1.118. 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 action of certain forces, one of which being F = 3i + 4j. Find the work performed by the force F. (Here r1, r2, and F are given in SI units).
1.119. A locomotive of mass m starts moving so that its velocity varies according to the law v = a sqrt(s), where a is a constant, and s is the distance covered. Find the total work performed by all the forces which are acting on the locomotive during the first t seconds after the beginning of motion.
1.120. The kinetic energy of a particle moving along a circle of radius R 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.
1.121. 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 hill is h, the length of its base l, and the coefficient of friction k.
1.123. Two bars of masses and m1 and 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 force has to be applied in the horizontal direction to the bar of mass m1 in order to shift the other bar?
1.124. 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 overhanging part equals η = 1/3 of the chain length. What will be the total work performed by the friction forces acting on the chain by the moment it slides completely off the table?
1.125. A body of mass m is thrown at an angle α to the horizontal with the initial velocity v0. Find the mean power developed by gravity over the whole time of motion of the body, and the instantaneous power of gravity as a function of time.
1.126. 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 forces acting on the particle, and the mean value of this power averaged over the first t seconds after the beginning of motion.
1.127. A small body of mass m is located on a horizontal plane at the point O. The body acquires a horizontal velocity v0. Find:
1.130. 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 force and the increment of the body's potential energy in the gravitational field of the Earth over the first half of the ascent.
1.131. 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:
1.133. 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.
1.134. A body of mass m is pushed with the initial velocity v0 up an inclined plane set at an angle α to the horizontal. The friction coefficient is equal to k. What distance will the body cover before it stops and what work do the friction forces perform over this distance?
1.135. 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 to ensure the maximum distance s covered by the disc? What is it equal to?
1.136. 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 the highest point of its trajectory (after breaking off the groove).
1.137. 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 that point? What will be the tension of the thread at the moment it will be passing the horizontal position?
1.141. 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 l0 = 40 cm to a point O (Fig,. 1.35). The coefficient of friction between the bar and the plank equals k = 0.20. The plank is slowly shifted to the right until the bar starts sliding over it. It occurs at the moment when the cord deviates from the vertical by an angle θ = 30°. Find the work that has been performed by that moment by the friction force acting on the bar in the reference frame fixed to the plane.
1.142. 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 l0 and stiffness χ. What work must be performed to slowly get this system going and reaching the angular velocity ω?
1.143. A pulley fixed to the ceiling carries a thread with bodies of masses m1 and m2 attached to its ends. The masses of the pulley and the thread are negligible, friction is absent. Find the acceleration wC of the centre of inertia of this system.
1.145. 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 angle θ = 45° with the vertical. Find the distance between the chain's centre of gravity and the rotation axis, and the tension of the thread.
1.147. In the reference frame K two particles travel along the x axis, one of mass m1 with velocity v1 and the other of mass m2 with velocity v2. Find:
1.149. 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 v1 and v2 whose directions are mutually perpendicular and lie in a horizontal plane. Find the total energy E of this system in the frame of the centre of inertia.
1.151. Two bars of masses m1 and m2 connected by a weightless spring of stiffness χ (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 velocity of the centre of inertia of the system after bar 1 breaks off the wall.
1.153. A system consists of two identical cubes, each of mass m, linked together by the compressed weightless spring of stiffness χ (Fig. 1.41). The cubes are also connected by a thread which is burned through at a certain moment. Find:
1.154. 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 places by jumping in the direction perpendicular to the motion direction. As a consequence, buggy 1 stops and buggy 2 keeps moving in the same direction, with its velocity becoming equal to v. Find the initial velocities of the buggies v1 and v2 if the mass of each buggy (without a man) equals M and the mass of each man m.
1.155. Two identical buggies move one after the other due to inertia (without friction) with the same velocity v0. A man of mass m rides the rear buggy. At a certain moment the man jumps into the front buggy with a velocity u relative to his buggy. Knowing that the mass of each buggy is equal to M, find the velocities with which the buggies will move after that.
1.156. 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 velocity u relative to the buggy: (1) simultaneously; (2) one after the other. In what case will the velocity of the buggy be greater and how many times?
1.157. 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 any moment is twice as great as the force of pressure exerted by the part already resting on the table.
1.161. A cannon of mass M starts sliding freely down a smooth inclined plane at an angle α to the horizontal. After the cannon covered the distance l, a shot was fired, the shell leaving the cannon in the horizontal direction with a momentum p. As a consequence, the cannon stopped. Assuming the mass of the shell to be negligible, as compared to that of the cannon, determine the duration of the shot.
1.162. 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 << M, find:
1.163. 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 (relative to the initial level) will the disc rise after breaking off the body M? The friction is assumed to be absent.
1.164. 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 friction between the disc and the plank the disc slows down and, beginning with a certain moment, moves in one piece with the plank.
1.166. 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 formed particle (both the vector v and its modulus), if the components of the vectors v1 and v2 are given in the SI units.
1.167. 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 equal to v1 and v2.
1.168. 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
1.169. Particle 1 experiences a perfectly elastic collision with a stationary particle 2. Determine their mass ratio, if
1.170. 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 balls and the direction of the initial motion of the striking ball is equal to α = 45°. Assuming the balls to be smooth, find the fraction η of the kinetic energy of the striking ball that turned into potential energy at the moment of the maximum deformation.
1.171. 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 obtain?
1.172. 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 η = 1.0%. Find the magnitude and direction of the velocity of particle 1 after the collision.
1.173. 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 motion of the particle m. How much (in per cent) and in what way has the kinetic energy of this system changed after the collision, if M/m = 5.0?
1.174. 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:
1.175. 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.
1.179. 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, if at the initial moment it possessed the mass m0 and its velocity was equal to zero. Make use of the formula given in the foregoing problem.
1.180. 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 u relative to the rocket, and its mass at the initial moment equals m0.
1.181. A spaceship of mass m0 moves in the absence of external forces with a constant velocity v0. To change the motion direction, a jet engine is switched on. It starts ejecting a gas jet with velocity u which is constant relative to the spaceship and directed at right angles to the spaceship motion. The engine is shut down when the mass of the spaceship decreases to m. Through what angle α did the motion direction of the spaceship deviate due to the jet engine operation?
1.182. 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 constant velocity μ kg/s. Find the acceleration and the velocity of the cart at the moment t, if at the initial moment t = 0 the cart with loaded sand had the mass m0 and its velocity was equal to zero. The friction is to be neglected.
1.184. 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 end A of the chain is set free. With what velocity will this end of the chain slip out of the tube?
1.186. A ball of mass m is thrown at an angle α to the horizontal with the initial velocity v0. Find the time dependence of the magnitude of the ball's angular momentum vector relative to the point from which the ball is thrown. Find the angular momentum M at the highest point of the trajectory if m = 130 g, α = 45°, and v0 = 25 m/s. The air drag is to be neglected.
1.187. 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 motion direction of the disc and the normal of the wall is equal to α. Find:
1.188. 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 angular momentum M of the ball remain constant? Find the magnitude of the increment of the vector of the ball's angular momentum relative to the point O picked up during half a revolution.
1.189. 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 to the point O of the reference frame moving translationally in a horizontal direction with a velocity V). The ball starts falling from the point O. The air drag is to be neglected.
1.190. A smooth horizontal disc rotates with a constant angular velocity ω 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 point with velocity v0. Find the angular momentum M(t) of the disc relative to the point O in the reference frame fixed to the disc. Make sure that this angular momentum is caused by the Coriolis force.
1.191. 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 the field). Find the mass of the particle if its minimum distance from the point O equals r1 and its velocity at the point farthest from O equals v2.
1.192. 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 horizontal direction at right angles to the vertical plane in which the thread is located. What is the initial velocity that has to be imparted to the ball so that it could deviate through the maximum angle π/2 in the process of motion?
1.193. 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 thread tension as a function of the distance r between the body and the hole if at r = r0 the angular velocity of the thread is equal to ω0.
1.195. A uniform sphere of mass m and radius R starts rolling without slipping down an inclined plane at an angle α to the horizontal. Find the time dependence of the angular momentum of the sphere relative to the point of contact at the initial moment. How will the obtained result change in the case of a perfectly smooth inclined plane?
1.198. A ball of mass m moving with velocity v0 experiences a head-on elastic collision with one of the spheres of a stationary rigid dumbbell as shown in Fig. 1.50. The mass of each sphere equals m/2, and the distance between them is l. Disregarding the size of the spheres, find the proper angular momentum M of the dumbbell after the collision, i.e. the angular momentum in the reference frame moving translationally and fixed to the dumbbell's centre of inertia.