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121A 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.
122A 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, the disc stops. Find the work performed by the friction forces over the whole distance, assuming the friction coefficient k = 0.15 for both inclined and horizontal planes.
123Two 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?
124A 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?
125A 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.
126A 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.
127A small body of mass m is located on a horizontal plane at the point O. The body acquires a horizontal velocity v0. Find: (a) the mean power developed by the friction force during the whole time of motion, if the friction coefficient k = 0.27, m = 1.0 kg, and v0 = 1.5 m/s; (b) the maximum instantaneous power developed by the friction force, if the friction coefficient varies as k = αx, where α is a constant, and x is the distance from the point O.
128A small body of mass m = 0.10 kg moves in the reference frame rotating about a stationary axis with a constant angular velocity w = 5.0 rad/s. What work does the centrifugal force of inertia perform during the transfer of this body along an arbitrary path from point 1 to point 2 which are located at the distances r1 = 30 cm and r2 = 50 cm from the rotation axis?
129A 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 dL.
130A 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.
131The 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 r0 corresponding to the equilibrium position of the particle; examine whether this position is steady; (b) the maximum magnitude of the attraction force; draw the plots U(r) and Fr(r) (the projections of the force on the radius vector r).
132In a certain two-dimensional field of force the potential energy of a particle has the form U = ax^2+by^2, where a and b are positive constants whose magnitudes are different. Find out: (a) whether this field is central; (b) what is the shape of the equipotential surfaces and also of the surfaces for which the magnitude of the vector of force F = const.
133There 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.
134A 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?
135A 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?
136A 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).
137A 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?
138A 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 v0 is imparted to the disc as shown in the figure. How long will it move along the plane until it strikes against the cylinder? The friction is assumed to be absent.
139A smooth rubber cord of length L whose coefficient of elasticity is k is suspended by one end from the point 0 (Fig. 1.33). The other end is fitted with a catch B. A small sleeve A of mass m starts falling from the point 0. Neglecting the masses of the thread and the catch, find the maximum elongation of the cord.
140A 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. Besides, the bar is also attached to a point O by means of a light nondeformed spring of length L0 = 50 cm and stiffness x = 5mg/L0, where m is the mass of the bar. The thread PA having been burned, the bar starts moving. Find its velocity at the moment when it is breaking off the plane.
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