Vector 1 - Lecture notes 1 PDF

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Chapter-1

Vectors Test Your Understanding (TYU) 󰇍 have 1. (Test Your Understanding of Section 1.7) Two displacement vectors, 𝑆 and 𝑇 magnitudes S = 3 m and T = 4 m. Which of the following could be the magnitude of the 󰇍 ? (There may be more than one correct answer.) (i) 9 m; (ii) 7 m; difference vector 𝑆 − 𝑇 (iii) 5 m; (iv) 1 m; (v) 0 m; (vi) -1 m. 󰇍 has 2. Test Your Understanding of Section 1.10 Vector 𝐴 has magnitude 2 and vector 𝐵 󰇍 is known to be 0°, 90°, or 180°. For each of the magnitude 3. The angle 𝜙 between 𝐴 and 𝐵 following situations, state what the value of 𝜙 must be. (In each situation there may be 󰇍 = 0 (b) 𝐴 × 𝐵 󰇍 = 0 (c) 𝐴. 𝐵 󰇍 = 6 (d) 𝐴. 𝐵󰇍 = −6 (e) more than one correct answer.) (a) 𝐴. 𝐵 󰇍 | = 6 |𝐴 × 𝐵

In Class Problems 1. 2. 3.

󰇍 = (6𝑖 + 3𝑗 − 𝑘)𝑚 and (Refer text book example-1.9) Given the two displacements 𝐷  )𝑚 , find the magnitude of the displacement 2𝐷 󰇍 = (4𝑖 − 5𝑗 + 8𝑘 󰇍 − 𝐸 󰇍 𝐸 (Refer text book example-1.11) Find the angle between the vectors 𝐴 = 2𝑖 + 3𝑗 + 𝑘 and 󰇍 = −4𝑖 + 2𝑗 − 𝑘 𝐵 (Refer text book example-1.12) Vector 𝐴 has magnitude 6 units and is in the direction of 󰇍 has magnitude 4 units and lies in the xy-plane, making an angle of +x-axis. The vector 𝐵 𝐵. 30° with the +x-axis. Find the vector product 𝐶 = 𝐴 × 󰇍󰇍󰇍 Assignment Questions

1. 2.

3.

(Refer text book exercises -1.40) The head of a vector at the coordinate (3, 4, 7) and its tail is at (-2, 5, 1). Write the vector. (Refer text book exercises -1.44) (a) Is the vector (𝑖 + 𝑗 + 𝑘) a unit vector?. Justify your answer. (b) Can a unit vector have any components with magnitude greater than unity? Can it have any negative components? In each case justify your answer. (c) 𝐴 = 𝑎(3.0𝑖 + 4.0𝑗) , where a is a constant, determine the value of a that makes 𝐴 a unit vector. 󰇍 = 3𝑖 + (Refer text book exercises-1.53) Given two vectors 𝐴 = −2𝑖 + 3𝑗 + 4𝑘 and 𝐵  𝑗 − 3𝑘, do the following. (a) Find the magnitude of each vector. (b) Write an expression 󰇍 using unit vectors. (c) Find the magnitude of the vector for the vector difference 𝐴 − 𝐵 󰇍 . Is this the same as the magnitude of 𝐵󰇍 − 𝐴. Explain? difference 𝐴 − 𝐵

Chapter-2

Motion along a straight line Test your Understanding (TYU)

1.

Test Your Understanding of Section 2.1 Each of the following automobile trips takes one hour. The positive x-direction is to the east. (i) Automobile A travels 50 km due east. (ii) Automobile B travels 50 km due west. (iii) Automobile C travels 60 km due east, then turns around and travels 10 km due west. (iv) Automobile D travels 70 km due east. (v) Automobile E travels 20 km due west, then turns around and travels 20 km due east. (a) Rank the five trips in order of average x-velocity from most positive to most negative. (b) Which trips, if any, have the same average x-velocity? (c) For which trip, if any, is the average x-velocity equal to zero?

2.

Test Your Understanding of Section 2.3 Look again at the x-t graph in Fig. 2.9 at the end of Section 2.2. (a) At which of the points P, Q, R, and S is the x-acceleration ax positive? (b) At which points is the x-acceleration negative? (c) At which points does the x-acceleration appear to be zero? (d) At each point state whether the velocity is increasing, decreasing, or not changing.

In-class problem: 1. Refer text book-example-2.4 A motorcyclist heading east through a small town accelerates at a constant 4 m/s2 after he leaves the city limits. At time t = 0, he is 5.0 m east of the citylimits signpost, moving east at 15 m/s. (a) Find his position and velocity at t = 2 s (b) Where is he when his velocity is 25 m/s?

2. Refer text book-example-2.7 You throw a ball vertically upward from the roof of a tall building. The ball leaves your hand at a point even with the roof railing with an upward speed of 15 m/s; the ball is then in free fall. On its way back down, it just misses the railing. Find (a) the ball’s position and velocity 1 s and 4 s after leaving your hand; (b) the ball’s velocity when it is 5 m above the railing; (c) the maximum height reached; (d) the ball’s acceleration when it is at its maximum height.

3. Refer text book-example-2.9 Sally is driving along a straight highway in her 1965 Mustang. At t = 0, when she is moving at 10 m/s in the positive x-direction, she passes a signpost at x = 50 m. Her x-acceleration as a function of time is 𝑎𝑥 = 2 𝑚 ⁄𝑠2 − (0.1 𝑚⁄ 𝑠 3)𝑡 (a) Find her x-velocity 𝑣𝑥 and position x as functions of time. (b) When is her x-velocity greatest? (c) What is that maximum x-velocity? (d) Where is the car when it reaches that maximum x-velocity?

Assignment problem: 1.

Refer text book- Exercise -2.10 An electric train leaves a station starting from rest and attains a speed of 72 km/h in 10 s. It travels at that speed for 120 s. Then it undergoes uniform retardation for 20 s to come to a halt at the next station. Calculate: i) the distance between the two stations and ii) the average velocity of the train.

2.

Refer text book- Exercise -2.18 The position of the front bumper of a test car under microprocessor control is given by 𝑥(𝑡) = 2.17 𝑚 + (4.8 𝑚⁄ 𝑠2)𝑡 2 − (0.1 𝑚⁄ 𝑠6)𝑡 6 . (a) Find its position and acceleration at the instants when the car has zero velocity. (b) Draw x-t, 𝑣𝑥 ~𝑡 and 𝑎𝑥 ~𝑡 graphs for the motion of the bumper between and t = 0 and t = 2s.

3.

Refer text book- Exercise -2.38 A stone is thrown up vertically with a velocity of 20 m/s. Find out the instances at which the magnitudes of its, (i) momentum and (ii) kinetic energy will be half its initial value (g = 9.8 m/s2).

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Chapter-3

Motion in 2-D and 3-D Test your Understanding (TYU)

1. Test Your Understanding of Section 3.1: In which of these situations would the average velocity vector 𝑣𝑎𝑣 over an interval be equal to the instantaneous velocity 𝑣 at the end of the interval? (i) a body moving along a curved path at constant speed; (ii) a body moving along a curved path and speeding up; (iii) a body moving along a straight line at constant speed; (iv) a body moving along a straight line and speeding up. 2. Test Your Understanding of Section 3.2: A sled travels over the crest of a snow-covered hill. The sled slows down as it climbs up one side of the hill and gains speed as it descends on the other side. Which of the vectors (1 through 9) in the figure correctly shows the direction of the sled’s acceleration at the crest? (Choice 9 is that the acceleration is zero.)

3. Test Your Understanding of Section 3.3: In Example 3.10, suppose the tranquilizer dart as a relatively low muzzle velocity so that the dart reaches a maximum height at a point P before striking the monkey, as shown in the figure. When the dart is at point P, will the monkey be (i) at point A (higher than P), (ii) at point B (at the same height as P), or (iii) at point C (lower than P)? Ignore air resistance.

4. Test Your Understanding of Section 3.4: Suppose that the particle in Fig. 3.30 experiences four times the acceleration at the bottom of the loop as it does at the top of the loop. Compared to its speed at the top of the loop, is its speed at the bottom of the loop (i) √2 times as great; (ii) 2 times as great; (iii) 2√2 times as great; (iv) 4 times as great; or (v) 16 times as great? 5. Test Your Understanding of Section 3.5: Suppose the nose of an airplane is pointed due east and the airplane has an air speed of 150 km/h. Due to the wind, the airplane is moving due north relative to the ground and its speed relative to the ground is 150 km/s. What is the velocity of the air relative to the earth? (i) 150 km/h from east to west; (ii) 150 km/h from south to north; (iii) 150 km/h from southeast to northwest; (iv) 212 km/h from east to west; (v) 212 km/h from south to north; (vi) 212 km/s from southeast to northwest; (vii) there is no possible wind velocity that could cause this.

In-class problem 1.

Refer text book-Examle-3.1: A robotic vehicle, or rover, is exploring the surface of Mars. The stationary Mars lander is the origin of coordinates, and the surrounding Martian surface lies in the xy-plane. The rover, which we represent as a point, has x- and ycoordinates that vary with time: 𝑥 = 2.0 𝑚 − (0.25 𝑚⁄ 𝑠 2)𝑡 2 𝑦 = (1.0 𝑚/𝑠)𝑡 + (0.025 𝑚⁄ 𝑠3)𝑡 3 (a) Find the rover’s coordinates and distance from the lander at t = 2 s (b) Find the rover’s displacement and average velocity vectors for the interval t = 0 s to t =2s (c) Find a general expression for the rover’s instantaneous velocity vector 𝑣. Express 𝑣 at t = 2s in component form and in terms of magnitude and direction.

2.

Refer text book-Examle-3.7: A batter hits a baseball so that it leaves the bat at speed v0 = 37 m/s at an angle 53.10. (a) Find the position of the ball and its velocity (magnitude and direction) at t = 2 s (b) Find the time when the ball reaches the highest point of its flight, and its height h at this time. (c) Find the horizontal range R—that is, the horizontal distance from the starting point to where the ball hits the ground.

3.

Refer text book-Examle-3.12: Passengers on a carnival ride move at constant speed in a horizontal circle of radius 5.0 m, making a complete circle in 4.0 s. What is their acceleration?

4.

Refer text book-Examle-3.14: An airplane’s compass indicates that it is headed due north, and its airspeed indicator shows that it is moving through the air at 240 km/h. If there is a 100 km/h wind from west to east, what is the velocity of the airplane relative to the earth?

Assignment problem 1.

2.

Refer text book-Exercise-3.4 A ladder of 6 m length, which is in contact with a vertical wall and horizontal ground slides down the vertical plane. When the lower end is at a distance of 3 m from the wall, its velocity is 4 m/s. What is the velocity of the upper end at that instant? Refer text book-Exercise-3.9 Two particles are thrown up simultaneously with a velocity of 30 m/s, one thrown vertically and another at 450 with respect to the horizon. Find out the distance between them at t = 1.5s.

3.

Refer text book-Exercise-3.29 A model of a helicopter rotor has four blades, each 3.4 m long from the central shaft to the blade tip. The model is rotated in a wind tunnel at 550 rev/min. (a) What is the linear speed of the blade tip, in m/s? (b) What is the radial acceleration of the blade tip expressed as a multiple of the acceleration of gravity, g?

4.

Refer text book-Exercise-3.36 A canoe has a velocity of 0.4 m/s southeast relative to the earth. The canoe is on a river that is flowing 0.5 m/s east relative to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

Chapter-4

Newton’s laws of Motion

Test your Understanding (TYU)

1.

Test Your Understanding of Section 4.2: In which of the following situations is there zero net force on the body? (i) an airplane flying due north at a steady 120 m/s and at a constant altitude; (ii) a car driving straight up a hill with 30 a slope at a constant 90 km/h (iii) a hawk circling at a constant 20 km/h at a constant height of 15 m above an open field; (iv) a box with slick, frictionless surfaces in the back of a truck as the truck accelerates forward on a level road at 5 m/s2.

2.

Test Your Understanding of Section 4.3: Rank the following situations in order of the magnitude of the object’s acceleration, from lowest to highest. Are there any cases that have the same magnitude of acceleration? (i) a 2.0-kg object acted on by a 2.0-N net force; (ii) a 2.0-kg object acted on by an 8.0-N net force; (iii) an 8.0-kg object acted on by a 2.0N net force; (iv) an 8.0-kg object acted on by a 8.0-N net force.

3.

Test Your Understanding of Section 4.4: Suppose an astronaut landed on a planet where g = 19.6 m/s2. Compared to earth, would it be easier, harder, or just as easy for her to walk around? Would it be easier, harder, or just as easy for her to catch a ball that is moving horizontally at 12 m/s. (Assume that the astronaut’s spacesuit is a lightweight model that doesn’t impede her movements in any way.)

In-class problem 1.

2.

Refer text book-Examle-4.5 A waitress shoves a ketchup bottle with mass 0.45 kg to her right along a smooth, level lunch counter. The bottle leaves her hand moving at 2.8 m/s then slows down as it slides because of a constant horizontal friction force exerted on it by the countertop. It slides for 1.0 m before coming to rest. What are the magnitude and direction of the friction force acting on the bottle? Refer text book-Exercise-4.7 A 2.49 × 104 𝑁 Rolls-Royce Phantom travelling in the +xdirection makes an emergency stop; the x-component of the net force acting on it is −1.83 × 104 𝑁 .What is its acceleration?

Assignment problem

1.

Refer text book-Exercise-4.12 A crate with mass 32.5 kg initially at rest on a warehouse floor is acted on by a net horizontal force of 140 N. (a) What acceleration is produced? (b) How far does the crate travel in 10.0 s? (c) What is its speed at the end of 10.0 s?

2.

Refer text book-Exercise-4.24 The upward normal force exerted by the floor is 620 N on an elevator passenger who weighs 650 N. What are the reaction forces to these two forces? Is the passenger accelerating? If so, what are the magnitude and direction of the acceleration?

3.

Refer text book-Exercise-4.30 A .22 rifle bullet, travelling at 350 m/s strikes a large tree, which it penetrates to a depth of 0.130 m. The mass of the bullet is 1.80 g. Assume a constant retarding force. (a) How much time is required for the bullet to stop? (b) What force, in newtons, does the tree exert on the bullet?

Chapter-5 Applying Newton’s laws

Test your Understanding (TYU) 1. Test Your Understanding of Section 5.1: A traffic light of weight w hangs from two lightweight cables, one on each side of the light. Each cable hangs at a 45° angle from the horizontal. What is the tension in each cable? (i) w/2 (ii) 𝑤 ⁄√2 ; (iii) w; (iv) 𝑤√2 ; (v) 2w 2. Test Your Understanding of Section 5.3: Consider a box that is placed on different surfaces. (a) In which situation(s) is there no friction force acting on the box? (b) In which situation(s) is there a static friction force acting on the box? (c) In which situation(s) is there a kinetic friction force on the box? (i) The box is at rest on a rough horizontal surface. (ii) The box is at rest on a rough tilted surface. (iii) The box is on the rough-surfaced flat bed of a truck—the truck is moving at a constant velocity on a straight, level road, and the box remains in the same place in the middle of the truck bed. (iv) The box is on the rough-surfaced flat bed of a truck—the truck is speeding up on a straight, level road, and the box remains in the same place in the middle of the truck bed. (v) The box is on the rough-surfaced flat bed of a truck—the truck is climbing

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a hill, and the box is sliding toward the back of the truck 4. Test Your Understanding of Section 5.4: Satellites are held in orbit by the force of our planet’s gravitational attraction. A satellite in a small-radius orbit moves at a higher speed than a satellite in an orbit of large radius. Based on this information, what you can conclude about the earth’s gravitational attraction for the satellite? (i) It increases with increasing distance from the earth. (ii) It is the same at all distances from the earth. (iii) It decreases with increasing distance from the earth. (iv) This information by itself isn’t enough to answer the question.

In-class problem

1.

Refer text book-Examle-5.1 A gymnast with mass mG = 50kg suspends herself from the lower end of a hanging rope of negligible mass. The upper end of the rope is attached to the gymnasium ceiling. (a) What is the gymnast’s weight? (b) What force (magnitude and direction) does the rope exert on her? (c) What is the tension at the top of the rope?

2.

Refer text book-Examle-5.8 An elevator and its load have a combined mass of 800 kg (Fig.). The elevator is initially moving downward at 10 m/s, it slows to a stop with constant acceleration in a distance of 25.0 m. What is the tension T in the supporting cable while the elevator is being brought to rest?

3.

Refer text book-Examle-5.13: You want to move a 500-N crate across a level floor. To start the crate moving, you have to pull with a 230-N horizontal force. Once the crate “breaks loose” and starts to move, you can keep it moving at constant velocity with only 200 N. What are the coefficients of static and kinetic friction?

4.

Refer text book-Examle-5.16: Let’s go back to the toboggan we studied in Example 5.10 (Question-5.10 A toboggan loaded with students (total weight w) slides down a snowcovered slope. The hill slopes at a constant angle α, and the toboggan is so well waxed that there is virtually no friction. What is its acceleration?). The wax has worn off, so there is now a nonzero coefficient of kinetic friction 𝜇𝑘 . The slope has just the right angle to make the toboggan slide with constant velocity. Find this angle in terms of w and 𝜇𝑘 Refer text book-Examle-5.21 The sports car is rounding a flat, unbanked curve with radius R. If the coefficient of static friction between tires and road is 𝜇𝑠 , what is the maximum speed 𝑣𝑚𝑎𝑥 at which the driver can take the curve without sliding?

5.

6.

Refer text book-Examle-5.22 For a car travelling at a certain speed, it is possible to bank a curve at just the right angle so that no friction at all is needed to maintain the car’s turning radius. Then a car can safely round the curve even on wet ice. (Bobsled racing depends on this same idea.) Your engineering firm plans to rebuild the curve so that a car moving at a chosen speed can safely make the turn even with no friction. At what angle 𝝱 should the curve be banked?

7.

Refer text book-Examle-5.23 A passenger on a carnival Ferris wheel moves in a vertical circle of radius R with constant speed v. The seat remains upright during the motion. Find expressions for the force the seat exerts on the passenger at the top of the circle and at the bottom.

Assignment problem

1. Refer text book-Exercise-5.10 In the given Fig. the weight w is 60.0 N. (a) What is the tension in the diagonal string? (b) Find the magnitudes of the horizontal forces 𝐹󰇍󰇍󰇍 1 󰇍 󰇍2 that must be applied to hold the system in the position shown. and 𝐹

2. Refer text book-Exercise-5.25 A person of mass 60 kg who is jumping from a height of 4 m comes to rest in 0.12 s. What will be the force acting on him that has took 6 s to comes to rest (g = 9.8 m/s2). 3. Refer text book-Exercise-5.28 A stockroom worker pushes a box with mass 11.2 kg on a horizontal surface with a constant speed of 3.5 m/s. The coefficient of kinetic friction between the box and the surface is 0.20. (a) What horizontal force must the worker apply to maintain the motion? (b) If the force calculated in part (a) is removed, how far does the box slide before coming to rest?

Chapter-6

Work and Energy Test your Understanding (TYU)

1.

Test Your Understanding of Section 6.1: In An electron moves in a straight line toward the east with a constant speed of 8 x 107 m/s. It has electric, magnetic, and gravitational forces acting on it. During a 1-m displacement, the total work done on the electron is (i) positive; (ii) negative; (iii) zero; (iv) not enough information given to decide. [Ans- (iii)]

2.

Test Your Understanding of Section 6.2: Rank the following bodies in order of their kinetic e...