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irodov_problems_in_general_physics_2011_7.pdf, Study Guides, Projects, Research of Physics

Pasadena City CollegePhysics

irodov_problems_in_general_physics_2011_7.pdf

Typology: Study Guides, Projects, Research

2010/2011

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bg1
both bodies began moving with constant accelerations. Find the
friction force between the ball and the thread if
t
seconds after the
beginning of motion the ball got opposite the upper end of the rod.
The rod length equals
1.
1.75. In the arrangement shown in Fig. 1.18 the mass of ball
1
is =
1.8 times as great as that of rod
2.
The length of the latter is
1 =
100 cm. The masses of the pulleys and the threads, as well as
the friction, are negligible. The ball is set on the same level as the
lower end of the rod and then released. How soon will the ball be
opposite the upper end of the rod?
1.76. In the arrangement shown in Fig. 1.19 the mass of body
1
is z =
4.0 times as great as that of body
2.
The height
h =
20 cm.
The masses of the pulleys and the threads, as well as the friction,
are negligible. At a certain moment body
2
is released and the arrange-
ment set in motion. What is the maximum height that body
2
will
go up to?
1.77. Find the accelerations of rod
A
and wedge
B
in the arrange-
ment shown in Fig. 1.20 if the ratio of the mass of the wedge to that
of the rod equals 11, and the friction between all contact surfaces is
negligible.
1.78. In the arrangement shown in Fig. 1.21 the masses of the
wedge
M
and the body m are known. The appreciable friction exists
Fig. 1.20.
Fig. 1.21.
only between the wedge and the body m, the friction coefficient being
equal to
k.
The masses of the pulley and the thread are negligible.
Find the acceleration of the body m relative to the horizontal surface
on which the wedge slides.
1.79. What is the minimum acceleration with which bar
A
(Fig. 1.22)
should be shifted horizontally to keep bodies
1
and
2
stationary
relative to the bar? The masses of the bodies are equal, and the coef-
ficient of friction between the bar and the bodies is equal to
k.
The
masses of the pulley and the threads are negligible, the friction in
the pulley is absent.
1.80. Prism
1
with bar
2
of mass m placed on it gets a horizontal
acceleration w directed to the left (Fig. 1.23). At what maximum
value of this acceleration will the bar be still stationary relative to
the prism, if the coefficient of friction between them
k<
cot a?
24
pf3
pf4
pf5

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both bodies began moving with constant accelerations. Find the friction force between the ball and the thread if t seconds after the beginning of motion the ball got opposite the upper end of the rod. The rod length equals 1. 1.75. In the arrangement shown in Fig. 1.18 the mass of ball 1 is = 1.8 times as great as that of rod 2. The length of the latter is 1 = 100 cm. The masses of the pulleys and the threads, as well as the friction, are negligible. The ball is set on the same level as the lower end of the rod and then released. How soon will the ball be opposite the upper end of the rod? 1.76. In the arrangement shown in Fig. 1.19 the mass of body 1 is z = 4.0 times as great as that of body (^) 2. The height h = 20 cm. The masses of the pulleys and the threads, as well as the friction, are negligible. At a certain moment body (^) 2 is released and the arrange- ment set in motion. What is the maximum height that body 2 will go up to? 1.77. Find the accelerations of rod A and wedge B in the arrange- ment shown in Fig. 1.20 if the ratio of the mass of the wedge to that of the rod equals 11, and the friction between all contact surfaces is negligible. 1.78. In the arrangement shown in Fig. 1.21 the masses of the wedge M and the body m are known. The appreciable friction exists

Fig. 1.20. Fig. 1.21.

only between the wedge and the body m, the friction coefficient being equal to k. The masses of the pulley and the thread are negligible. Find the acceleration of the body m relative to the horizontal surface on which the wedge slides. 1.79. What is the minimum acceleration with which bar A (Fig. 1.22) should be shifted horizontally to keep bodies 1 and 2 stationary relative to the bar? The masses of the bodies are equal, and the coef- ficient of friction between the bar and the bodies is equal to k. The masses of the pulley and the threads are negligible, the friction in the pulley is absent. 1.80. Prism 1 with bar 2 of mass m placed on it gets a horizontal acceleration w directed to the left (Fig. 1.23). At what maximum value of this acceleration will the bar be still stationary relative to the prism, if the coefficient of friction between them k< cot a?

Fig. 1.24.

1.81. Prism 1 of mass ml and with angle a (see Fig. 1.23) rests on a horizontal surface. Bar 2 of mass m2is placed on the prism. Assum- ing the friction to be negligible, find the acceleration of the prism. 1.82. In the arrangement shown in Fig. 1.24 the masses m of the bar and M^ of the wedge, as well as the wedge angle a, are known.

Fig. 1.22. (^) Fig. 1.23.

The masses of the pulley and the thread are negligible. The friction is absent. Find the acceleration of the wedge M. 1.83. A particle of mass m moves along a circle of radius (^) R. Find the modulus of the average vector of the force acting on the particle over the distance equal to a quarter of the circle, if the particle moves (a) uniformly with velocity v; (b) with constant tangential acceleration iv.„ the initial velocity being equal to zero. 1.84. An aircraft loops the loop of radius R = 500 m with a constant velocity v = 360 km per hour. Find the weight of the flyer of mass m = 70 kg in the lower, upper, and middle points of the loop. 1.85. A small sphere of mass m suspended by a thread is first taken aside so that the thread forms the right angle with the vertical and then released. Find: (a) the total acceleration of the sphere and the thread tension as a function of 0, the angle of deflection of the thread from the vertical; (b) the thread tension at the moment when the vertical component of the sphere's velocity is maximum; (c) the angle 0 between the thread and the vertical at the moment when the total acceleration vector of the sphere is directed horizon- tally. 1.86. A ball suspended by a thread swings in a vertical plane so that its acceleration values in the extreme and the lowest position are equal. Find the thread deflection angle in the extreme position. 1.87. A small body A starts sliding off the top of a smooth sphere of radius R. Find the angle 0 (Fig. 1.25) corresponding to the point at which the body breaks off the sphere, as well as the break-off veloc- ity of the body. 1.88. A device (Fig. 1.26) consists of a smooth L-shaped rod locat- ed in a horizontal plane and a sleeve A of mass m attached by a weight-

1.95. Find the magnitude and direction of the force acting on the particle of mass in during its motion in the plane xy according to the law x =^ a^ sin cot, y =^ b^ cos cot, where a,^ b,^ and co are constants. 1.96. A body of mass in is thrown at an angle to the horizontal with the initial velocity v0. Assuming the air drag to be negligible, find: (a) the momentum increment Op that the body acquires over the first t^ seconds of motion; (b) the modulus of the momentum increment ip during the total time of motion. 1.97. At the moment t = 0 a stationary particle of mass in expe- riences a time-dependent force F = at (r — t), where a is a constant vector, r is the time during which the given force acts. Find: (a) the momentum of the particle when the action of the force dis- continued; (b) the distance covered by the particle while the force acted. 1.98. At the moment t = 0 a particle of mass m starts moving due to a force F = F, sin cot, where F0and co are constants. Find the distance covered by the particle as a function of t. Draw the approx- imate plot of this function. 1.99. At the moment t = 0 a particle of mass m starts moving due to a force F = F, cos cot, where F, and co are constants. How long will it be moving until it stops for the first time? What distance will it traverse during that time? What is the maximum velocity of the particle over this distance? 1.100. A motorboat of mass m moves along a lake with velocity v0. At the moment t = 0 the engine of the boat is shut down. Assuming the resistance of water to be proportional to the velocity of the boat F = —rv, find: (a) how long the motorboat moved with the shutdown engine; (b) the velocity of the motorboat as a function of the distance cov- ered with the shutdown engine, as well as the total distance covered till the complete stop; (c) the mean velocity of the motorboat over the time interval (beginning with the moment t = 0), during which its velocity de- creases ittimes. 1.101. Having gone through a plank of thickness h, a bullet changed its velocity from v, to v. Find the time of motion of the bullet in the plank, assuming the resistance force to be proportional to the square of the velocity. 1.102. A small bar starts sliding down an inclined plane forming an angle cc with the horizontal. The friction coefficient depends on the distance x covered as k = ax, where a is a constant. Find the distance covered by the bar till it stops, and its maximum velocity over this distance. 1.103. A body of mass m rests on a horizontal plane with the fric- tion coefficient (^) lc. At the moment t = 0 a horizontal force is applied to it, which varies with time as F = at, where a is a constant vector.

Fig. 1.27.

Find the distance traversed by the body during the first t seconds after the force action began. 1.104. A body of mass m is thrown straight up with velocity vo. Find the velocity v' with which the body comes down if the air drag equals kv2, where k is a constant and v is the velocity of the body. 1.105. A particle of mass m moves in a certain plane P^ due to a force F whose magnitude is constant and whose vector rotates in that plane with a constant angular velocity co. Assum- ing the particle to be stationary at the moment t = 0, find: (a) its velocity as a function of time; (b) the distance covered by the particle between two successive stops, and the mean velocity over this time. 1.106. A small disc A is placed on an inclined plane forming an angle a with the horizontal (Fig. 1.27) and is imparted an initial velocity v0. Find how the velocity of the disc depends on the angle y if the friction coefficient k = tan a and at the initial moment yo = = nI2. 1.107. A chain of length 1 is placed on a smooth spherical surface of radius R with one of its ends fixed at the top of the sphere. What will be the acceleration w of each element of the chain when its upper

end is released? It is assumed that the length of the chain 1<--1nR. 2. 1.108. A small body is placed on the top of a smooth sphere of radius R.^ Then the sphere is imparted a constant acceleration wo in the horizontal direction and the body begins sliding down. Find: (a) the velocity of the body relative to the sphere at the moment of break-off; (b) the angle 00between the vertical and the radius vector drawn from the centre of the sphere to the break-off point; calculate 00 for w0 = g. 1.109. A particle moves in a plane under the action of a force which is always perpendicular to the particle's velocity and depends on a distance to a certain point on the plane as 1/rn, where n is a constant. At what value of n will the motion of the particle along the circle be steady? 1.110. A sleeve A can slide freely along a smooth rod bent in the shape of a half-circle of radius R (Fig. 1.28). The system is set in rota- tion with a constant angular velocity co about a vertical axis 00'. Find the angle 0 corresponding to the steady position of the sleeve. 1.111. A rifle was aimed at the vertical line on the target located precisely in the northern direction, and then fired. Assuming the air drag to be negligible, find how much off the line, and in what direc- tion, will the bullet hit the target. The shot was fired in the horizontal