Find approximately the third cosmic
velocity v3, i.e. the minimum velocity that has to
be imparted to a body relative to the Earth’s surface to drive it out of the
Solar system. The rotation of the Earth about its own axis is to be neglected.
I understand Physics because I fight with problem unless I solve it.............RKH SIR
Tuesday, 11 September 2012
PROBLEM:1.232
What is the minimum work that has to be
performed to bring a spaceship of mass m = 2.0*103 kg from the surface of the Earth to
the Moon?
PROBLEM:1.231
At
what distance from the centre of the Moon is the point at which the strength of
the resultant of the Earth’s and Moon’s gravitational fields is equal to zero?
The Earth’s mass is assumed to be n = 81 times that of the Moon, and the
distance between the centres of these planets 60 times greater than the radius of the Earth R.
PROBLEM:1.230
A
spaceship is launched into a circular orbit close to the Earth’s surface. What additional
velocity has to be imparted to the spaceship to overcome the gravitational
pull?
PROBLEM:1.229
A
spaceship approaches the Moon along a parabolic trajectory which is almost
tangent to the Moon’s surface. At the moment of the maximum approach the brake
rocket was fired for a short time interval, and the spaceship was transferred
into a circular orbit of a Moon satellite. Find how the spaceship velocity
modulus increased in the process of braking.
PROBLEM:1.228
Calculate
the orbital and escape velocities for the Moon. Compare the results obtained
with the corresponding velocities for the Earth.
PROBLEM:1.227
An
artificial satellite of the Moon revolves in a circular orbit whose radius
exceeds the radius of the Moon r) times. In the process of motion the satellite
experiences a slight resistance due to cosmic dust. Assuming the resistance
force to depend on the velocity of the satellite as F = av2, where a is a constant, find how
long the satellite will stay in orbit until it falls onto the Moon’s surface.
PROBLEM:1.226
A
satellite must move in the equatorial plane of the Earth close to its surface
either in the Earth’s rotation direction or against it. Find how many times the
kinetic energy of the satellite in the latter case exceeds that in the former
case (in the reference frame fixed to the Earth).
PROBLEM:1.225
A
satellite revolves from east to west in a circular equatorial orbit of radius R = 1.00*104 km around the Earth. Find the velocity
and the acceleration of the satellite in the reference frame fixed to the
Earth.
PROBLEM:1.224
A
satellite revolving in a circular equatorial orbit of radius R = 2.00*104 km from west to east appears over a
certain point at the equator every t=11.6 hours. Using these data, calculate the mass
of the Earth. The gravitational constant is supposed to be known.
PROBLEM:1.223
Calculate
the radius of the circular orbit of a stationary Earth’s satellite, which
remains motionless with respect to its surface. What are its velocity and
acceleration in the inertial reference frame fixed at a given moment to the
centre of the Earth?
PROBLEM:1.222
An artificial satellite is launched into a circular
orbit around the Earth with velocity v relative
to the reference frame moving transitionally and fixed to the Earth’s
rotation axis. Find the distance from the satellite to the Earth’s surface. The
radius of the Earth and the free-fall acceleration on its surface are supposed
to be known.
PROBLEM:1.221
On
the pole of the Earth a body is imparted velocity v0 directed vertically up. Knowing the
radius of the Earth and the free- fall acceleration on its surface, find the
height to which the body will ascend. The air drag is to be neglected.
PROBLEM:1.220
At
what height over the Earth’s pole the free-fall acceleration decreases by one
per cent; by half?
PROBLEM:1.219
Calculate
the ratios of the following accelerations: the acceleration w1 due to the gravitational force on the
Earth’s surface the
acceleration w2 due to the centrifugal force of
inertia on the Earth’s equator, and the acceleration w3 caused by the Sun to the bodies on the
Earth.
PROBLEM:1.218
Two
Earth’s satellites move in a common plane along circular orbits. The orbital
radius of one satellite r = 7000 km while that of the other
satellite is 70 km less. What time interval
separates the periodic approaches of the satellites to each other over the
minimum distance?
PROBLEM:1.217
Find
the proper potential energy of gravitational interaction of matter forming
(a)
a
thin uniform spherical layer of mass m and radius R;
(b) a
uniform sphere of mass m and radius R (make use of the answer to Problem 1.214).
PROBLEM:1.216
A
uniform sphere has a mass M and radius R. Find the pressure p inside the sphere, caused by
gravitational compression, as a function of the distance r from its centre.
Evaluate p at the centre of the Earth, assuming it
to be a uniform sphere.
PROBLEM:1.215
Inside
a uniform sphere of density p there is a spherical cavity whose centre is at a
distance I from the centre of the sphere. Find
the strength G of the gravitational field inside the cavity.
PROBLEM:1.214
There
is a uniform sphere of mass M and radius R. Find the strength G and the potential of the
gravitational field of this sphere as a function of the distance r from its
centre (with r is less than R and grater than R). Draw the approximate plots of the functions G (r) and V(r).
PROBLEM:1.213
A particle of mass m was
transferred from the centre of the base of a uniform hemisphere of mass M and radius
R into infinity.What work
was performed in the process by the gravitational force exerted on the particle
by the hemisphere?
PROBLEM:1.212
Demonstrate that the gravitational force acting on
a particle A inside a uniform spherical
layer of matter is equal to zero.
PROBLEM:1.211
A particle of mass m is located
outside a uniform sphere of mass M at a
distance r from its centre. Find:`
(a)
the potential energy of gravitational interaction
of the particle and the sphere;
(b)
the gravitational force which the sphere exerts on
the particle.
PROBLEM:1.210
A cosmic body A moves to
the Sun with velocity v0 (when far
from the Sun) and aiming parameter L the arm of
the vector v0 relative to the centre of the Sun .
Find the minimum distance
by which this body will get to the Sun.
PROBLEM:1.209
A planet A moves along an elliptical orbit around
the Sun. At the moment when it was at the distance r0 from the Sun its velocity was equal
to v0 and the angle between the radius vector r0 and the velocity vector v0 was equal to a. Find the maximum and minimum
distances that will separate this planet from the Sun during its orbital
motion.
PROBLEM:1.208
Using
the conservation laws, demonstrate that the total mechanical energy of a planet
of mass m moving around the Sun along an ellipse depends
only on its semi-major axis a. Find this energy as a function of a
PROBLEM:1.207
A planet of mass m moves along an ellipse around the Sun
so that its maximum and minimum distances from the Sun are equal to r1 and r2 respectively. Find the angular momentum M of this planet relative to the centre
of the Sun.
PROBLEM:1.206
Find the potential energy of the
gravitational interaction
(a)
of two mass points of masses m1 and m2 located at a distance r from each
other;
(b)
of a mass point of mass m and a thin uniform rod of mass M and length L, if they are located along
a straight line at a distance a from each
other; also find the force of their interaction.
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