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physics
mechanics
Fundamentals of Physics 8th Extended edition Jearl Walker, Halliday Resnick - Solutions
A car starts moving rectilinearly, first with acceleration w = 5.0 m/s2 (the initial velocity is equal to zero), then uniformly, and finally, decelerating at the same rate w, comes to a stop. The total time of motion equals τ 25 s. The average velocity during that time is equal to (v) = 72 km
A point moves rectilinearly in one direction. Fig.1.1 showsThe distance s traversed by the point as a function of the time t. Using the plot find:(a) The average velocity of the point during the time of motion;(b) The maximum velocity;(c) The time moment t0 at which the instantaneous velocity is
Two particles, 1 and 2, move with constant velocities v1 and v2. At the initial moment their radius vectors are equal to r1 and r2. How must these four vectors be interrelated for the particles to collide?
A ship moves along the equator to the east with velocity v0 = 30 km/hour. The southeastern wind blows at an angle φ = 60° to the equator with velocity v = 15 km/hour. Find the wind velocity v' relative to the ship and the angle φ' between the equator and the wind direction in the
Two swimmers leave point A on one bank of the river to reach point B lying right across on the other bank. One of them crosses the river along the straight line AB while the other swims at right angles to the stream and then walks the distance that he has been carried away by the stream to get
Two boats, A and B, move away from a buoy anchored at the middle of a river along the mutually perpendicular straight lines: the boat A along the river, and the boat B across the river. Having moved off an equal distance from the buoy the boats returned. Find the ratio of times of motion of boats
A boat moves relative to water with a velocity which is n = 2.0 times less than the river flow velocity. At what angle to the stream direction must the boat move to minimize drifting?
Two bodies were thrown simultaneously from the same point: one, straight up, and the other, at an angle of θ = 60° to the horizontal. The initial velocity of each body is equal to v0 =- 25 m/s. Neglecting the air drag, find the distance between the bodies t = 1.70 s later.
Two particles move in a uniform gravitational field with an acceleration g. At the initial moment the particles were located at one point and moved with velocities v1 = 3.0 m/s and v2 = 4.0 m/s horizontally in opposite directions. Find the distance between the particles at the moment when their
Three points are located at the vertices of an equilateral triangle whose side equals a. They all start moving simultaneously with velocity v constant in modulus, with the first point heading continually for the second, the second for the third, and the third for the first. How soon will the points
Point A moves uniformly with velocity v so that the vector v is continually “aimed” at point B which in its turn moves rectilinearly and uniformly with velocity u < v. At the initial moment of time v ┴ u and the points are separated by a distance l. How soon will the points converge?
A train of length l = 350 m starts moving rectilinearly with constant acceleration w = 3.0. 10-2 m/s2; t = 30 s after the start the locomotive headlight is switched on (event 1), and τ= 60 s after that event the tail signal light is switched on (event 2). Find the distance between these events
An elevator car whose floor-to-ceiling distance is equal to 2.7 m starts ascending with constant acceleration 1.2 m/s2; 2.0 s after the start a bolt begins falling from the ceiling of the car. Find: (a)The bolt's free fall time; (b)The displacement and the distance covered by the bolt during the
Two particles, 1 and 2, move with constant velocities vl and v2 along two mutually perpendicular straight lines toward the intersection point 0. At the moment t = 0 the particles were located at the distances l1 and l2 from the point 0. How soon will the distance between the particles become the
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 l from the highway. It is known that the car moves in the field η times slower than on the highway. At what distance from point D one must turn off the highway?
A point travels along the x axis with a velocity whose projection vx is presented as a function of time by the plot in 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
A point traversed hall a circle of radius R = 160 cm during time interval τ = 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) |; (c) The modulus of the mean vector of the total acceleration
A radius vector of a particle varies with time t as r = at (1 - at), 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 Δt taken by the particle to return to the initial
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 = v0 (1 – t /τ), where v0 is the initial velocity vector whose modulus equals v0 = 10.0 cm/s; τ = 5.0 s. Find: (a) The x coordinates of the particle
The velocity of a particle moving in the positive direction of the x axis varies as v = a √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;
A point moves rectilinearly with deceleration whose modulus depends on the velocity v of the particle as w = a √v where a is a positive constant. At the initial moment the velocity of the point is equal to v0. What distance will it traverse before it stops? What time will it take to cover
A radius vector of a point A relative to the origin varies with time t as r = ati - bt2j, 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
A point moves in the plane xy according to the law x = at, y = at (1- at), where a and α 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
A point moves in the plane xy according to the law x = a sin ωt, y = a (1 - cos ωt), where a and ω are positive constants. Find: (a) The distance s traversed by the point during the time τ; (b) The angle between the point's velocity and acceleration vectors.
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 constants. Find the velocity of the particle at the origin of coordinates.
A small body is thrown at an angle to the horizontal with the initial velocity v0. 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.
A body is thrown from the surface of the Earth at an angle α to the horizontal with the initial velocity vo. assuming the air drags to be negligible, find: (a) The time of motion; (b) The maximum height of ascent and the horizontal range; at what value of the angle a they will be equal to
Using the conditions of the foregoing problem, draw the approximate time dependence of moduli of the normal wn and tangent wτ acceleration vectors, as well as of the projection of the total acceleration vector wv on the velocity vector direction
A ball starts falling with zero initial velocity on a smooth inclined plane forming an angle α with the horizontal. Having fall - en the distance h, the ball rebounds elastically off the inclined plane. At what distance from the impact point will the ball rebound for the second time?
A cannon and a target are 5.10 km apart and located at the same level. How soon will the shell launched with the initial velocity 240 m/s reach the target in the absence of air drag?
A cannon fires successively two shells with velocity vo = 250 m/s; the first at the angle θ1 = 60° and the second at the angle θ2 = 45° to the horizontal, the azimuth being the same. Neglecting the air drag, find the time interval between firings leading to the collision of the shells
A balloon starts rising from the surface of the Earth. The ascension rate is constant and equal to vo. Due to the wind the bal- loon gathers the horizontal velocity component vx = ay, where a is a constant and y is the height of ascent. Find how the following quantities depend on the height of
A particle moves in the plane xy with velocity v = ai + bxj, where i and j are the unit vectors of the x and y axes, and a and b are constants. At the initial moment of time the particle was located at the point x = y = 0. Find: (a) The equation of the particle's trajectory y (x); (b) The curvature
A particle A moves in one direction along a given trajectory with a tangential acceleration wτ = aτ, where a is a constant vector coinciding in direction with the x axis (Fig. 1.4), and τ is a unit vector coinciding in direction with the velocity vector at a given point. Find how the
A point moves along a circle with a velocity v = at, where a = 0.50 m/s2. Find the total acceleration of the point at the moment when it covered the n-th (n = 0.10) fraction of the circle after the beginning of motion.
A point moves with deceleration along the circle of radius R so that at any moment of time its tangential and normal accelerationsare 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
A point moves along an arc of a circle of radius R. Its velocity depends on the distance covered s as v = a√ s, where α is a constant. Find the angle a between the vector of the total acceleration and the vector of velocity as a function of s.
A particle moves along an arc of a circle of radius R according to the law l = α sin ωt, where l is the displacement from the initial position measured along the arc, and α and ω are constants. Assuming R = 1.00 m, α = 0.80 m, and ω = 2.00 rad /s, find: (a) The
A point moves in the plane so that its tangential acceleration wτ = α, and its normal acceleration wn = bt4, where α 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
A particle moves along the plane trajectory y (x) with velocity v whose modulus is constant. Find the acceleration of the particle 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 (x/a)2 +
A particle A moves along a circle of radius R = 50 cm so that its radius vector r relative to the point O (Fig. 1.5) rotates with the constant angular velocity ω = 0.40 rad /s. Find the modulus of the velocity of the particle, and the modulus and direction of its total acceleration.
A wheel rotates around a stationary axis so that the rotation angle φ varies with time as φ = at2, where α = 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.
A shell acquires the initial velocity v = 320 m/s, having made n = 2.0 turns inside the barrel whose length is equal to l = 2.0 m. Assuming that the shell moves inside the barrel with a uniform acceleration, find the angular velocity of its axial rotation at the moment when the shell escapes the
A solid body rotates about a stationary axis according to the law φ = at - bt3, where α = 6.0 rad/s and b = 2.0 rad/s3. Find: (a) The mean values of the angular velocity and angular acceleration averaged over the time interval between t = 0 and the complete stop; (b) The angular
A solid body starts rotating about a stationary axis with an angular acceleration β = at, where α = 2.0.10-2 rad/s3. How soon after the beginning of rotation will the total acceleration vector of an arbitrary point of the body form an angle α = 60° with its velocity vector?
A solid body rotates with deceleration about a stationary axis with an angular deceleration β ∞ √ω where ω 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
A solid body rotates about a stationary axis so that its angular velocity depends on the rotation angle φ as ω = ωo - αφ, where ωo and α are positive constants. At the moment t = 0 the angle φ = 0. Find the time dependence of (a) The rotation angle; (b)
A solid body starts rotating about a stationary axis with an angular acceleration β = βo cos φ, where βo is a constant vector and φ is an angle of rotation from the initial position. Find the angular velocity of the body as a function of the angle φ. Draw the plot of
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 O after which it moved(a) With a constant velocity v, while the
A point A is located on the rim of a wheel of radius R = 0.50 m which rolls without slipping along a horizontal surface with velocity v = 1.00 m/s. Find: (a) The modulus and the direction of the acceleration vector of the point A; (b) The total distance s traversed by the point A between the two
A ball of radius R = 10.0 cm rolls without slipping down an inclined plane so that its centre moves with constant accelerationw = 2.50 cm/s2; t = 2.00 s after the beginning of motion its position corresponds to that shown in Fig. i.7. Find:(a) The velocities of the points A, B, and O;(b) The
A cylinder rolls without slipping over a horizontal plane. The radius of the cylinder is equal to r. Find the curvature radii of trajectories traced out by the points A and B (see Fig, 1.7).
Two solid bodies rotate about stationary mutually perpendicular intersecting axes with constant angular velocities ω1 = 3.0 rad/s and ω2 = 4.0 rad/s. Find the angular velocity and angular acceleration of one body relative to the other.
A solid body rotates with angular velocity ω = ati + bt2j, where α = 0.50 rad/s2, b = 0.060 rad/s3, and i and j are the unit vectors of the x and y axes. Find: (a) The moduli of the angular velocity and the angular acceleration at the moment t = 10.0 s; (b) The angle between the
A round cone with half-angle α = 30° and the radius of the base R = 5.0 cm rolls uniformly and without slipping over a horizontal plane as shown in Fig. 1.8. The cone apex is hinged at the point O which is on the same level with the point C, the cone base centre. The velocity of point C
A solid body rotates with a constant angular velocity ωo = 0.50 rad/s about a horizontal axis AB. At the moment t = 0 the axis AB starts turning about the vertical with a constant angular acceleration βo = 0.10 rad/s s. Find the angular velocity and angular acceleration of the body after
An aerostat of mass m starts coming down with a constant acceleration w. Determine the ballast mass to be dumped for the aerostat to reach the upward acceleration of the same magnitude. The air drag is to he neglected.
In the arrangement of Fig. the masses m0, m1, and m2 of bodies are equal, the masses of the pulley and the threads are negligible, and there is no friction in the pulley. Find the acceleration w with which the body m0 comes down, and the tension of the thread binding together the bodies m1 and m2,
Two touching bars 1 and 2 are placed on an inclined plane forming an angle α with the horizontal (Fig). The masses of the bars are equal to m1 and m2, and the coefficients of friction between the inclined plane and these bars are equal to k1 and k2 respectively, with k1 > k2. Find:(a) The
A small body was launched up an inclined plane set at an angle α = 15° against the horizontal. Find the coefficient of friction, if the time of the ascent of the body is η = 2.0 times less than the time of its descent.
The following parameters of the arrangement of Fig are available: the angle α which the inclined plane forms with the horizontal, and the coefficient of friction k between the body m1 and the inclined plane. The masses of the pulley and the threads, as well as the friction in the pulley, are
The inclined plane of Fig. forms an angle α = 30° with the horizontal. The mass ratio m2/m1 = η = 2/3. The coefficient of friction between the body m1 and the inclined plane is equal to k = 0.10. The masses of the pulley and the threads are negligible. Find the magnitude and the
A plank of mass m1 with a bar of mass m2 placed on it lies on a smooth horizontal plane. A horizontal force growing with time t as F = at (α is constant) is applied to the bar. Find how the accelerations of the plank w1 and of the bar w2 depend on t, if the coefficient of friction between the
A small body A starts sliding down from the top of a wedge (Fig.) whose base is equal to l = 2.10 m. The coefficient of friction between the body and the wedge surface is k = 0.140. At what value of the angle α will the time of sliding be the least? What will it be equal to?
A bar of mass m is pulled by means of a thread up an inclined plane forming an angle α with the horizontal (Fig). The coefficient of friction is equal to k. Find the angle β which the thread must form with the inclined plane for the tension of the thread to be minimum. What is it equal
At the moment t = 0 the force F = at is applied to a small body of mass m resting on a smooth horizontal plane (α is a constant). The permanent direction of this force forms an angle α with the horizontal (Fig.). Find:(a) The velocity of the body at the moment of its breaking off the
A bar of mass m resting on a smooth horizontal plane starts moving due to the force F = mg/3 of constant magnitude. In the process of its rectilinear motion the angle α between the direction of this force and the horizontal varies as α = as, where α is a constant, and s is the
A horizontal plane with the coefficient of friction k supports two bodies: a bar and an electric motor with a battery on a block. A thread attached to the bar is wound on the shaft of the electric motor. The distance between the bar and the electric motor is equal to l. When the motor is switched
A pulley fixed to the ceiling of an elevator car carries a thread whose ends are attached to the loads of masses m1 and m2. The car starts going up with an acceleration Wo. Assuming the masses of the pulley and the thread, as well as the friction, to be negligible find: (a) The acceleration of the
Find the acceleration w of body 2 in the arrangement shown in Fig., if its mass is η times as great as the mass of bar 1 and the angle that the inclined plane forms with the horizontal is equal to α. The masses of the pulleys and the threads, as well as the friction, are assumed to be
In the arrangement shown in Fig the bodies have masses m0, m1, m2, the friction is absent, the masses of the pulleys and the threads are negligible. Find the acceleration of the body m1. Look into possible cases.
In the arrangement shown in Fig. the mass of the rod M exceeds the mass m of the ball. The ball has an opening permitting it to slide along the thread with some friction. The mass of the pulley and the friction in its axle are negligible. At the initial moment the ball was located opposite the
In the arrangement shown in Fig. 1.18 the mass of ball 1 is η = 1.18 times as great as that of rod 2. The length of the latter is l = 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
In the arrangement shown in Fig. the mass of body 1 is η = 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 arrangement set in motion. What is the
Find the accelerations of rod A and wedge B in the arrangement shown in Fig. if the ratio of the mass of the wedge to that of the rod equals η, and the friction between all contact surfaces is negligible.
In the arrangement shown in Fig. the masses of the wedge M and the body m are known. The appreciable friction exists 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
What is the minimum acceleration with which bar A (Fig) should be shifted horizontally to keep bodies 1 and 2 stationary relative to the bar? The masses of the bodies are equal, and the coefficient of friction between the bar and the bodies is equal to k. The masses of the pulley and the threads
Prism 1 with bar 2 of mass m placed on it gets a horizontal acceleration w directed to the left (Fig.). 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
Prism 1 of mass m1 and with angle a (see Fig) rests on a horizontal surface. Bar 2 of mass m2 is placed on the prism. Assuming the friction to be negligible, find the acceleration of the prism.
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. The masses of the pulley and the thread are negligible. The friction is absent. Find the acceleration of the wedge M.
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 wτ, the
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.
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 θ, the angle of deflection of the thread from the vertical;
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.
A small body A starts sliding off the top of a smooth sphere of radius R Find the angle θ (Fig. 125) corresponding to the point at which the body breaks off the sphere, as well as the break-off velocity of the body.
A device (Fig.) consists of a smooth L-shaped rod located in a horizontal plane and a sleeve A of mass m attached by a weight- less spring to a point B. The spring stiffness is equal to x. The whole system rotates with a constant angular velocity ω about a vertical axis passing through the
A cyclist rides along the circumference of a circular horizontal plane of radius R, the friction coefficient being dependent only on distance r from the centre O of the plane as k = ko (1-r/R), where ko is a constant. Find the radius of the circle with the centre at the point along which the
A car moves with a constant tangential acceleration wτ = 0.62 m/s2 along a horizontal surface circumscribing a circle of radius R = 40 m. The coefficient of sliding friction between the wheels of the car and the surface is k = 0.20. What distance will the car ride without sliding if at the
A car moves uniformly along a horizontal sine curve y = a sin (x/a), where a and a are certain constants. The coefficient of friction between the wheels and the road is equal to k. At what velocity will the car ride without sliding?
A chain of mass m forming a circle of radius R is slipped on a smooth round cone with half-angle θ. Find the tension of the chain if it rotates with a constant angular velocity ω about a vertical axis coinciding with the symmetry axis of the cone.
A fixed pulley carries a weightless thread with masses m1 and m2 at its ends. There is friction between the thread and the pulley. It is such that the thread starts slipping when the ratio m2/ml =ηo. Find: (a) The friction coefficient; (b) The acceleration of the masses when m2/ml = η
A particle of mass m moves along the internal smooth surface of a vertical cylinder of radius R. Find the force with which the particle acts on the cylinder wall if at the initial moment of time its velocity equals vo and forms an angle a with the horizontal.
Find the magnitude and direction of the force acting on the particle of mass m during its motion in the plane xy according to the law x = asin ωt, y = bcos ωt, where a, b, and ω are constants.
A body of mass m is thrown at an angle to the horizontal with the initial velocity vo. Assuming the air drag to be negligible, find: (a) The momentum increment Δp that the body acquires over the first t seconds of motion; (b) The modulus of the momentum increment Δp during the total
At the moment t = 0 a stationary particle of mass m experiences a time-dependent force F = at (τ - t), where a is a constant vector, τ is the time during which the given force acts. Find: (a) The momentum of the particle when the action of the force discontinued: (b) The distance
At the moment t = 0 a particle of mass m starts moving due to a force F = Fo sin ωt, where Fo and ω are constants. Find the distance covered by the particle as a function of t. Draw the approximate plot of this function.
At the moment t = 0 a particle of mass m starts moving due to a force F = Fo cos ωt, where F 0 and ω 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
A motorboat of mass m moves along a lake with velocity vo. 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
Having gone through a plank of thickness h, a bullet changed its velocity from vo 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.
A small bar starts sliding down an inclined plane forming an angle α with the horizontal. The friction coefficient depends on the distance x covered as k = αx, where α is a constant. Find the distance covered by the bar till it stops and its maximum velocity over this distance.
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