№ 
Condition 
free/or 0.5$ 
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? 

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. 

183  A flatcar of mass m0 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 n kg/s. Find the time dependence of the velocity and the acceleration of the flatcar in the process of loading. The friction is negligibly small. 

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? 

185  The angular momentum of a particle relative to a certain point O varies with time as M = a + bt^2 , where a and b are constant vectors, with a __ b. Find the force moment N relative to the point 0 acting on the particle when the angle between the vectors N and M equals 45°. 

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. 

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: (a) the points relative to which the angular momentum M of the disc remains constant in this process; (b) the magnitude of the increment of the vector of the disc s angular momentum relative to the point O* which is located in the plane of the disc motion at the distance l from the point O. 

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. 

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. 

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. 

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. 

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? 

193  A small body of mass m tied to a nonstretchable 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. 

194  A light nonstretchable 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 moving. Find its angular momentum relative to the pulley axle as a function of time t. 

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? 

196  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 point 0 is determined by the radius vector r0 . Find out when the angular momentum of the system of particles does not depend on the choice of the point O. 

197  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 = M1+rcp, where M is its proper angular momentum (in the reference frame moving translationally and fixed to the centre of inertia), r0 is the radius vector of the centre of inertia relative to the point O, p is the total momentum of the system of particles in the reference frame K. 

198  A ball of mass m moving with velocity v0 experiences a headon 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. 

199  Two small identical discs, each of mass m, lie on a smooth horizontal plane. The discs are interconnected by a light nondeformed spring of length L0 and stiffness x. At a certain moment one of the discs is set in motion in a horizontal direction perpendicular to the spring with velocity v0 . Find the maximum elongation of the spring in the process of motion, if it is known to be considerably less than unity. 

200  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. 
