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physics
mechanics
Vector Mechanics For Engineers Statics And Dynamics 8th Edition Ferdinand Beer, E. Russell Johnston, Jr., Elliot Eisenberg, William Clausen, David Mazurek, Phillip Cornwell - Solutions
A small body starts falling onto the Sun from a distance equal to the radius of the Earth's orbit. The initial velocity of the body is equal to zero in the heliocentric reference frame. Making use of Kepler's laws, find how long the body will be falling.
Suppose we have made a model of the Solar system scaled down in the ratio η but of materials of the same mean density as the actual materials of the planets and the Sun. How will the orbital periods of revolution of planetary models change in this case?
A double star is a system of two stars moving around the centre of inertia of the system due to gravitation. Find the distance between the components of the double star, if its total mass equals M and the period of revolution T.
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
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.
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.
A planet A moves along an elliptical orbit around the Sun. At the moment when it was at the distance ro from the Sun its velocity 'was equal to vo and the angle between the radius vector ro and the velocity vector vo was equal to a. Find the maximum and minimum distances that will separate this
A cosmic body A moves to the Sun with velocity vo (when far from the Sun) and aiming parameter l the arm of the vector vo relative to the centre of the Sun (Fig. 1.51). Find the minimum distance by which this body will get to the Sun.
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.
Demonstrate that the gravitational force acting on a particle A inside a uniform spherical layer of matter is equal to zero.
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?
There is a uniform sphere of mass M and radius R. Find the strength G and the potential φ of the gravitational field of this sphere as a function of the distance r from its centre (with r < R and r > R). Draw the approximate plots of the functions G (r) and φ (r).
Inside a uniform sphere of density p there is a spherical cavity whose centre is at a distance l from the centre of the sphere. Find the strength G of the gravitational field inside the cavity.
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.
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).
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 Δr = 70 km less. What time interval separates the periodic approaches of the satellites to each other over the minimum distance?
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.
At what height over the Earth's pole the free-fall acceleration decreases by one per cent; by half?
On the pole of the Earth a body is imparted velocity vo 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.
An artificial satellite is launched into a circular orbit around the Earth with velocity v relative to the reference frame moving translationally 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
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?
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 τ = 11.6 hours. Using these data, calculate the mass of the Earth. The gravitational constant is supposed to be known.
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.
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).
An artificial satellite of the Moon revolves in a circular orbit whose radius exceeds the radius of the Moon η 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,
Calculate the orbital and escape velocities for the Moon. Compare the results obtained with the corresponding velocities for the Earth.
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
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?
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 η = 81 times that of the Moon, and the distance between the centres of these planets n = 60 times
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?
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.
A thin uniform rod AB of mass m = 1.0 kg moves translationally with acceleration w = 2.0 m/s2 due to two antiparallel forces F1 and F2 (Fig. 1.52). The distance between the points at which these forces are applied is equal to a = 20 cm. Besides, it is known that F2 = 5.0 N. Find the length of the
A force F = Ai +0 Bj is applied to a point whose radius vector relative to the origin of coordinates O is equal to r = ai + bj, where a, b, A, B are constants, and i, j are the unit vectors of the x and y axes. Find the moment N and the arm l of the force F relative to the point O.
A force F1 = Aj is applied to a point whose radius vector r1 = ai, while a force F2 = Bi is applied to the point whose radius vector r2 = bj. Both radius vectors are determined relative to the origin of coordinates O, i and j are the unit vectors of the x and y axes, a, b, A, B are constants. Find
Three forces are applied to a square plate as shown in Fig. 1.53. Find the modulus, direction, and the point of application of the resultant force, if this point is taken on the side BC.
Find the moment of inertia (a) of a thin uniform rod relative to the axis which is perpendicular to the rod and passes through its end, if the mass of the rod is m and its length l; (b) of a thin uniform rectangular plate relative to the axis passing perpendicular to the plane of the plate through
Calculate the moment of inertia (a) of a copper uniform disc relative to the symmetry axis perpendicular to the plane of the disc, if its thickness is equal to b = 2.0 mm and its radius to R = 100 mm; (b) Of a uniform solid cone relative to its symmetry axis, if the mass of the cone is equal to m
A uniform disc of radius R = 20 cm has a round cut as shown in Fig. 1.54. The mass of the remaining (shaded) portion of the disc equals m = 7.3 kg. Find the moment of inertia of such a disc relative to the axis passing through its centre of inertia and perpendicular to the plane of the disc.
Demonstrate that in the case of a thin plate of arbitrary shape there is the following relationship between the moments of inertia: I1 + I2 = I3, where subindices 1, 2, and 3 define three mutually perpendicular axes passing through one point, with axes 1 and 2 lying in the plane of the plate. Using
Using the formula for the moment of inertia of a uniform sphere, find the moment of inertia of a thin spherical layer of mass m and radius R relative to the axis passing through its centre.
A light thread with a body of mass m tied to its end is wound on a uniform solid cylinder of mass M and radius R (Fig. 1.55). At a moment t = 0 the system is set in motion. Assuming the friction in the axle of the cylinder to be negligible, find the time dependence of(a) The angular velocity of the
The ends of thin threads tightly wound on the axle of radius r of the Maxwell disc are attached to a horizontal bar. When the disc unwinds, the bar is raised to keep the disc at the same height. The mass of the disc with the axle is equal to m, the moment of inertia of the arrangement relative to
A thin horizontal uniform rod AB of mass m and length l can rotate freely about a vertical axis passing through its end A. At a certain moment the end B starts experiencing a constant force F which is always perpendicular to the original position of the stationary rod and directed in a horizontal
In the arrangement shown in Fig. 1.56 the mass of the uniform solid cylinder of radius R is equal to m and the masses of two bodies are equal to m1 and m2. The thread slipping and the friction in the axle of the cylinder are supposed to be absent. Find the angular acceleration of the cylinder and
In the system shown in Fig. 1.57 the masses of the bodies are known to be ml and m2, the coefficient of friction between the body m1 and the horizontal plane is equal to k, and a pulley of mass m is assumed to be a uniform disc. The thread does not slip over the pulley. At the moment t = 0 the body
A uniform cylinder of radius R is spinned about its axis to the angular velocity ωo and then placed into a corner (Fig. 1.58).The coefficient of friction between the corner walls and the cylinder is equal to k. How many turns will the cylinder accomplish before it stops?
A uniform disc of radius R is spinned to the angular velocity ω and then carefully placed on a horizontal surface. How long will the disc be rotating on the surface if the friction coefficient is equal to k? The pressure exerted by the disc on the surface can be regarded as uniform.
A flywheel with the initial angular velocity ωo decelerates due to the forces whose moment relative to the axis is proportional to the square root of its angular velocity. Find the mean angular velocity of the flywheel averaged over the total deceleration time.
A uniform cylinder of radius R and mass M can rotate freely about a stationary horizontal axis O (Fig. 1.59). A thin cord of length l and mass m is wound on the cylinder in a single layer. Find the angular acceleration of the cylinder as a function of the length x of the hanging part of the cord.
A uniform sphere of mass m and radius R rolls without slipping down an inclined plane set at an angle a to the horizontal. Find: (a) The magnitudes of the friction coefficient at which slipping is absent; (b) The kinetic energy of the sphere t seconds after the beginning of motion.
A uniform cylinder of mass m = 8.0 kg and radius R = 1.3 cm (Fig.1.60) starts descending at a moment t = 0 due to gravity. Neglecting the mass of the thread, find:(a) The tension of each thread and the angular acceleration of the cylinder;(b) The time dependence of the instantaneous power developed
Thin threads are tightly wound on the ends of a uniform solid cylinder of mass m. The free ends of the threads are attached to the ceiling of an elevator car. The car starts going up with an acceleration wo. Find the acceleration w' of the cylinder relative to the car and the force F exerted by the
A spool with a thread wound on it is placed on an inclined smooth plane set at an angle a = 30° to the horizontal. The free end of the thread is attached to the wall as shown in Fig. 1.61. The mass of the spool is m = 200 g, its moment of inertia relative to its own axis I = 0.45 g.m2, the
A uniform solid cylinder of mass m rests on two horizontal planks. A thread is wound on the cylinder. The hanging end of the thread is pulled vertically down with a constant force F (Fig. 1.62).Find the maximum magnitude of the force F which still does not bring about any sliding of the cylinder,
A spool with thread wound on it, of mass m, rests on a rough horizontal surface. Its moment of inertia relative to its own axis is equal to I = ymR2, where y, is a numerical factor, and R is the outside radius of the spool. The radius of the wound thread layer is equal to r. The spool is pulled
The arrangement shown in Fig. 1.64 consists of two identical uniform solid cylinders, each of mass m, on which two light threads are wound symmetrically. Find the tension of each thread in the process of motion. The friction in the axle of the upper cylinder is assumed to be absent.
In the arrangement shown in Fig. 1.65 a weight A possesses mass m. a pulley B possesses mass M. Also known are the moment of inertia I of the pulley relative to its axis and the radii of the pulley R and 2R. The mass of the threads is negligible. Find the acceleration of the weight A after the
A uniform solid cylinder A of mass m1 can freely rotate about a horizontal axis fixed to a mount B of mass m2 (Fig. 1.66). A constant horizontal force F is applied to the end K of a light thread tightly wound on the cylinder. The friction between the mount and the supporting horizontal plane is
A plank of mass m1 with a uniform sphere of mass m2 placed on it rests on a smooth horizontal plane. A constant horizontal force F is applied to the plank. With what accelerations will the plank and the centre of the sphere move provided there is no sliding between the plank and the sphere?
A uniform solid cylinder of mass m and radius R is set in rotation about its axis with an angular velocity ωo, then lowered with its lateral surface onto a horizontal plane and released. The coefficient of friction between the cylinder and the plane is equal to k. Find: (a) How long the
A uniform ball of radius r rolls without slipping down from the top of a sphere of radius R. Find the angular velocity of the ball at the moment it breaks off the sphere. The initial velocity of the ball is negligible. ,
A uniform solid cylinder of radius R =15 cm rolls over a horizontal plane passing into an inclined plane forming an angle a = 30° with the horizontal (Fig. 1.67). Find the maximum value of the velocity vo which still permits the cylinder to roll onto the inclined plane section without a jump.
A small body A is fixed to the inside of a thin rigid hoop of radius R and mass equal to that of the body A. The hoop rolls without slipping over a horizontal plane; at the moments when the body A gets into the lower position, the centre of the hoop moves with velocity vo (Fig. 1.68). At what
Determine the kinetic energy of a tractor crawler belt of mass m if the tractor moves with velocity v (Fig. 1.69).
A uniform sphere of mass m and radius r rolls without sliding over a horizontal plane, rotating about a horizontal axle OA (Fig. 1.70). In the process, the centre of the sphere moves with velocity v along a circle of radius R. Find the kinetic energy of the sphere.
Demonstrate that in the reference frame rotating with a constant angular velocity ω about a stationary axis a body of mass m experiences the resultant (a) Centrifugal force of inertia Fef = mω2Rc, where Rc is the radius vector of the body's centre of inertia relative to the rotation
A midpoint of a thin uniform rod AB of mass m and length l is rigidly fixed to a rotation axle OO' as shown in Fig. 1.71 The rod is set into rotation with a constant angular velocity ω. Find the resultant moment of the centrifugal forces of inertia relative to the point C in the reference
A conical pendulum, a thin uniform rod of length l and mass m, rotates uniformly about a vertical axis with angular velocity ω (the upper end of the rod is hinged). Find the angle θ between the rod and the vertical.
A uniform cube with edge a rests on a horizontal plane whose friction coefficient equals k. The cube is set in motion with an initial velocity, travels some distance over the plane and comes to a standstill. Explain the disappearance of the angular momentum of the cube relative to the axis lying in
A smooth uniform rod AB of mass M and length l rotates freely with an angular velocity ωo in a horizontal plane about a stationary vertical axis passing through its end A. A small sleeve of mass m starts sliding along the rod from the point A. Find the velocity v' of the sleeve relative to the
A uniform rod of mass m = 5.0 kg and length l = 90 cm rests on a smooth horizontal surface. One of the ends of the rod is struck with the impulse J = 3.0 N. s in a horizontal direction perpendicular to the rod. As a result, the rod obtains the momentum p = 3.0 N-s. Find the force with which one
A thin uniform square plate with side l and mass M can rotate freely about a stationary vertical axis coinciding with one of its sides. A small ball of mass m flying with velocity v at right angles to the plate strikes elastically the centre of it. Find: (a) The velocity of the ball v' after the
A vertically oriented uniform rod of mass M and length l can rotate about its upper end. A horizontally flying bullet of mass m strikes the lower end of the rod and gets stuck in it; as a result, the rod swings through an angle a. Assuming that m
A horizontally oriented uniform disc of mass M and radius .R rotates freely about a stationary vertical axis passing through its centre. The disc has a radial guide along which can slide without friction a small body of mass m. A light thread running down through the hollow axle of the disc is tied
A man of mass m1 stands on the edge of a horizontal uniform disc of mass m2 and radius R which is capable of rotating freely about a stationary vertical axis passing through its centre. At a certain moment the man starts moving along the edge of the disc; he shifts over an angle φ` relative to
Two horizontal discs rotate freely about a vertical axis passing through their centres. The moments of inertia of the discs relative to this axis are equal to I1 and I2, and the angular velocities to ω1 and ω2. When the upper disc fell on the lower one, both discs began rotating, after
A small disc and a thin uniform rod of length l, whose mass is η times greater than the mass of the disc, lie on a smooth horizontal plane. The disc is set in motion, in horizontal direction and perpendicular to the rod, with velocity v, after which it elastically collides with the end of the
A stationary platform which can rotate freely about a vertical axis (Fig. 1.72) supports a motor M and a balance weight N. The moment of inertia of the platform with the motor and the balance weight relative to this axis is equal to I. A light frame is fixed to the motor's shaft with a uniform
A horizontally oriented uniform rod AB of mass m = 1.40 kg and length lo = 100 cm rotates freely about a stationary vertical axis OO' passing through its end A. The point A is located at the middle of the axis OO' whose length is equal to l = 55 cm. At what angular velocity of the rod the
The middle of a uniform rod of mass m and length l is rigidly fixed to a vertical axis OO' so that the angle between the rod and the axis is equal to θ (see Fig. 1.71). The ends of the axis OO' are provided with bearings. The system rotates without friction with an angular velocity ω.
A top of mass m = 0.50 kg, whose axis is tilted by an angle θ = 30° to the vertical, precesses due to gravity. The moment of inertia of the top relative to its symmetry axis is equal to I = 2.0 g.m2, the angular velocity of rotation about that axis is equal to ω = 350 rad/s, the distance
A gyroscope, a uniform disc of radius R = 5.0 cm at the end of a rod of length l = 10 cm (Fig. 1.73), is mounted on the floor of an elevator car going up with a constant acceleration w = 2.0 m/s2. The other end of the rod is hinged at the point O. The gyroscope precesses with an angular velocity n
A top of mass m = 1.0 kg and moment of inertia relative to its own axis I = 4.0 g-m2 spins with an angular velocity ω = 310 rad/s. Its point of rest is located on a block which is shifted in a horizontal direction with a constant acceleration w = 1.0 m/s2. The distance between the point of
A uniform sphere of mass m = 5.0 kg and radius R = 6.0 cm rotates with an angular velocity ω = 1250 rad/s about a horizontal axle passing through its centre and fixed on the mounting base by means of bearings. The distance between the bearings equals l = 15 cm. The base is set in rotation
A cylindrical disc of a gyroscope of mass m = 15 kg and radius r = 5.0 cm spins with an angular velocity ω = 330 rad/s. The distance between the bearings in which the axle of the disc is mounted is equal to l = 15 cm. The axle is forced to oscillate about a horizontal axis with a period T =
A ship moves with velocity v = 36 km per hour along an arc of a circle of radius R = 200 m. Find the moment of the gyroscopic forces exerted on the bearings by the shaft with a flywheel whose moment of inertia relative to the rotation axis equals I = 3.8.103 kg.m2 and whose rotation velocity n =
A locomotive is propelled by a turbine whose axle is parallel to the axes of wheels. The turbine's rotation direction coincides with that of wheels. The moment of inertia of the turbine rotor relative to its own axis is equal to I = 240 kg.m2. Find the additional force exerted by the gyroscopic
What pressure has to be applied to the ends of a steel cylinder to keep its length constant on raising its temperature by 100 °C?
What internal pressure (in the absence of an external pressure) can be sustained? (a) By a glass tube; (b) By a glass spherical flask, if in both cases the wall thickness is equal to Δr = 1.0 mm and the radius of the tube and the flask equals r = 25 mm?
A horizontally oriented copper rod of length l =1.0 m is rotated about a vertical axis passing through its middle. What is the number of rps at which this rod ruptures?
A ring of radius r = 25 cm made of lead wire is rotated about a stationary vertical axis passing through its centre and perpendicular to the plane of the ring. What is the number of rps at which the ring ruptures?
A steel wire of diameter d = 1.0 mm is stretched horizontally between two clamps located at the distance l = 2.0 m from each other. A weight of mass m = 0.25 kg is suspended from the mid- point O of the wire. What will the resulting descent of the point O be in centimetres?
A uniform elastic plank moves over a smooth horizontal plane due to a constant force Fo distributed uniformly over the end face. The surface of the end face is equal to S, and Young's modulus of the material to E. Find the compressive strain of the plank in the direction of the acting force.
A thin uniform copper rod of length l and mass m rotates uniformly with an angular velocity ω in a horizontal plane about a vertical axis passing through one of its ends. Determine the tension in the rod as a function of the distance r from the rotation axis. Find the elongation of the rod.
A solid copper cylinder of length l = 65 cm is placed on a horizontal surface and subjected to a vertical compressive force F = 1000 N directed downward and distributed uniformly over the end face. What will be the resulting change of the volume of the cylinder in cubic millimetres?
A copper rod of length l is suspended from the ceiling by one of its ends. Find: (a) The elongation Δl of the rod due to its own weight; (b) The relative increment of its volume ΔV/V.
A bar made of material whose Young's modulus is equal to E and Poisson's ratio to µ is subjected to the hydrostatic pressure p. Find: (a) The fractional decrement of its volume; (b) The relationship between the compressibility β and the elastic constants E and µ. Show that Poisson's
One end of a steel rectangular girder is embedded into a wall (Fig. 1.74). Due to gravity it sags slightly. Find the radius of curvature of the neutral layer (see the dotted line in the figure) in the vicinity of the point O if the length of the protruding section of the girder is equal to l = 6.0
The bending of an elastic rod is described by the elastic curve passing through centres of gravity of rod's cross-sections. At small bendings the equation of this curve takes the form N (x) = EI d2y/dx2, where N (x) is the bending moment of the elastic forces in the cross. section corresponding to
A steel girder of length l rests freely on two supports (Fig. 1.77). The moment of inertia of its cross-section is equal to I (see the foregoing problem). Neglecting the mass of the girder and assuming the sagging to be slight, find the deflection λ, due to the force F applied to the middle
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