Classical mechanics problems - MIT course

Massachusetts Institute of Technology Physics 8.09 Fall 2006 Homework 1 due Wednesday, September 13, 2006 before 12:30 PM

Suggested Reading Refresh your mechanics knowledge by reading from your favorite Mechanics book. For example the book by Marion & Thornton is very helpful. Goldstein is a bit too dry for this initial “Newtonian mechanics refresher”.

Problems Problem 1.1 (20 pts) A block of mass of mass m1 rests on top of a of a block of mass of mass m2 . The static and kinetic coefficients of friction of friction between the two blocks and the bottom block and the table are µs and µk . A fixed force F is applied to the bottom block. friction

m

1

friction m

F 2

Describe the motion and find the accelerations of the two blocks a1 and a2 for all possible values of F from F = 0 → ∞.

Problem 1.2 (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 1.3 (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 of the velocity and the force on the mass at the bottom of the 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

h

R

45

B A

x

Problem 1.4 (20 pts) A particle moves in a two dimensional orbit defined by x(t) = A(2αt − sin αt) y (t) = A(1 − 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 1.5 (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 = kmv 2 , show that the time required to move a distance d after starting from rest is cosh 1 (ekd ) √ kg sin θ −

t=

2

Massachusetts Institute of Technology Physics 8.09 Fall 2006 Homework 2 due Wednesday, September 20, 2006 at 12:30 PM

Suggested Reading Goldstein Sections 1.1-1.2 and 2.1-2.5

Problems Problem 2.1 (20 pts) Goldstein, Problem 6, Chapter 2, Page 64

Problem 2.2 (20 pts) Goldstein, Problem 7, Chapter 2, Page 65


Problem 2.4 (20 pts) Consider a region of space divided by a plane. The potential energy of a particle in region 1 is U 1 and in region 2 is U 2 . 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 = sin θ2 where

T 1

=

1 2

2 . mv1

�

1+

U 1

−

U 2

T 1

What is the optical analog of this problem ?

Problem 2.5 (20 pts) A pendulum consists of a mass m suspended by a massless spring with unextended length k . Find Lagrange’s equations of motion.

1

b

and spring constant

Massachusetts Institute of Technology Physics 8.09 Fall 2006 Homework 3 due Wednesday, September 27, 2006 at 12:30 PM

Suggested Reading Goldstein Chapter 1.3, 1.4, 2.4, 2.6, 2.7






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Massachusetts Institute of Technology Physics 8.09 Fall 2006 Homework 4 due Wednesday, October 11, 2005 at 12:30 PM

Suggested Reading Goldstein Sections 8.1 and 8.2





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Massachusetts Institute of Technology Physics 8.09 Fall 2006 Homework 5 due Monday, October 18, 2006 at 12:30 PM

Suggested Reading Goldstein Chapter.Section Canonical Transformations: 9.1-9.3,9.6,9.7

Problems Problem 5.1 (20 pts) Goldstein, Chapter 9, Problem 4, page 422.

Problem 5.2 (20 pts) Goldstein, Chapter 9, Problem 9, page 423.




1

Massachusetts Institute of Technology Physics 8.09 Fall 2006 Homework 6 due Wednesday, October 25, 2006 at 12:30 PM

Suggested Reading Goldstein Kepler Problem 3.1-3.3

Problems Problem 6.1 (25 pts) Goldstein, Chapter 3, Problem 11, page 128.


Problem 6.3 (25 pts) Goldstein, Chapter 3, Exercise 19, page 129.

Problem 6.4 (25 pts) Goldstein, Chapter 3, Exercise 21, page 130.

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Massachusetts Institute of Technology Physics 8.09 Fall 2006 Homework 7 due Monday, November 6, 2006 at 12:30 PM (Note unusual date)

Suggested Reading Goldstein Collisions 3.10-3.11, lecture material, Marion and Thornton 9.6-9.10 (Goldstein doesn’t contain a lot about scattering)

Problems Problem 7.1 (20 pts) Marion & Thornton 9-26

Problem 7.2 (20 pts) Marion & Thornton 9-33




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Massachusetts Institute of Technology Physics 8.09 Fall 2006 Homework 8 due Monday, November 8, 2006 at 12:30 PM

Suggested Reading Lecture Material, Goldstein Sections 4.9, 4.10, 5.3 and 5.4. Also Chapter 10, Marion and Thronton.

Problems Problem 8.1 (25 pts) Goldstein, Exercise 22, page 183

Problem 8.2 (25 pts) Marion and Thornton 10-12

Problem 8.3 (25 pts) Marion and Thornton 10-15

Problem 8.4 (25 pts) Goldstein, Exercise 17, page 235

1

Massachusetts Institute of Technology Physics 8.09 Fall 2006 Homework 9 due Wednesday, November 29, 2006 at 12:30 PM

Suggested Reading Goldstein Chapter 5

Problems Problem 9.1 (25 pts) Goldstein, Derivation 6, page 233. Ignore the last sentence of question c) (Poinsot construction)




1

Massachusetts Institute of Technology Physics 8.09 Fall 2006 Homework 10 due Wednesday, December 6, 2006 at 12:30 PM

Suggested Reading Goldstein Sections 1.5, 8.5 and Chapter 13

Problems Problem 10.1 (25 pts) Goldstein, Exercise 1, page 598




1

Massachusetts Institute of Technology Department of Physics Course: 8.09 Classical Mechanics Term: Fall 2006

Quiz 1 October 4, 2006

Instructions •

Do not start until you are told to do so.

•

Solve all problems.

•

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

•

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

•

Mark the final answers.

•

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

2

Useful Formulae Newton and Basic Kinematics:  = p˙ = ma for v F v = v0 +

�

c



�

t =t

t� =0

�

dt a

�

t� =t

r = r0 + v0 t +

�

t =0 �

dt

�

t�� =t�

t =0 ��

 (t )/m dt F ��

��

Lagrangian and Hamiltonian: L(q, q˙) = T − U ;

H (p, q) = T + U = pq˙ − L

Euler-Lagrange (without and with constraints): ∂L ∂x

−

d ∂L = 0; dt ∂x˙

∂L ∂x

−

d ∂L + dt ∂x˙

 λ ∂g a

a

a

∂x

=0

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 = arctan(x); 1 + x2

�

dx = arctanh(x) 1 − 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

∂L ∂t

, is it equal to zero ?

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 Hamilto nian. 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 gravi tational 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 Multipli ers.

4

m

Ω

R0 g

Figure 1: Particle on a rotating wire

d) Obtain the expressions for the forces of constraints. Give the physical interpre tation 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 m1

L

m

2

2

Figure 2: Double Pendulum

a) How many degrees of freedom does the system have? b) Write the Lagrangian for the system. c) Write equations of motion for the system. Do not solve.

Massachusetts Institute of Technology Department of Physics Course: 8.09 Classical Mechanics Term: Fall 2006

Quiz 2 November 15, 2006

Instructions

• Do not start until you are told to do so. • Solve all problems. • Put your name on the covers of all notebooks you are using. • Show all work neatly in the white book, label the problem you are working on.

• Mark the final answers. • Books and notes are not to be used. You may use your calculator.

2

Useful Formulae Newton and Basic Kinematics:  = p˙ = ma for v F v = r =

�

t

� v +

=t

c



dt a

0

t� =0

t� =t

� r + v t + 0

0

t

�

=0

t�� =t�

� dt 

t

��

=0

 (t )/m dt F 



Gravitational Law:  = F

− Gmr m rˆ 1

2

2 12

12

Lagrangian and Hamiltonian: L(q, q˙) = T

− U ;

H (p, q) = T + U = p

∂q ∂t

−L

Hamilton Equation of Motion: ∂H = ∂q

∂H = q˙ ∂p

−p˙;

Generating function: ∂F (Q, q) = p; ∂q

∂F (Q, q) = ∂Q

−P

Poisson Brackets: [g, f ] =

∂g ∂f ∂q ∂p

∂f − ∂g ∂p ∂q

Euler-Lagrange (without and with constraints): ∂L ∂x

− dtd ∂L = 0; ∂x˙

∂L ∂x

∂g − dtd ∂L +λ =0 ∂x˙ ∂x

Polar Coordinates: x = r sin θ cos φ; y = r sin θ sin φ; z = r cos θ Orbit Equation: 

u +u=

−  µu F (u) 2

2

with u =

1 r

3 Effective Potential: 2 2µr2

V (r) = U (r) + Keplerian Orbits:

α = ε cos θ + 1 r 2E 2 4π2 µ 3 2 2 ; ε = 1+ ; τ = α= a µk µk2 k α α ; rmax = a(1 + ε) = rmin = a(1 ε) = 1+ε 1 ε U (r) =

− kr ;



−

−

Spherical Coordinates: x = r sin θ cos φ; y = r sin θ sin φ; z = r cos θ Inelastic Scattering (coefficient of restitution): =

|v − v | |u − u | 2

1

2

1

Scattering:

 

dσ dσ b db = = σ(θ) = sin θ dθ dΩ dφ sin θdθ sin θ π tan ψ = ; ζ = cos θ + (m1 /m2 ) 2

 

− 2θ ;

Vector in fixed frame expressed in terms of vector in rotating frame and rotation velocity.   dX dX )inertial = ( )rotating + ( ω dt dt Acceleration in accelerated and rotating frame: (

a = g

× X  )

− V  ˙ − (ω˙ × r) − 2(ω × v) − (ω × ( ω × r))

Useful Trigonometrical Formulas sin(A + B) = sin A cos B + cos A sin B; cos(A + B) = cos A cos B sin A sin B;

−

4

Problem 1:

Canonical Transformation (35 points)

Consider a harmonic oscillator with Hamiltonian: p2 mω2q 2 + 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) = z 3 for z > 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 m/s 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 = 6 106 m and that the angular velocity of Earth’s rotation is Ω = 7 10 5 s 1 . Gravitational acceleration is g = 10 m/s2 . 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 expres   sion for acceleration ai = (ax , ay , az ) in terms of components of Ω, r = (x,y,z) and 2 2 ˙ ˙ z˙). Ignore very small terms proportional to Ω x, Ω y and replace z with R. v = (x,y, 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 E,N,W,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 Earth’s surface at that lattitude. d) Based on the magnitudes and signs of the terms of a obtained in c) estimate the effect of the earth’s 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

North

y

East

x West

South

Figure 1: Impact Point Graph

Classical mechanics problems - MIT course

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