Part IMechanics
M03M.1Lagrange points and WMAP
M03M.1Lagrange points and WMAP
Problem
The Earth is ins a circular orbit of angular frequency about the Sun. The Sun is so much
more massive that the Earth that, for our purposes, it may take to sit at rest a
Part I - Mechanics
J06M.1 - Gyroscope
J06M.1 - Gyroscope
Problem
A gyroscope, illustrated in the gures below, is free to pivot about point O under the eect of gravity. Its total
mass is M and its center of mass is located at point P at a distance R from O
Part I - Mechanics
J09M.1 - Coupled Pendula
J09M.1 - Coupled Pendula
Problem
Two simple pendula, each of length l and mass m, are coupled by a spring of force constant k. The spring is attached
to the rods of the pendula, which are massless and inextensib
Part I - Mechanics
J05M.1 - Rope Around a Cylinder
J05M.1 - Rope Around a Cylinder
Problem
A long rope is wound around a cylinder of radius r so that a length, l, of the rope is in contact with the cylinder.
The coecient of static friction between te rope
Part I - Mechanics
J10M.1 - Rod on a Rail (M93M.2)
J10M.1 - Rod on a Rail (M93M.2)
Problem
s
g
z
x
A uniform rod of length and mass m moves in the x-z plane. One end of the rod is suspended
from a straight rail that slopes downwards with an angle relative
Part I - Mechanics
M07M.1 - Planetary Orbits
M07M.1 - Planetary Orbits
Problem
A satellite in a low Earth circular orbit with Radius R0 has an orbital period T0 .
a)
How long does it take to transfer the satellite into a new circular orbit with a larger r
Part I - Mechanics
M08M.1 - Bead on a Hoop
M08M.1 - Bead on a Hoop
Problem
A bead of mass m slides without friction on a circular loop of radius a and mass M . The loop lies in a vertical plane
and rotates about a vertical diameter with angular velocity .
Part I - Mechanics
M09M.1 - Bubble in an Incompressible Fluid (J07M.3, J94M.1)
M09M.1 - Bubble in an Incompressible Fluid (J07M.3, J94M.1)
Problem
An ideal incompressible uid of density contains a bubble of radius R(t). The uid pressure is
held constant a
Part I - Mechanics
J08M.1 - Pendulum on a Sled
J08M.1 - Pendulum on a Sled
Problem
A plane pendulum consists of a bob of mass m suspended by a massless rigid rod of length l that is hinged to a sled
of mass M . The sled slides without friction on a horizo
Part IMechanics
M99M.1Ball Rolling in a Cylinder
M99M.1Ball Rolling in a Cylinder
Problem
A solid ball of radius r and mass m is rolling without slipping inside a long hollow vertical cylinder
of radius R > r under the inuence of gravity. Initially the ve
Part IMechanics
J04M.1Bead on a Wire (J06M.3)
J04M.1Bead on a Wire (J06M.3)
Problem
A bead of mass m slides without friction on a wire whose shape is
z(r) = a
r
a
4
The wire rotates about the z axis with constant angular velocity . Earths gravity causes a
Part IMechanics
J02M.1Flapping Toy
J02M.1Flapping Toy
Problem
Deduce the frequency of small oscillations of the apping toy shown in the gure below, supposing
the central mass m moves only vertically, and the motion of the others masses is only in the x-y
Part IMechanics
J03M.1Scattering from an Attractive Potential
J03M.1Scattering from an Attractive Potential
Problem
This problem is about scattering by an attractive potential.
a)
Consider a particle with energy E and z < 0 approaching the z = 0 plane at
Part IMechanics
J98M.1Hanging Rope
J98M.1Hanging Rope
Problem
A piece of thin uniform unstretchable rope has length 2L and mass M . Its ends are attached to
points at the same height separated by distance 2w, and the rope hangs between them under the
inue
Part IMechanics
M00M.1Precession of the Perihelion
M00M.1Precession of the Perihelion
Problem
With Newtonian mechanics, we wish to compute the rate of precession of the perihelion (point of
closest approach) of a planet in orbit around a stationary ring-s
Part IMechanics
M02M.1Particle in a Cone
M02M.1Particle in a Cone
Problem
A small particle of mass m is constrained to slide, without friction, on the inside of a circular cone
whose vertex is at the origin and whose axis is along the z-axis. The half ang
Part IMechanics
M04M.1Particles on a Line
M04M.1Particles on a Line
Problem
Two elastic spherical particles with masses m and M (m
M ) are constrained to move along a
straight line with an elastically reecting wall at its end. At t = 0 they are in motion
Part IMechanics
M98M.1Mass on a Rope and Cylinder
M98M.1Mass on a Rope and Cylinder
Problem
A mass m is lifted by means of a rope drawn across a cylinder as sketched in the gure. The
cylinder is xed so that it does not rotate. A steady horizontal tension
Part IMechanics
J00M.1Shape of an Arch
J00M.1Shape of an Arch
Problem
The shape of an arch is determined by the condition that each brick is held in place by the normal
force of its neighbors, with no need for mortar or glue. To model this consider a thin