

sion for acceleration ai = (ax ay az) in terms of components of r = (x y z) and Ω v = (x y˙ z) Ignore very small terms proportional to Ω2x Ω2y and replace radicz with R Keep all variables in algebraic form but replace sin 45o and cos 45o with 1 2

Note the following sections can be quite time consuming Start them only after you have finalized the other problems and sections in this quiz

c) The gun is being shot in the four geographical directions ENW S Write down the expressions for acceleration individually for each direction In each case remove the very small terms and find cancellations of terms of roughly equal size Justify your choices Note that some numerical cancellations are possible since the speed of the projectile happens to be very close to the linear speed of the Earthrsquos surface at that lattitude

d) Based on the magnitudes and signs of the terms of a obtained in c) estimate the effect of the earthrsquos rotation on the position of the impact parameter of the projectile for each of the four cases Mark the location of the new impact point as compared to no-rotation impact point (as obtained in a)) on graphs similar to Figure 1 The center of the graph corresponds to the no-rotation impact position Do not do the precise calculations try only to estimate the direction and relative magnitude of the deviation All four graphs should be oriented with North at the top and East to the right

7

x

yNorth

East

West

South

Figure 1 Impact Point Graph

m

R

B 45

h

xA

Problem 14 (20 pts)

A particle moves in a two dimensional orbit defined by

x(t) = A(2αt minus sin αt)

y(t) = A(1 minus cos αt)

a Find the tangential acceleration at and normal acceleration an as a function of time where the tangential and normal components are taken with respect to the velocity

b Determine at what times in the orbit an has a maximum

Problem 15 (20 pts)

A particle of mass m slides down an inclined plane with inclination angle θ under the influence of gravity If the motion is resisted by a force f = kmv2 show that the time required to move a distance d after starting from rest is

coshminus1(ekd)t = radic

kg sin θ

2



Suggested Reading


Problems

Problem 21 (20 pts)


Problem 22 (20 pts)


Problem 23 (20 pts)


Problem 24 (20 pts)




Problem 25 (20 pts)


1



Suggested Reading


Problems

Problem 31 (20 pts)


Problem 32 (20 pts)


Problem 33 (20 pts)


Problem 34 (20 pts)


Problem 35 (20 pts)


1



Suggested Reading


Problems

Problem 41 (25 pts)


Problem 42 (25 pts)


Problem 43 (25 pts)


Problem 44 (25 pts)


1



Suggested Reading


Problems

Problem 51 (20 pts)


Problem 52 (20 pts)


Problem 53 (20 pts)


Problem 54 (20 pts)


Problem 55 (20 pts)


1



Suggested Reading


Problems

Problem 61 (25 pts)


Problem 62 (25 pts)


Problem 63 (25 pts)


Problem 64 (25 pts)


1



Suggested Reading


Problems

Problem 71 (20 pts)


Problem 72 (20 pts)


Problem 73 (20 pts)


Problem 74 (20 pts)


Problem 75 (20 pts)


1



Suggested Reading


Problems

Problem 81 (25 pts)


Problem 82 (25 pts)


Problem 83 (25 pts)


Problem 84 (25 pts)


1



Suggested Reading

Goldstein Chapter 5

Problems

Problem 91 (25 pts)


Problem 92 (25 pts)


Problem 93 (25 pts)


Problem 94 (25 pts)


1



Suggested Reading


Problems









1




Instructions







2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South




Suggested Reading


Problems

Problem 21 (20 pts)


Problem 22 (20 pts)


Problem 23 (20 pts)


Problem 24 (20 pts)




Problem 25 (20 pts)


1



Suggested Reading


Problems

Problem 31 (20 pts)


Problem 32 (20 pts)


Problem 33 (20 pts)


Problem 34 (20 pts)


Problem 35 (20 pts)


1



Suggested Reading


Problems

Problem 41 (25 pts)


Problem 42 (25 pts)


Problem 43 (25 pts)


Problem 44 (25 pts)


1



Suggested Reading


Problems

Problem 51 (20 pts)


Problem 52 (20 pts)


Problem 53 (20 pts)


Problem 54 (20 pts)


Problem 55 (20 pts)


1



Suggested Reading


Problems

Problem 61 (25 pts)


Problem 62 (25 pts)


Problem 63 (25 pts)


Problem 64 (25 pts)


1



Suggested Reading


Problems

Problem 71 (20 pts)


Problem 72 (20 pts)


Problem 73 (20 pts)


Problem 74 (20 pts)


Problem 75 (20 pts)


1



Suggested Reading


Problems

Problem 81 (25 pts)


Problem 82 (25 pts)


Problem 83 (25 pts)


Problem 84 (25 pts)


1



Suggested Reading

Goldstein Chapter 5

Problems

Problem 91 (25 pts)


Problem 92 (25 pts)


Problem 93 (25 pts)


Problem 94 (25 pts)


1



Suggested Reading


Problems









1




Instructions







2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South




Suggested Reading


Problems

Problem 31 (20 pts)


Problem 32 (20 pts)


Problem 33 (20 pts)


Problem 34 (20 pts)


Problem 35 (20 pts)


1



Suggested Reading


Problems

Problem 41 (25 pts)


Problem 42 (25 pts)


Problem 43 (25 pts)


Problem 44 (25 pts)


1



Suggested Reading


Problems

Problem 51 (20 pts)


Problem 52 (20 pts)


Problem 53 (20 pts)


Problem 54 (20 pts)


Problem 55 (20 pts)


1



Suggested Reading


Problems

Problem 61 (25 pts)


Problem 62 (25 pts)


Problem 63 (25 pts)


Problem 64 (25 pts)


1



Suggested Reading


Problems

Problem 71 (20 pts)


Problem 72 (20 pts)


Problem 73 (20 pts)


Problem 74 (20 pts)


Problem 75 (20 pts)


1



Suggested Reading


Problems

Problem 81 (25 pts)


Problem 82 (25 pts)


Problem 83 (25 pts)


Problem 84 (25 pts)


1



Suggested Reading

Goldstein Chapter 5

Problems

Problem 91 (25 pts)


Problem 92 (25 pts)


Problem 93 (25 pts)


Problem 94 (25 pts)


1



Suggested Reading


Problems









1




Instructions







2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South




Suggested Reading


Problems

Problem 41 (25 pts)


Problem 42 (25 pts)


Problem 43 (25 pts)


Problem 44 (25 pts)


1



Suggested Reading


Problems

Problem 51 (20 pts)


Problem 52 (20 pts)


Problem 53 (20 pts)


Problem 54 (20 pts)


Problem 55 (20 pts)


1



Suggested Reading


Problems

Problem 61 (25 pts)


Problem 62 (25 pts)


Problem 63 (25 pts)


Problem 64 (25 pts)


1



Suggested Reading


Problems

Problem 71 (20 pts)


Problem 72 (20 pts)


Problem 73 (20 pts)


Problem 74 (20 pts)


Problem 75 (20 pts)


1



Suggested Reading


Problems

Problem 81 (25 pts)


Problem 82 (25 pts)


Problem 83 (25 pts)


Problem 84 (25 pts)


1



Suggested Reading

Goldstein Chapter 5

Problems

Problem 91 (25 pts)


Problem 92 (25 pts)


Problem 93 (25 pts)


Problem 94 (25 pts)


1



Suggested Reading


Problems









1




Instructions







2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South




Suggested Reading


Problems

Problem 51 (20 pts)


Problem 52 (20 pts)


Problem 53 (20 pts)


Problem 54 (20 pts)


Problem 55 (20 pts)


1



Suggested Reading


Problems

Problem 61 (25 pts)


Problem 62 (25 pts)


Problem 63 (25 pts)


Problem 64 (25 pts)


1



Suggested Reading


Problems

Problem 71 (20 pts)


Problem 72 (20 pts)


Problem 73 (20 pts)


Problem 74 (20 pts)


Problem 75 (20 pts)


1



Suggested Reading


Problems

Problem 81 (25 pts)


Problem 82 (25 pts)


Problem 83 (25 pts)


Problem 84 (25 pts)


1



Suggested Reading

Goldstein Chapter 5

Problems

Problem 91 (25 pts)


Problem 92 (25 pts)


Problem 93 (25 pts)


Problem 94 (25 pts)


1



Suggested Reading


Problems









1




Instructions







2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South




Suggested Reading


Problems

Problem 61 (25 pts)


Problem 62 (25 pts)


Problem 63 (25 pts)


Problem 64 (25 pts)


1



Suggested Reading


Problems

Problem 71 (20 pts)


Problem 72 (20 pts)


Problem 73 (20 pts)


Problem 74 (20 pts)


Problem 75 (20 pts)


1



Suggested Reading


Problems

Problem 81 (25 pts)


Problem 82 (25 pts)


Problem 83 (25 pts)


Problem 84 (25 pts)


1



Suggested Reading

Goldstein Chapter 5

Problems

Problem 91 (25 pts)


Problem 92 (25 pts)


Problem 93 (25 pts)


Problem 94 (25 pts)


1



Suggested Reading


Problems









1




Instructions







2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South




Suggested Reading


Problems

Problem 71 (20 pts)


Problem 72 (20 pts)


Problem 73 (20 pts)


Problem 74 (20 pts)


Problem 75 (20 pts)


1



Suggested Reading


Problems

Problem 81 (25 pts)


Problem 82 (25 pts)


Problem 83 (25 pts)


Problem 84 (25 pts)


1



Suggested Reading

Goldstein Chapter 5

Problems

Problem 91 (25 pts)


Problem 92 (25 pts)


Problem 93 (25 pts)


Problem 94 (25 pts)


1



Suggested Reading


Problems









1




Instructions







2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South




Suggested Reading


Problems

Problem 81 (25 pts)


Problem 82 (25 pts)


Problem 83 (25 pts)


Problem 84 (25 pts)


1



Suggested Reading

Goldstein Chapter 5

Problems

Problem 91 (25 pts)


Problem 92 (25 pts)


Problem 93 (25 pts)


Problem 94 (25 pts)


1



Suggested Reading


Problems









1




Instructions







2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South




Suggested Reading

Goldstein Chapter 5

Problems

Problem 91 (25 pts)


Problem 92 (25 pts)


Problem 93 (25 pts)


Problem 94 (25 pts)


1



Suggested Reading


Problems









1




Instructions







2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South




Suggested Reading


Problems









1




Instructions







2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South





Instructions







2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South


2

Useful Formulae



v = v0 +

dta t=0

t=t =t

r = r0 + v0t +

dt t

dtF (t)m t=0 t =0





Polar Coordinates






1 + x2 1 minus x2

3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South


3



dL = α = constant













4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South


4

m

g

R0 Ω






L1

g

m 1

L 2

m 2








Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South





Instructions







2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South


2

Useful Formulae



t=0 t=t t=t


Gravitational Law

F = minus Gm1m2

r122r12



partt



partq partp

Generating function


partq partQ

Poisson Brackets





Polar Coordinates


Orbit Equation


2u2 r

ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South


ε =

3

Effective Potential

2


Keplerian Orbits


r r

α = 2

ε = 1 +2E2

τ 2 =4π2micro

a 3







Scattering











4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South


4



p2 mω2q2

H = + 2m 2







5







6










7

x

yNorth

East

West

South


5







6










7

x

yNorth

East

West

South


6










7

x

yNorth

East

West

South


7

x

yNorth

East

West

South


Documents

Classical mechanics problems - MIT course