№ 
Condition 
free/or 0.5$ 
261  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? 

262  A uniform solid cylinder of mass m and radius R is set in rotation about its axis with an angular velocity ω0, 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. 

263  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. 

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 v0 which still permits the cylinder to roll onto the inclined plane section without a jump. The sliding is assumed to be absent. 

265  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 v0 (Fig. 1.68). At what values of v0 will the hoop move without bouncing? 

266  Determine the kinetic energy of a tractor crawler belt of mass m if the tractor moves with velocity v (Fig. 1.69). 

267  A uniform sphere of mass In 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. 

268  Demonstrate that in the reference frame rotating with a constant angular velocity o about a stationary axis a body of mass m experiences the resultant (a) centrifugal force of inertia Fcf = mw^2Rc, where Rc is the radius vector of the body's centre of inertia relative to the rotation axis; (b) Coriolis force Fcor = 2m[vcw], where vc is the velocity of the body's centre of inertia in the rotating reference frame. 

269  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 w. Find the resultant moment of the centrifugal forces of inertia relative to the point C in the reference frame fixed to the axle OO' and to the rod. 

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. 

271  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 the plane at right angles to the cube's motion direction. Find the distance between the resultants of gravitational forces and the reaction forces exerted by the supporting plane. 

272  A smooth uniform rod AB of mass M and length l rotates freely with an angular velocity ω0 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. 

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. 

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. 

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 "bulletrod" 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 "bulletrod" to remain constant during the impact. 

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 ω0. 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. 

277  A man of mass ml 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 f' relative to the disc and then stops. In the process of motion the velocity of the man varies with time as v* (t). Assuming the dimensions of the man to be negligible, find: (a) the angle through which the disc had turned by the moment the man stopped; (b) the force moment (relative to the rotation axis) with which the man acted on the disc in the process of motion. 

278  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 oh and co,. When the upper disc fell on the lower one, both discs began rotating, after some time, as a single whole (due to friction). Find: (a) the steadystate angular rotation velocity of the discs; (b) the work performed by the friction forces in this process. 

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? 

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 ω0 about a shaft BB' coinciding with the axis OO'. The moment of inertia of the sphere relative to the rotation axis is equal to I0. 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°. 
