Download irodov_problems_in_general_physics_2011_5.pdf and more Study Guides, Projects, Research Physics in PDF only on Docsity!
(^2) 5 6 7t
1.16. (^) Two particles, 1 and 2, move with constant velocities vi and v2along two mutually perpendicular straight lines toward the
intersection point 0. At the moment t = 0 the particles were located
at the distances 11and 12 from the point 0. How soon will the distance between the particles become the smallest? What is it equal to? 1.17. From point A located on a highway (Fig. 1.2) one has to get by car as soon as possible to point B located in the field at a distance 1 from the highway. It is known that the car moves in the field ri times slower than on the highway. At what distance from point D one must turn off the highway? 1.18. A point travels along the x axis with a velocity whose pro-
jection vx is presented as a function of time by the plot in Fig. 1.3.
vs 1 0
- -
Fig. 1.2. Fig. 1.3.
Assuming the coordinate of the point x^ =^ 0 at the moment^ t =^ 0,
draw the approximate time dependence plots for the acceleration wx, the x coordinate, and the distance covered s. 1.19. A point traversed half a circle of radius R = 160 cm during time interval x = 10.0 s. Calculate the following quantities aver- aged over that time: (a) the mean velocity (v);
(b) the modulus of the mean velocity vector (v) I;
(c) the modulus of the mean vector of the total acceleration I (w)I if the point moved with constant tangent acceleration. 1.20. A radius vector of a particle varies with time t as r =
= at (1 — cct), where a is a constant vector and a is a positive factor.
Find:
(a) the velocity v and the acceleration w of the particle as functions
of time; (b) the time interval At taken by the particle to return to the ini- tial points, and the distance s covered during that time.
1.21. At the moment t = 0 a particle leaves the origin and moves
in the positive direction of the x axis. Its velocity varies with time
as v = vc, (1 — tit), where v(, is the initial velocity vector whose
modulus equals vo = 10.0 cm/s; i = 5.0 s. Find:
(a) the x coordinate of the particle at the moments of time 6.0, 10, and 20 s; (b) the moments of time when the particle is at the distance 10.0 cm from the origin;
(c) the distance^ s^ covered by the particle during the first 4.0 and 8.0 s; draw the approximate plot s (t). 1.22. The velocity of a particle moving in the positive direction of the x axis varies as v = al/x, where a is a positive constant. Assuming that at the moment t^ = 0 the particle was located at the point x = 0, find: (a) the time dependence of the velocity and the acceleration of the particle; (b) the mean velocity of the particle averaged over the time that the particle takes to cover the first s metres of the path. 1.23. A point moves rectilinearly with deceleration whose modulus depends on the velocity v of the particle as w = al/ v, where a is a positive constant. At the initial moment the velocity of the point is equal to va. 'What distance will it traverse before it stops? What time will it take to cover that distance? 1.24. A radius vector of a point A^ relative to the origin varies with time t as r =^ ati — bt2 j,^ where a and^ b^ are positive constants, and i and j are the unit vectors of the x^ and y axes. Find: (a) the equation of the point's trajectory y^ (x);^ plot this function; (b) the time dependence of the velocity v and acceleration w vec- tors, as well as of the moduli of these quantities; (c) the time dependence of the angle a between the vectors w and v; (d) the mean velocity vector averaged over the first t seconds of motion, and the modulus of this vector. 1.25. A point moves in the plane xy according to the law x = at, y = at (1. — at), where a and a are positive constants, and t is time. Find: (a) the equation of the point's trajectory y (x); plot this function; (b) the velocity v and the acceleration w of the point as functions of time; (c) the moment t, at which the velocity vector forms an angle It/ with the acceleration vector_ 1.26. A point moves in the plane xy according to the law x = = a sin cot, y = a (1 — cos wt), where a and co are positive constants. Find: (a) the distance s traversed by the point during the time T; (b) the angle between the point's velocity and acceleration vectors. 1.27. A particle moves in the plane xy with constant acceleration w directed along the negative y axis. The equation of motion of the particle has the form y = ax — bx2, where a and b are positive con- stants. Find the velocity of the particle at the origin of coordinates. 1.28. A small body is thrown at an angle to the horizontal with the initial velocity vo. Neglecting the air drag, find: {a) the displacement of the body as a function of time r (t); (b) the mean velocity vector (v) averaged over the first t seconds and over the total time of motion. 1.29. A body is thrown from the surface of the Earth at an angle a 15
merit when it covered the n-th (n -= 0.10) fraction of the circle after the beginning of motion. 1.38. A point moves with deceleration along the circle of radius (^) R so that at any moment of time its tangential and normal accelerations
Fig. 1.4.
are equal in moduli. At the initial moment t = 0 the velocity of the point equals vo. Find: (a) the velocity of the point as a function of time and as a function of the distance covered s; (b) the total acceleration of the point as a function of velocity and the distance covered. 1.39. A point moves along an arc of a circle of radius R. Its velocity depends on the distance covered s as v = aYi, where a is a constant. Find the angle a between the vector of the total acceleration and the vector of velocity as a function of s. 1.40. A particle moves along an arc of a circle of radius R according to the law 1 = a sin cot, where 1 is the displacement from the initial position measured along the arc, and a and co are constants. Assum- ing R = 1.00 m, a = 0.80 m, and co = 2.00 rad/s, find: (a) the magnitude of the total acceleration of the particle at the points 1 = 0 and 1 = ±a; (b) the minimum value of the total acceleration wminand the cor- responding displacement lm. 1.41. A point moves in the plane so that its tangential acceleration w, = a, and its normal acceleration wn = bt4, where a and b are positive constants, and t is time. At the moment t = 0 the point was at rest. Find how the curvature radius R of the point's trajectory and the total acceleration w depend on the distance covered s. 1.42. A particle moves along the plane trajectory y (x) with velo- city v whose modulus is constant. Find the acceleration of the par- ticle at the point x = 0 and the curvature radius of the trajectory at that point if the trajectory has the form (a) of a parabola y = ax2; (b) of an ellipse (xla)2 (y/b)2= 1; a and b are constants here. 1.43. A particle A moves along a circle of radius R = 50 cm so that its radius vector r relative to the point 0 (Fig. 1.5) rotates with the constant angular velocity w = 0.40 rad/s. Find the modulus of the velocity of the particle, and the modulus and direction of its total a^ cceleration.
2-
Fig. 1.5.
1.44. A wheel rotates around a stationary axis so that the rotation angle pvaries with time as cp = ate, where a = 0.20 rad/s2. Find the
total acceleration w of the point A at the rim at the moment t = 2.5 s
if the linear velocity of the point A at this moment v = 0.65 m/s.
1.45. A shell acquires the initial velocity v = 320 m/s, having made n = 2.0 turns inside the barrel whose length is equal to 1 = = 2.0 m. Assuming that the shell moves
inside the barrel with a uniform accelera- A
tion, find the angular velocity of its axial rotation at the moment when the shell escapes the barrel. 1.46. A solid body rotates about a station- ary axis according to the law IT = at -
- bt3, where a = 6.0 rad/s and b = 2. rad/s3. Find: (a) the mean values of the angular velo- city and angular acceleration averaged over the time interval between t = 0 and the complete stop; (b) the angular acceleration at the moment when the body stops. 1.47. A solid body starts rotating about a stationary axis with an angular acceleration 13 = (^) at, where a = (^) 2.0.10-2rad/s3. How soon after the beginning of rotation will the total acceleration vector of an arbitrary point of the body form an angle a = 60° with its velo- city vector? 1.48. A solid body rotates with deceleration about a stationary axis with an angular deceleration f3 oc -troT, where co is its angular velocity. Find the mean angular velocity of the body averaged over the whole time of rotation if at the initial moment of time its angular velocity was equal to co,. 1.49. A solid body rotates about a stationary axis so that its angu- lar velocity depends on the rotation angle cp as co = coo— acp, where coo and a are positive constants. At the moment t = 0 the angle = 0. Find the time dependence of (a) the rotation angle; (b) the angular velocity. 1.50. A solid body starts rotating about a stationary axis with an
angular acceleration it = 1 0cos p, where Pois a constant vector and cp
is an angle of rotation from the initial position. Find the angular velocity of the body as a function of the angle cp. Draw the plot of this dependence. 1.51. A rotating disc (Fig. 1.6) moves in the positive direction of the x axis. Find the equation y (x) describing the position of the
instantaneous axis of rotation, if at the initial moment the axis C
of the disc was located at the point 0 after which it moved (a) with a constant velocity v, while the disc started rotating coun- terclockwise with a constant angular acceleration 13 (the initial angu- lar velocity is equal to zero);
18
