1.234. A thin uniform rod AB of mass m = 1.0 kg moves translationally with acceleration w = 2.0 m/s^{2} due to two antiparallel forces F_{1} and F_{2} (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 F_{2} = 5.0 N. Find the length of the rod.

1.236. A force F_{1} = Aj is applied to a point whose radius vector r_{1} = ai, while a force F_{2} = Bi is applied to the point whose radius vector r_{2} = 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 the arm l of the resultant force relative to the point O.

1.238. 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 one of its vertices, if the sides of the plate are equal to a and b, and its mass is m.

1.240. Demonstrate that in the case of a thin plate of arbitrary shape there is the following relationship between the moments of inertia: I_{1} + I_{2} = I_{3}, 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 this relationship, find the moment of inertia of a thin uniform round disc of radius R and mass m relative to the axis coinciding with one of its diameters.

1.241. 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.

1.242. 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.

1.243. 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 cylinder;
(b) the kinetic energy of the whole system.

1.245. 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 plane. Find the angular velocity of the rod as a function of its rotation angle φ counted relative to the initial position.

1.246. 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 m_{1} and m_{2}. 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 the ratio of tensions T_{1}/T_{2} of the vertical sections of the thread in the process of motion.

1.247. In the system shown in Fig. 1.57 the masses of the bodies are known to be m_{1} and m_{2}, the coefficient of friction between the body m_{1} 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 m_{2} starts descending. Assuming the mass of the thread and the friction in the axle of the pulley to be negligible, find the work performed by the friction forces acting on the body m_{1} over the first t seconds after the beginning of motion.

1.248. A uniform cylinder of radius R is spinned about its axis to the angular velocity ω_{0} 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?

1.249. 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.

1.250. A flywheel with the initial angular velocity ω_{0} 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.

1.252. A uniform sphere of mass m and radius R rolls without slipping down an inclined plane set at an angle α 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.

1.253. 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 by the gravitational force.

1.255. A spool with a thread wound on it is placed on an inclined smooth plane set at an angle α = 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*m^{2}, the radius of the wound thread layer r = 3.0 cm. Find the acceleration of the spool axis.

1.256. 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, if the coefficient of friction between the cylinder and the planks is equal to k. What is the acceleration w_{max} of the axis of the cylinder rolling down the inclined plane?

1.257. 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 = γmR^{2}, where γ 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 without sliding by the thread with a constant force F directed at an angle α to the horizontal (Fig. 1.63). Find:
(a) the projection of the acceleration vector of the spool axis on the x-axis;
(b) the work performed by the force F during the first t seconds after the beginning of motion.

1.258. 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.

1.259. 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 system is set free.

1.261. A plank of mass m_{1} with a uniform sphere of mass m_{2} 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?

1.262. A uniform solid cylinder of mass m and radius R is set in rotation about its axis with an angular velocity ω_{0}, 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 cylinder will move with sliding;
(b) the total work performed by the sliding friction force acting on the cylinder.

1.264. A uniform solid cylinder of radius R = 15 cm rolls over a horizontal plane passing into an inclined plane forming an angle α = 30° with the horizontal (Fig. 1.67). Find the maximum value of the velocity v_{0} which still permits the cylinder to roll onto the inclined plane section without a jump. The sliding is assumed to be absent.

1.270. 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.

1.272. A smooth uniform rod AB of mass M and length l rotates freely with an angular velocity ω_{0} 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 rod at the moment it reaches its other end B.

1.273. 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 half of the rod will act on the other in the process of motion.

1.274. 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 impact;
(b) the horizontal component of the resultant force which the axis will exert on the plate after the impact.

1.275. 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 α. Assuming that m << M, find:
(a) the velocity of the flying bullet;
(b) the momentum increment in the system "bullet-rod" during the impact; what causes the change of that momentum;
(c) at what distance x from the upper end of the rod the bullet must strike for the momentum of the system "bullet-rod" to remain constant during the impact.

1.276. 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 to the body. Initially the body was located at the edge of the disc and the whole system rotated with an angular velocity ω_{0}. Then by means of a force F applied to the lower end of the thread the body was slowly pulled to the rotation axis. Find:
(a) the angular velocity of the system in its final state;
(b) the work performed by the force F.

1.279. 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 rod. Find the velocity of the disc and the angular velocity of the rod after the collision. At what value of η will the velocity of the disc after the collision be equal to zero? reverse its direction?

1.280. A stationary platform P 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 sphere A rotating freely with an angular velocity ω_{0} about a shaft BB' coinciding with the axis OO'. The moment of inertia of the sphere relative to the rotation axis is equal to I_{0}. Find:
(a) the work performed by the motor in turning the shaft BB' through 90°; through 180°;
(b) the moment of external forces which maintains the axis of the arrangement in the vertical position after the motor turns the shaft BB' through 90°.

1.283. 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*m^{2}, the angular velocity of rotation about that axis is equal to ω = 350 rad/s, the distance from the point of rest to the centre of inertia of the top is l = 10 cm. Find:
(a) the angular velocity of the top's precession;
(b) the magnitude and direction of the horizontal component of the reaction force acting on the top at the point of rest.

1.286. 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 about a vertical axis with an angular velocity ω' = 5.0 rad/s. Find the modulus and direction of the gyroscopic forces.

1.288. 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*10^{3} kg*m^{2} and whose rotation velocity n = 300 rpm. The rotation axis is oriented along the length of the ship.