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physics
mechanics

Questions and Answers of Mechanics

A body of mass m rests on a horizontal plane with the friction coefficient k. At the moment t = 0 a horizontal force is applied to it, which varies with time as F = at, where a is a constant vector.
A body of mass m is thrown straight up with velocity vo. Find the velocity v' with which the body comes down if the air drag equals kv2, where k is a constant and v is the velocity of the body.
A particle of mass m moves in a certain plane P due to a force F whose magnitude is constant and whose vector rotates in that plane with a constant angular velocity ω. Assuming the particle to
A small disc A is placed on an inclined plane forming an angle a with the horizontal (Fig.) and is imparted an initial velocity vo. Find how the velocity of the disc depends on the angle if the
A chain of length l is placed on a smooth spherical surface of radius R with one of its ends fixed at the top of the sphere. What will be the acceleration w of each element of the chain when its
A small body is placed on the top of a smooth sphere of radius R. Then the sphere is imparted a constant acceleration we in the horizontal direction and the body begins sliding down. Find: (a) The
A particle moves in a plane under the action of a force which is always perpendicular to the particle's velocity and depends on a distance to a certain point on the plane as 1/m, where n is a
A sleeve A can slide freely along a smooth rod bent in the shape of a half-circle of radius R (Fig). The system is set in rotation with a constant angular velocity ω about a vertical axis OO'.
A rifle was aimed at the vertical line on the target located precisely in the northern direction, and then fired. Assuming the air drag to be negligible, find how much off the line, and in what
A horizontal disc rotates with a constant angular velocity ω = 6.0 rad/s about a vertical axis passing through its centre. A small body of mass m = 0.50 kg moves along a diameter of the disc
A horizontal smooth rod AB rotates with a constant angular velocity ω = 2.00 rad/s about a vertical axis passing through its end A. A freely sliding sleeve of mass m = 0.50 kg moves along the
A horizontal disc of radius R rotates with a constant angular velocity ω about a stationary vertical axis passing through its edge. Along the circumference of the disc a particle of mass m moves
A small body of mass m = 0.30 kg starts sliding down from the top of a smooth sphere of radius R = 1.00 m. The sphere rotates with a constant angular velocity ω = 6.0 rad/s about a vertical axis
A train of mass m = 2000 tons moves in the latitude ω = 60° North. Find: (a) The magnitude and direction of the lateral force that the train exerts on the rails if it moves along a meridian
At the equator a stationary (relative to the Earth) body falls down from the height h = 500 m. Assuming the air drag to be negligible, find how much off the vertical, and in what direction, the body
A locomotive of mass m starts moving so that its velocity varies according to the law v = a√ s, where a is a constant, and s is the distance covered. Find the total work performed by all the
The kinetic energy of a particle moving along a circle of radius B depends on the distance covered s as T = as2, where a is a constant. Find the force acting on the particle as a function of s
A body of mass m was slowly hauled up the hill (Fig.1.29) by a force F which at each point was directed along a tangent to the trajectory. Find the work performed by this force, if the height of the
A disc of mass m = 50 g slides with the zero initial velocity down an inclined plane set at an angle a = 30° to the horizontal; having traversed the distance l = 50 cm along the horizontal plane,
Two bars of masses m1and m2 connected by a non-deformed light spring rest on a horizontal plane. The coefficient of friction .between the bars and the surface is equal to k. What minimum constant
A chain of mass m = 0.80 kg and length l = 1.5 m rests on a rough-surfaced table so that one of its ends hangs over the edge. The chain starts sliding off the table all by itself provided the over-
A body of mass m is thrown at an angle a to the horizontal with the initial velocity vo. Find the mean power developed by gravity over the whole time of motion of the body, and the instantaneous
A particle of mass m moves along a circle of radius R with a normal acceleration varying with time as wn = at2, where a is a constant. Find the time dependence of the power developed by all the
A small body of mass m is located on a horizontal plane at the point O. The body acquires a horizontal velocity vo. Find: (a) The mean power developed by the friction force during the whole time of
A small body of mass m = 0.10 kg moves in the reference frame rotating about a stationary axis with a constant angular velocity ω = 5.0 rad/s. What work does the centrifugal force of inertia
A system consists of two springs connected in series and having the stiffness coefficients k1 and k2. Find the minimum work to be performed in order to stretch this system by Δ1
A body of mass m is hauled from the Earth's surface by applying a force F varying with the height of ascent y as F = 2 (ay = 1) mg, where a is a positive constant. Find the work performed by this
The potential energy of a particle in a certain field has the form U = a/r2 - b/r, where a and b are positive constants, r is the distance from the centre of the field. Find: (a) The value of ro
In a certain two-dimensional field of force the potential energy of a particle has the form U = ax2 + βy2, where a and β are positive constants whose magnitudes are different. Find out:
There are two stationary fields of force F = ayi and F = axi + byj where i and j are the unit vectors of the x and y axes, and a and b are constants. Find out whether these fields are potential.
A body of mass m is pushed with the initial velocity vo up an inclined plane set at an angle a to the horizontal. The friction coefficient is equal to k. What distance will the body cover before it
A small disc A slides down with initial velocity equal to zero from the top of a smooth hill of height H having a horizontal portion (Fig. 1.30). What must be the height of the horizontal portion h
A small body A starts sliding from the height h down an inclined groove passing into a half-circle of radius h/2 (Fig. 1.31). Assuming the friction to be negligible, find the velocity of the body at
A ball of mass m is suspended by a thread of length l. With what minimum velocity has the point of suspension to be shifted in the horizontal direction for the ball to move along the circle about
A horizontal plane supports a stationary vertical cylinder of radius R and a disc A attached to the cylinder by a horizontal thread AB of length l0 (Fig. 1.32, top view). An initial velocity vo is
A smooth rubber cord of length l whose coefficient of elasticity is k is suspended by one end from the point O (Fig. 1.33). The other end is fitted with a catch B. A small sleeve A of mass m starts
A small bar A resting on a smooth horizontal plane is attached by threads to a point P (Fig. 1.34) and, by means of a weightless pulley, to a weight B possessing the same mass as the bar itself.
A horizontal plane supports a plank with a bar of mass m = 1.0 kg placed on it and attached by a light elastic non-deformed cord of length lo = 40 cm to a point O (Fig,. 1.35). The coefficient of
A smooth light horizontal rod AB can rotate about a vertical axis passing through its end A. The rod is fitted with a small sleeve of mass m attached to the end A by a weightless spring of length lo
A pulley fixed to the ceiling carries a thread with bodies of masses ml and m2 attached to its ends. The masses of the pulley and the thread are negligible, friction is absent. Find the acceleration
Two interacting particles form a closed system whose centre of inertia is at rest. Fig. 1.36 illustrates the positions of both particles at a certain moment and the trajectory of the particle of mass
A closed chain A of mass m = 0.36 kg is attached to a vertical rotating shaft by means of a thread (Fig. 1.37), and rotates with a constant angular velocity ω = 35 rad/s. The thread forms an
A round cone A of mass m = 3.2 kg and half- angle a = l0° rolls uniformly and without slipping along a round conical surface B so that its apex O remains stationary (Fig. 1.38). The centre of
In the reference frame K two particles travel along the x axis, one of mass m1 with velocity vl, and the other of mass m1 with velocity v2. Find: (a) The velocity V of the reference frame K' in which
The reference frame, in which the centre of inertia of a given system of particles is at rest, translates with a velocity V relative to an inertial reference frame K. The mass of the system of
Two small discs of masses m1 and m2 interconnected by a weightless spring rest on a smooth horizontal plane. The discs are set in motion with initial velocities vl and v2 whose directions are
A system consists of two small spheres of masses m1 and m2 interconnected by a weightless spring. At the moment t = 0 the spheres are set in motion with the initial velocities v1 and v2 after which
Two bars of masses ml and m2 connected by a weightless spring of stiffness x (Fig. 1.39) rest on a smooth horizontal plane. Bar 2 is shifted a small distance x to the left and then released. Find the
Two bars connected by a weightless spring of stiffness x and length (in the non-deformed state) lo rest on a horizontal plane. A constant horizontal force F starts acting on one of the bars as shown
A system consists of two identical cubes, each of mass m, linked together by the compressed weightless spring of stiffness x (Fig. 1.41). The cubes are also connected by a thread which is burned
Two identical buggies 1 and 2 with one man in each move without friction due to inertia along the parallel rails toward each other. When the buggies get opposite each other, the men exchange their
Two identical buggies move one after the other due to inertia (without friction) with the same velocity vo A man of mass m rides the rear buggy. At a certain moment the man jumps into the front buggy
Two men, each of mass m, stand on the edge of a stationary buggy of mass M. Assuming the friction to be negligible, find the velocity of the buggy after both men jump off with the same horizontal
A chain hangs on a thread and touches the surface of a table by its lower end. Show that after the thread has been burned through, the force exerted on the table by the falling part of the chain at
A steel ball of mass m = 50 g falls from the height h = 1.0 m on the horizontal surface of a massive slab. Find the cumulative momentum that the ball imparts to the slab after numerous bounces, if
A raft of mass M with a man of mass m aboard stays motion-less on the surface of a lake. The man moves a distance l' relative to the raft with velocity v'(t) and then stops. Assuming the water
A stationary pulley carries a rope whose one end supports a ladder with a man and the other end the counterweight of mass M. The man of mass m climbs up a distance l' with respect to the ladder and
A cannon of mass M starts sliding freely down a smooth inclined plane at an angle a to the horizontal. After the cannon covered the distance l, a shot was fired, the shell leaving the cannon in the
A horizontally flying bullet of mass m gets stuck in a body of mass M suspended by two identical threads of length l (Fig.1.42) As a result, the threads swerve through an angle θ. Assuming m
A body of mass M (Fig. 1.43) with a small disc of mass m placed on it rests on a smooth horizontal plane. The disc is set in motion in the horizontal direction with velocity v. To what height
A small disc of mass m slides down a smooth hill of height h without initial velocity and gets onto a plank of mass M lying on the horizontal plane at the base of the hill (Fig. 1.44). Due to
A stone falls down without initial velocity from a height h onto the Earth's surface. The air drag assumed to be negligible, the stone hits the ground with velocity vo = √2gh relative to the
A particle of mass 1.0 g moving with velocity v1 = 3.0i - 2.0j experiences a perfectly inelastic collision with another particle of mass 2.0 g and velocity v2 = 4.0j - 6.0k. Find the velocity of the
Find the increment of the kinetic energy of the closed system comprising two spheres of masses m1 and m2 due to their perfectly inelastic collision, if the initial velocities of the spheres were
A particle of mass m1 experienced a perfectly elastic collision with a stationary particle of mass m2. What fraction of the kinetic energy does the striking particle lose, if? (a) It recoils at right
Particle 1 experiences a perfectly elastic collision with a stationary particle 2. Determine their mass ratio, if (a) After a head-on collision the particles fly apart in the opposite directions
A ball moving translationally collides elastically with another, stationary, ball of the same mass. At the moment of impact the angle between the straight line passing through the centres of the
A shell flying with velocity v = 500 m/s bursts into three identical fragments so that the kinetic energy of the system increases η = 1.5 times. What maximum velocity can one of the fragments
Particle 1 moving with velocity v = 10 m/s experienced a head-on collision with a stationary particle 2 of the same mass. As a result of the collision, the kinetic energy of the system decreased by
A particle of mass m having collided with a stationary particle of mass M deviated by an angle π/2 whereas the particle M recoiled at an angle θ = 30° to the direction of the initial
A closed system consists of two particles of masses m1 and m2 which move at right angles to each other with velocities v1 and v2. Find: (a) The momentum of each particle and (b) The total kinetic
A particle of mass m1 collides elastically with a stationary particle of mass m2 (m1 > m2). Find the maximum angle through which the striking particle may deviate as a result of the collision.
Three identical discs A, B, and C (Fig. 1.45) rest on a smooth horizontal plane. The disc A is set in motion with velocity v after which it experiences an elastic collision simultaneously with the
A molecule collides with another, stationary, molecule of the same mass. Demonstrate that the angle of divergence (a) equals 90° when the collision is ideally elastic; (b) Differs from 90° when
A rocket ejects a steady jet whose velocity is equal to u relative to the rocket. The gas discharge rate equals µ kg/s. Demonstrate that the rocket motion equation in this case takes the form mw =
A rocket moves in the absence of external forces by ejecting a steady jet with velocity u constant relative to the rocket. Find the velocity v of the rocket at the moment when its mass is equal to m,
Find the law according to which the mass of the rocket varies with time, when the rocket moves with a constant acceleration w, the external forces are absent, the gas escapes with a constant velocity
A spaceship of mass mo moves in the absence of external forces with a constant velocity vo. To change the motion direction, a jet engine is switched on. It starts ejecting a gas jet with velocity u
A cart loaded with sand moves along a horizontal plane due to a constant force F coinciding in direction with the cart's velocity vector. In the process, sand spills through a hole in the bottom with
A flatcar of mass mo starts moving to the right due to a constant horizontal force F (Fig.1.46). Sand spills on the flatcar from a stationary hopper. The velocity of loading is constant and equal to
A chain AB of length l is located in a smooth horizontal tube so that its fraction of length h hangs freely and touches the surface of the table with its end B (Fig. 1.47). At a certain moment the
The angular momentum of a particle relative to a certain point O varies with time as M = a +bt2, where a and b are constant vectors, with a┴b. Find the force moment N relative to the point O
A ball of mass m is thrown at an angle a to the horizontal with the initial velocity vo. Find the time dependence of the magnitude of the ball's angular momentum vector relative to the point from
A disc A of mass m sliding over a smooth horizontal surface with velocity v experiences a perfectly elastic collision with a smooth stationary wall at a point O (Fig. 1.48). The angle between the
A small ball of mass m suspended from the ceiling at a point O by a thread of length l moves along a horizontal circle with a constant angular velocity ω. Relative to which points does the
A ball of mass m falls down without initial velocity from a height h over the Earth's surface. Find the increment of the ball's angular momentum vector picked up during the time of falling (relative
A smooth horizontal disc rotates with a constant angular velocity co about a stationary vertical axis passing through its centre, the point O. At a moment t = 0 a disc is set in motion from that
A particle moves along a closed trajectory in a central field of force where the particle's potential energy U = kr2 (k is a positive constant, r is the distance of the particle from the centre O of
A small ball is suspended from a point O by a light thread of length l. Then the ball is drawn aside so that the thread deviates through an angle θ from the vertical and set in motion in a
A small body of mass m tied to a non-stretchable thread moves over a smooth horizontal plane. The other end of the thread is being drawn into a hole O (Fig. 1.49) with a constant velocity. Find the
A light non-stretchable thread is wound on a massive fixed pulley of radius R. A small body of mass m is tied to the free end of the thread. At a moment t = 0 the system is released and starts
A uniform sphere of mass m and radius R starts rolling without slipping down an inclined plane at an angle a to the horizontal. Find the time dependence of the angular momentum of the sphere relative
A certain system of particles possesses a total momentum p and an angular momentum M relative to a point O. Find its angular momentum M' relative to a point O' whose position with respect to the
Demonstrate that the angular momentum M of the system of particles relative to a point O of the reference frame K can be represented as M = M + [rcp],Where M is its proper angular momentum (in the
A ball of mass m moving with velocity vo experiences a head-on elastic collision with one of the spheres of a stationary rigid dumbbell as whown in Fig. 1.50. The mass of each sphere equals m/2, and
Two small identical discs, each of mass m, lie on a smooth horizontal plane. The discs are interconnected by a light non-deformed spring of length lo and stiffness x. At a certain moment one of the
A particle has shifted along some trajectory in the plane xy from point 1 whose radius vector r1 = i + 2j to point 2 with the radius vector r2 =2i- 3j. During that time the particle experienced the
A planet of mass M moves along a circle around the Sun with velocity v = 34.9 km/s (relative to the heliocentric reference frame). Find the period of revolution of this planet around the Sun.
The Jupiter's period of revolution around the Sun is 12 times that of the Earth. Assuming the planetary orbits to be circular, find: (a) How many times the distance between the Jupiter and the Sun
A planet of mass M moves around the Sun along an ellipse so that its minimum distance from the Sun is equal to r and the maximum distance to R. Making use of Kepler's laws, find its period of

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