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Vectors - Enrichment

Vectors Problem 1

physics.fisikastudycenter.com, high school vectors tutorial. In this tutorial we'll see how to find the angle between two vectors.

Question
Given two vectors F1 and F2 that are forming an angle (θ). If F1 = F2 = R, where R is the resultant of the two vectors, find the magnitude of θ !
(Dua buah vektor yang besarnya F1 dan F2 memiliki titik tangkap sama. Jika F1 = F2 = R (dengan R adalah resultan kedua vektor tersebut), tentukan besarnya sudut apit antara dua vektor tersebut !)
a. 30°
b. 45°
c. 60°
d. 90°
e. 120°


Discussion
Take an attention that F1 = F2 = R, so we'll just use the magnitude of the three vectors as R. By applying the two vectors addition formula then we get



As we know that Arc. cos − 0.5 is 120°

prepd.by : Fisikastudycenter.com
Sources/Literatures
- Vektor, Fisika 1 untuk SMA / MA Kelas X, Setya Nurachmandani,
BSE Pusat Perbukuan, Departemen Pendidikan Nasional
- Vektor, Fisika untuk SMA dan MA Kelas X,
Sri Handayani, BSE Pusat Perbukuan, Departemen Pendidikan Nasional

 

Vectors Problem 2

physics.fisikastudycenter.com, high school vectors tutorial. In this problem another basic concepts are needed, trigonometric, and equilibrium conditions.

Question
Given two vectors F1 (the magnitude is unknown) and F2 = 7 N in a x-y system as shown in below figure. The resultant vector R = 14 N is in x-positive direction forms an angle β with F1. Find the magnitudes of sin β and cos β!
(Dua buah vektor masing-masing F1 dan F2 = 7 N dalam sistem koordinat x-y seperti tampak pada gambar. Resultan kedua vektor tersebut adalah R = 14 N searah sumbu x positif dengan sudut β terhadap gaya F1. Tentukan nilai dari sin β dan cos β !)


Discussion
This graph illustrates the configuration of the three vectors mentioned above:



An alternative method that we'll try in here is using equilibrium conditions so we put a force R' where R' has the same magnitude with R, but in opposite direction. Take a note if
F1 + F2 = R
so then
(F1 + F2) − R = 0, this is an equilibrium equation and in here − R is actually R'.



And by applying equilibrium conditions after projecting F1 in x and y axis to get its components:

 

we get equilibrium equations




Combining equations (1) and (2)





Using trigonometric concepts then we get:





prepd.by : Fisikastudycenter.com
Sources/Literatures
- Fisika SMA Kelas X, Marthen Kanginan
- BSE Fisika 1 untuk SMA / MA Kelas X, Setya Nurachmandani, Pusat Perbukuan, Departemen Pendidikan Nasional
- Fisika untuk SMA dan MA Kelas X,
Sri Handayani, Pusat Perbukuan, Departemen Pendidikan Nasional

Vectors Problem 3

physics.fisikastudycenter.com, high school vectors tutorial. In this discussion, we'll learn adding vectors by components using sketch and adding vectors geometrically.

Question
A car is driven east for a distance of 50 km, then north for 30 km, and then in a direction 30° east of north for 25 km. Determine the magnitude and the angle of the car's total displacement from its starting point!
(Sebuah mobil menempuh jarak 50 km ke arah timur kemudian membelok ke utara sejauh 30 km kemudian dilanjutkan 30° ke arah timur laut. Tentukan besar dan arah perpindahan mobil dari titik berangkatnya!)

Discussion
First thing that we've to do is to review compass directions. Learn the fig below:


Picture : wikipedia.org


Sketch the vector diagram of car movements:



Sum the three individual displacement vectors:
dnet = d1 + d2 + d3
Sum for the x components:
dnet, x = d1x + d2x + d3x

d1x = 50 km
d2x = 0
d3x = (25 km) cos 30° = 21.65 km

dnet, x = 50 km + 0 + 21.65 km = 71.65 km

Sum for the y components:
dnet, y = d1y + d2y + d3y

d1y = 0
d2y = 30 km
d3y = (25 km) sin 30° = 12.50 km

dnet, y = 0 + 12.50 km + 30 km = 42.50 km

To find the magnitude and angle of dnet :



The angle θ is measured from the positive direction of x

prepared.by : Fisikastudycenter.com
Sources/Literatures
- Fundamentals of Physics, Halliday and Resnick
- www.wikipedia.org

 

Vectors Problem 4

physics.fisikastudycenter.com, high school vectors tutorial. The solution of this problem shows how to use dot product / scalar product to find work done by a force in unit-vector notation.

Question
A force F = (2i + 3j) does work with it's displacement vector is R = (4i + aj) m where i and j are the unit-vectors in x and y cartesian respectively. The work done is 26 J, then the value of a equals ....
(Sebuah gaya F = (2i + 3j) N melakukan usaha dengan titik tangkapnya berpindah menurut R = (4i + aj) m dan vektor i dan j berturut-turut adalah vektor satuan yang searah dengan sumbu x dan sumbu y pada koordinat kartesian. Bila usaha itu bernilai 26 J, maka nilai a sama dengan .....)
a. 5
b. 6
c. 7
d. 8
e. 12
Question source: UMPTN 1991


Discussion
Work (W) is resulted from dot product or scalar product between the vector F (force) and vector d (displacement). See the folowing equation below:

W = F⋅d = F d cos θ

Remember that in dot product
a⋅b = a b cos θ

 

and can be expressed in unit-vector notation (two dimensions) as
a⋅b = (axi + ayj)⋅(bxi + byj)
a⋅b = axbx + ayby

So then :
W = F⋅d = (2i + 3j)⋅(4i + aj)
26 = 8 + 3a
3a = 26-8 = 18
a = 18/3 = 6

prepd.by : Fisikastudycenter.com
Sources/Literatures
- Fundamentals of Physics, Halliday and Resnick
- Seribu Pena Fisika Kelas 10 SMA, Marthen K

Vectors Problem 5

physics.fisikastudycenter.com, high school vectors tutorial. This problem shows how to find the resultant of three vectors using analityc method.

Question
Given three forces as following figure, find the magnitude of resultant!
(Tentukan resultan dari ketiga gaya pada gambar berikut!)



Discussion
Sketch and project F3 on x and y axis to get its components



Components in x axis :

Σ Fx = F1 − F3 cos 45°
= 30√2 − 150 cos 45°
= 30√2 − 150 (1/2 √2)
= −45√2 Newton

Components in y axis :
Σ Fy = F3 sin 45° − F2
= 150 sin 45° − 30√2
= 75√2 − 30√2
= 45√2 Newton

The resultant:
R = √[(Σ Fx)2 + (Σ Fy)2]
R = √[(−45√2 )2 + (45√2 )2]
R = 90 Newton

 

prepd.by : Fisikastudycenter.com
Literatures
- EBTANAS 1994 Questions

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