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physics
mechanics

Questions and Answers of Mechanics

A light rope passes over a light, frictionless pulley. One end is fastened to a bunch of bananas of mass M, and a monkey of mass M clings to the other end (Fig. P11.48). The monkey climbs the rope in
A puck of mass m is attached to a cord passing through a small hole in a frictionless, horizontal surface (Fig. P11.49), the puck is initially orbiting with speed vi in a circle of radius ri. The
A projectile of mass m moves to the right with a speed vi (Fig. P11.50a). The projectile strikes and sticks to the end of a stationary rod of mass M, length d, pivoted about a frictionless axle
Two astronauts (Fig P11.51), each having a mass of 75.0 kg, are connected by a 10.0-m rope of negligible mass. They are isolated in space, orbiting their center of mass at speeds of 5.00 m/s.
Two astronauts (Fig P11.51), each having a mass M, are connected by a rope of length d having negligible mass. They are isolated in space, orbiting their center of mass at speeds v. treating the
Global warming is a cause for concern because even small changes in the Earth’s temperature can have significant consequences. For example, if the Earth’s polar ice caps were to melt entirely,
A solid cube of wood of side 2a and mass M is resting on a horizontal surface. The cube is constrained to rotate about an axis AB (Fig. P11.54). A bullet of mass m and speed v is shot at the face
A solid cube of side 2a and mass M is sliding on a frictionless surface with uniform velocity v as in Figure P11.55a. It hits a small obstacle at the end of the table, which causes the cube to tilt
A uniform solid disk is set into rotation with an angular speed ώi about an axis through its center. While still rotating at this speed, the disk is placed into contact with a horizontal surface
Suppose a solid disk of radius R is given an angular speed ώi about an axis through its center and then lowered to a horizontal surface and released, as in Problem 56 (Fig. P11.56). Furthermore,
The polar coordinates of a point are r = 5.50 m and θ = 240°. What are the Cartesian coordinates of this point?
Two points in a plane have polar coordinates (2.50 m, 30.0°) and (3.80 m, 120.0°). Determine (a) the Cartesian coordinates of these points and (b) the distance between them.
A fly lands on one wall of a room. The lower left-hand corner of the wall is selected as the origin of a two-dimensional Cartesian coordinate system. If the fly is located at the point having
Two points in the xy plane have Cartesian coordinates (2.00, ─4.00) m and (─3.00, 3.00) m. Determine (a) the distance between these points and (b) their polar coordinates.
If the rectangular coordinates of a point are given by (2, y) and its polar coordinates are (r, 30°), determine y and r.
If the polar coordinates of the point (x, y) are (r, θ), determine the polar coordinates for the points: (a) (─x, y), (b) (─2x, ─2y), and (c) (3x, ─3y).
A surveyor measures the distance across a straight river by the following method: starting directly across from a tree on the opposite bank, she walks 100 m along the riverbank to establish a
A pedestrian moves 6.00 km east and then 13.0 km north. Find the magnitude and direction of the resultant displacement vector using the graphical method.
A plane flies from base camp to lake A, 280 km away, in a direction of 20.0° north of east. After dropping off supplies it flies to lake B, which is 190 km at 30.0° west of north from lake A.
Vector A has a magnitude of 8.00 units and makes an angle of 45.0° with the positive x axis. Vector B also has a magnitude of 8.00 units and is directed along the negative x axis. Using graphical
A skater glides along a circular path of radius 5.00 m. If he coasts around one half of the circle, find (a) the magnitude of the displacement vector and (b) how far the person skated. (c) What is
A force F1 of magnitude 6.00 units acts at the origin in a direction 30.0° above the positive x axis. A second force F2 of magnitude 5.00 units acts at the origin in the direction of the positive y
Arbitrarily define the “instantaneous vector height” of a person as the displacement vector from the point halfway between his or her feet to the top of the head. Make an order-of-magnitude
A dog searching for a bone walks 3.50m south, then runs 8.20 m at an angle 30.0° north of east, and finally walks 15.0 m west. Find the dog’s resultant displacement vector using graphical
Each of the displacement vectors A and B shown in Fig P3.15 has a magnitude of 3.00 m. Find graphically (a) A + B, (b) A─B, (c) B ─ A, (d) A ─ 2B. Report all angles counterclockwise
Three displacements are A = 200 m, due south; B = 250 m, due west; C = 150 m, 30.0° east of north. Construct a separate diagram for each of the following possible ways of adding these vectors: R1 =
A roller coaster car moves 200 ft horizontally, and then rises 135 ft at an angle of 30.0° above the horizontal. It then travels 135 ft at an angle of 40.0° downward. What is its displacement from
Find the horizontal and vertical components of the 100-m displacement of a superhero who flies from the top of a tall building following the path shown in Fig. P3.18
A vector has an x component of #25.0 units and a y component of 40.0 units. Find the magnitude and direction of this vector.
A person walks 25.0° north of east for 3.10 km. How far would she have to walk due north and due east to arrive at the same location?
Obtain expressions in component form for the position vectors having the following polar coordinates: (a) 12.8 m, 150° (b) 3.30 cm, 60.0° (c) 22.0 in., 215°.
A displacement vector lying in the xy plane has a magnitude of 50.0m and is directed at an angle of 120° to the positive x axis. What are the rectangular components of this vector?
A girl delivering newspapers covers her route by traveling 3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east. (a) What is her resultant displacement? (b) What is the total distance she
In 1992, Akira Matsushima, from Japan, rode a unicycle across the United States, covering about 4 800 km in six weeks. Suppose that, during that trip, he had to find his way through a city with
While exploring a cave, a spelunker starts at the entrance and moves the following distances. She goes 75.0 m north, 250 m east, 125 m at an angle 30.0° north of east, and 150 m south. Find the
A map suggests that Atlanta is 730 miles in a direction of 5.00° north of east from Dallas. The same map shows that Chicago is 560 miles in a direction of 21.0° west of north from Atlanta. Modeling
Given the vectors A = 2.00ˆi + 6.00ˆj and B = 3.00ˆi ─ 2.00ˆj, (a) draw the vector sum C = A + B and the vector difference D = A ─ B. (b) Calculate C and D, first in terms of unit
Find the magnitude and direction of the resultant of three displacements having rectangular components (3.00, 2.00) m, (─ 5.00, 3.00) m, and (6.00, 1.00) m.
A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude of 150 cm and makes an angle of 120° with the positive x axis. The resultant displacement has a
Vector A has x and y components of ─ 8.70 cm and 15.0 cm, respectively; vector B has x and y components of 13.2 cm and ─ 6.60 cm, respectively. If A ─ B + 3C = 0, what are the
Consider the two vectors A = 3ˆi ─ 2ˆj and B = ─ˆi ─ 4ˆj. Calculate (a) A + B, (b) A ─ B, (c) |A + B|, (d) |A ─ B|, and (e) the directions of A + B and A ─ B.
Consider the three displacement vectors A = (3ˆi ─ 3ˆj) m, B = (ˆi ─ 4ˆj) m, and C = (─2ˆi + 5ˆj) m. Use the component method to determine (a) the magnitude and direction of
A particle undergoes the following consecutive displacements: 3.50 m south, 8.20 m northeast, and 15.0 m west. What is the resultant displacement?
In a game of American football, a quarterback takes the ball from the line of scrimmage, runs backward a distance of 10.0 yards, and then sideways parallel to the line of scrimmage for 15.0 yards. At
The helicopter view in Fig P3.35 shows two people pulling on a stubborn mule. Find (a) the single force that is equivalent to the two forces shown, and (b) the force that a third person would have to
A novice golfer on the green takes three strokes to sink the ball. The successive displacements are 4.00 m to the north, 2.00 m northeast, and 1.00 m at 30.0° west of south. Starting at the same
Use the component method to add the vectors A and B shown in Figure P3.15. Express the resultant A + B in unit–vector notation.
In an assembly operation illustrated in Figure P3.38, a robot moves an object first straight upward and then also to the east, around an arc forming one quarter of a circle of radius 4.80 cm that
Vector B has x, y, and z components of 4.00, 6.00, and 3.00 units, respectively. Calculate the magnitude of B and the angles that B makes with the coordinate axes.
You are standing on the ground at the origin of a coordinate system. An airplane flies over you with constant velocity parallel to the x axis and at a fixed height of 7.60 x 103 m. At time t = 0 the
The vector A has x, y, and z components of 8.00, 12.0, and ─ 4.00 units, respectively. (a) Write a vector expression for A in unit–vector notation. (b) Obtain a unit–vector expression
Instructions for finding a buried treasure include the following: Go 75.0 paces at 240° turn to 135° and walk 125 paces, then travel 100 paces at 160°. The angles are measured counterclockwise
Given the displacement vectors A = (3ˆi ─ 4ˆj + 4ˆk) m and B = (2ˆi + 3ˆj ─ 7ˆk) m, find the magnitudes of the vectors (a) C = A + B and (b) D = 2A ─ B, also expressing
A radar station locates a sinking ship at range 17.3 km and bearing 136° clockwise from north. From the same station a rescue plane is at horizontal range 19.6 km, 153° clockwise from north, with
As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 60.0° north of west with a speed of 41.0 km/h. Three hours later, the course of the hurricane suddenly shifts
(a) Vector E has magnitude 17.0 cm and is directed 27.0° counterclockwise from the + x axis. Express it in unit–vector notation. (b) Vector F has magnitude 17.0 cm and is directed 27.0°
An airplane starting from airport A flies 300 km east, then 350 km at 30.0° west of north, and then 150 km north to arrive finally at airport B. (a) The next day, another plane flies directly from
Three displacement vectors of a croquet ball are shown in Figure P3.49, where |A| = 20.0 units, |B| = 40.0 units, and |C| = 30.0 units. Find(a) The resultant in unit€“vector notation and(b)
If A = (6.00ˆi ─ 8.00ˆj) units, B = (─8.00ˆi + 3.00ˆj) units, and C = (26.0ˆi +19.0ˆj) units, determine a and b such that aA + bB + C = 0.
Two vectors A and B have precisely equal magnitudes. In order for the magnitude of A + B to be one hundred times larger than the magnitude of A ─ B, what must be the angle between them?
Two vectors A and B have precisely equal magnitudes. In order for the magnitude of A + B to be larger than the magnitude of A ─ B by the factor n, what must be the angle between them?
A vector is given by R = 2ˆi + ˆj + 3ˆk. Find (a) the magnitudes of the x, y, and z components, (b) the magnitude of R, and (c) the angles between R and the x, y, and z axes.
The biggest stuffed animal in the world is a snake 420 m long, constructed by Norwegian children. Suppose the snake is laid out in a park as shown in Figure P3.54, forming two straight sides of a
An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 800 m, horizontal distance 19.2 km and 25.0° south of west. The second aircraft is at altitude 1 100 m,
A ferry boat transports tourists among three islands. It sails from the first island to the second island, 4.76 km away, in a direction 37.0° north of east. It then sails from the second island to
The rectangle shown in Figure P3.57 has sides parallel to the x and y axes. The position vectors of two corners are A = 10.0 m at 50.0° and B = 12.0 m at 30.0°.(a) Find the perimeter of the
Find the sum of these four vector forces: 12.0 N to the right at 35.0° above the horizontal, 31.0 N to the left at 55.0° above the horizontal, 8.40 N to the left at 35.0° below the horizontal and
The total trip consists of four straight-line paths. At the end of the walk, what is the person€™s resultant displacement measured from the starting point?
The instantaneous position of an object is specified by its position vector r leading from a fixed origin to the location of the point object. Suppose that for a certain object the position vector is
A jet airliner, moving initially at 300 mi/h to the east, suddenly enters a region where the wind is blowing at 100 mi/h toward the direction 30.0° north of east. What are the new speed and
Long John Silver, a pirate, has buried his treasure on an island with five trees, located at the following points: (30.0 m, ─20.0 m), (60.0 m, 80.0 m), (─10.0 m, ─10.0 m), (40.0 m,
Consider a game in which N children position themselves at equal distances around the circumference of a circle. At the center of the circle is a rubber tire. Each child holds a rope attached to the
A rectangular parallelepiped has dimensions a, b, and c, as in Figure P3.64.(a) Obtain a vector expression for the face diagonal vector R1. What is the magnitude of this vector?(b) Obtain a vector
Vectors A and B have equal magnitudes of 5.00. If the sum of A and B is the vector 6.00ˆj, determine the angle between A and B.
In Figure P3.66 a spider is resting after starting to spin its web. The gravitational force on the spider is 0.150 newton down. The spider is supported by different tension forces in the two strands
A point P is described by the coordinates (x, y) with respect to the normal Cartesian coordinate system shown in Fig. P3.67. Show that (x`, y`), the coordinates of this point in the rotated
Vector A has a negative x component 3.00 unit in length and a positive y component 2.00 units in length. (a) Determine an expression for A in unit–vector notation. (b) Determine the magnitude
A block of mass 2.50 kg is pushed 2.20 m along a frictionless horizontal table by a constant 16.0-N force directed 25.0° below the horizontal. Determine the work done on the block by (a) The
A shopper in a supermarket pushes a cart with a force of 35.0 N directed at an angle of 25.0° downward from the horizontal. Find the work done by the shopper on the cart as he moves down an aisle
Batman whose mass is 80.0 kg, is dangling on the free end of a 12.0-m rope, the other end of which is fixed to a tree limb above. He is able to get the rope in motion as only Batman knows how
A raindrop of mass 3.35 x 10─5 kg falls vertically at constant speed under the influence of gravity and air resistance. Model the drop as a particle. As it falls 100 m, what is the work done on
Vector A has a magnitude of 5.00 units, and B has a magnitude of 9.00 units. The two vectors make an angle of 50.0° with each other. Find A∙B.
For any two vectors A and B, show that A∙B = AxBx + AyBy + AzBz. (Suggestion: Write A and B in unit vector form and use Equations 7.4 and 7.5.)
A force F = (6i ─ 2j) N acts on a particle that undergoes a displacement Δr = (3i + j) m. Find (a) the work done by the force on the particle and (b) the angle between F and Δr.
Find the scalar product of the vectors in Figure P7.8.
Using the definition of the scalar product, find the angles between (a) A = 3i ─ 2j and B = 4i ─ 4j; (b) A = ─2i + 4j and B = 3i + 4j + 2k; (c) A = i ─ 2j + 2k and B =
For A = 3i + j ─ k, B = ─i + 2j + 5k, and C = 2j ─ 3k, find C∙ (A ─ B).
The force acting on a particle varies as in Figure P7.11. Find the work done by the force on the particle as it moves (a) From x = 0 to x = 8.00 m, (b) From x = 8.00 m to x = 10.0 m, and (c)
The force acting on a particle is Fx = (8x ─ 16) N, where x is in meters. (a) Make a plot of this force versus x from x = 0 to x = 3.00 m. (b) From your graph, find the net work done by
A particle is subject to a force Fx that varies with position as in Figure P7.13. Find the work done by the force on the particle as it moves(a) From x = 0 to x = 5.00 m,(b) From x = 5.00 m to x =
A force F = (4xi + 3yj) N acts on an object as the object moves in the x direction from the origin to x = 5.00 m. Find the work = ƒ F ∙dx done on the object by the force.
When a 4.00-kg object is hung vertically on a certain light spring that obeys Hooke’s law, the spring stretches 2.50 cm. If the 4.00-kg object is removed, (a) How far will the spring stretch if a
An archer pulls her bowstring back 0.400 m by exerting a force that increases uniformly from zero to 230 N. (a) What is the equivalent spring constant of the bow? (b) How much work does the archer do
Truck suspensions often have €œhelper springs€ that engage at high loads. One such arrangement is a leaf spring with a helper coil spring mounted on the axle, as in Figure P7.17.
A 100-g bullet is fired from a rifle having a barrel 0.600 m long. Assuming the origin is placed where the bullet begins to move, the force (in newtons) exerted by the expanding gas on the bullet is
If it takes 4.00 J of work to stretch a Hooke’s-law spring 10.0 cm from its unstressed length, determine the extra work required to stretch it an additional 10.0 cm.
A small particle of mass m is pulled to the top of a frictionless half-cylinder (of radius R) by a cord that passes over the top of the cylinder, as illustrated in Figure P7.20. (a) If the particle
A light spring with spring constant 1200 N/m is hung from an elevated support. From its lower end a second light spring is hung, which has spring constant 1 800 N/m. An object of mass 1.50 kg is hung
A light spring with spring constant k1 is hung from an elevated support. From its lower end a second light spring is hung, which has spring constant k2. An object of mass m is hung at rest from the
Express the units of the force constant of a spring in SI base units.

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