Unit, Quantities and Vector | Bài tập môn Vật lý đại cương 1 CTTT | Trường Đại học Bách Khoa Hà Nội

02/07/2021
Dot and Cross Product of vectors
Chapter 1. Unit, Quantities and Vector 1 2
1.69(73). A spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45° east Words for direction:
of south, and then 280 m at 30° east of north. After a fourth unmeasured displacement she finds herself back where
- Take the east direction to be x- direction and the north direction to y N
she started. Use the method of components to determine the magnitude and direction of the fourth displacement. be the y – direction
Draw the vector addition diagram and show that it is in qualitative agreement with your numerical solution.
(Chọn hướng đông làm hướng trục x, hướng bắc làm hướng trục y) A
- Use a coordinate system where east is in the + x-direction and
* Use a coordinate system where east is in the + x-direction and north is in the + y-direction north is in the + y-direction W 25o E N N
- Let +x be east and +y be north. x 55o 𝐵
- The net east displacement: Tổng khoảng cách dịch chuyển theo S D hướng đông W Straight west E W E A C 30o east of north 180 180
In the direction 30o West of South:
In the direction 25o North of East 280 280 A 210 210
In the direction 65o East of North 45o 30o 45o 30o B 55o West of North 45o east of south B 35o North of West S S 3 4 1 02/07/2021
1.69(73). A spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45°
east of south, and then 280 m at 30° east of north. After a fourth unmeasured displacement she finds herself back
1.79. You can go canoeing on a lake. You start at your camp on the shore, travelling 240m in the direction 32.0o south
where she started. Use the method of components to determine the magnitude and direction of the fourth
of east to reach a store to purchase supplies. You know the distance because you have located your both your camp and
displacement. Draw the vector addition diagram and show that it is in qualitative agreement with your numerical
the store in the lake map. On the return trip, you travel distance B in the direction 48.0o north of west, distance C in the solution.
direction 62.0o south of west and then return to your camp. Use vector method to calculate C and B.
* Use a coordinate system where east is in the + x-direction and north is in the + y-direction N
𝐴⃗ + 𝐵 + 𝐶⃗ + 𝐷 = 0 N 𝐴⃗ = −180𝒊 + 0𝒋 𝐴 + 𝐵 + 𝐶 + 𝐷 = 0
𝐵 = (210sin45 )𝒊 − (210cos45 )𝒋 𝐴 + 𝐵 + 𝐶 + 𝐷 = 0 𝐴⃗ + 𝐵 + 𝐶⃗ = 0 D C
𝐶⃗ = (280sin30 )𝒊 + (280cos30 )𝒋 W E A C
𝐴⃗ = 240cos32 𝚤⃗ − 240sin32 𝚥⃗ W 48o B E 180
𝐵 = −𝐵cos48 𝚤⃗ + 𝐵sin48 𝚥⃗ 280 32o 210 62.0o 240
𝐶⃗ = −𝐶cos62 𝚤⃗ − 𝐶sin62 𝚥⃗ 45o 30o B A 𝐷 = 𝐷 + 𝐷 = 144𝑚
⇒ 𝐵 = 255𝑚; 𝐶 = 70𝑚 𝐷 −94.0 S tan𝜃 = = ⇒ 𝜃 = 220.9 𝐷 −108.5 S 5 6
1.77(81). Bones and Muscles. A patient in therapy has a forearm that weighs 20.5 N and that lifts a 112.0-N weight. These
1.59(62). The Hydrogen Maser. You can use the radio waves generated by a hydrogen maser as a standard of frequency. The frequency of these waves is
two forces have direction vertically downward. The only other significant forces on his forearm come from the biceps
1,420,405,751.786 hertz. (A hertz is another name for one cycle per second.) A clock controlled by a hydrogen maser is off by only 1 second in 100,000
muscle (which acts perpendicularly to the forearm) and the force at the elbow. If the biceps produce a pull of 232 N
years. For the following questions, use only three significant figures. (The large number of significant figures given for the frequency simply illustrates
the remarkable accuracy to which it has been measured). (a) What is the time for one cycle of the radio wave? (b) How many cycles occur in 1 h? (c) How
when the forearm is raised 43o above the horizontal, find the magnitude and direction of the force that the elbow exerts on
many cycles would have occurred during the age of the earth, which is estimated to be 4.6x109 years? By how many seconds would a hydrogen maser
the forearm. (The sum of the elbow force and the biceps force must balance the weight of the arm and the weight it is
clock be off after a time interval equal to the age of the earth?
carrying, so their vector sum must be 132.5 N, upward.)
(a) What is the time for one cycle of the radio wave? 1 T = 3s. f. f y 𝐸 𝑔⃗
𝐵 is the force the biceps exerts
(b) How many cycles occur in 1 h? E tan y      270o
𝐸 is the force the elbow exerts t E 4s. f. n = n = 𝐵 𝑅 x = 3s. f. T 3s. f. E + B = R R = 20.5 + 112.0 =132.5 N
(c) How many cycles would have occurred during the age of the earth, which is estimated to be 4.6x109 years? forearm 43o t 𝐸 + 𝐵 = 𝑅 n = 2s. f.× 4s. f. n = = 2s. f. Elbow x T 3s. f. 𝐸 + 𝐵 = 𝑅
(d) How many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth?    𝐸 E  158.2i – 37.2 j tan𝛼 = ⇒ 𝛼 = 270o 𝐸
E = 160 N,  = 30o below horizontal 7 8 2 02/07/2021
1.67(71). You are to program a robotic arm on an assembly line to move in the xy-plane. Its first displacement is A ; its second displacement
1.71. A cross-country skier skis 2.80 km in the direction 45o west of south, then 7.40 km in the direction
is B, of magnitude 6.40 cm and direction 63.0o measured in the sense from the +x-axis toward the -y-axis. The resultant C = A + B of the
30.0o north of east, and finally 3.30 km in the direction 22.0o south of west. (a) Show these displacement
two displacements should also have a magnitude of 6.40 cm, but a direction 22.0o measured in the sense from the + x-axis toward the + y-
in a diagram. (b) How far and in what direction is she from the starting point?
axis. (a) Draw the vector addition diagram for these vectors, roughly to scale. (b) Find the components of A . (c) Find the magnitude and direction of A . N
Cross-country skiers in western Norway. y 3.3 22o B W E 0 A C 7.4 45o 2.8 22o 0 30o 63o x B S The net east displacement is The net north displacement is The distance traveled is 9 10
1.75(79). A ship leaves the island of Guam and sails 285 km at 40.0o north of west. In which direction must it now head and how
1.89(93). A cube is placed so that one corner is at the origin and three edges are along the x-,
far must it sail so that its resultant displacement will be 115 km directly east of Guam?
y-, and z-axes of a coordinate system. Use vectors to compute: (a) the angle between the
edge along the z-axis (line ab) and the diagonal from the origin to the opposite corner
(line ad), and (b) the angle between line ac (the diagonal of a face) and line ad.
Let +x be east and +y be north. N
Let 𝐴⃗ be the displacement 285 km at 40o north of west and let B be the unknown displacement. 285 km B W A 40o E Guam R ; 115 km S 11 12 3 02/07/2021
1.85(89). Given two vectors A = -2.00 𝚤⃗ + 3.00 𝚥⃗ + 4.00 𝑘 and B= 3.00 𝚤⃗ + 1.00 𝚥⃗ - 3.00𝑘, do the
following. (a) Find the magnitude of each vector. (b) Write an expression for the vector difference A -
B, using unit vectors. (c) Find the magnitude of the vector difference A - B. Is this the same as the magnitude of B - A?
1.91(95). You are given vectors A = 5.0 𝚤⃗ - 6.5 𝚥⃗ and B = -3.5 𝚤⃗ + 7.0 𝚥⃗. A third vector C lies in the xy-
plane. Vector C is perpendicular to vector A , and the scalar product of C with B is 15.0. From this
information, find the components of vector C.
We have two equations in two unknowns Cx and Cy. Solving gives Cx =8.0 and Cy = 6.1 13 4