Resultan dari penjumlahan dua buah vektor yang sama besarnya 3 satuan. Jika kedua vektor tersebut membentuk sudut , nilai masing-masing vektor tersebut adalah ….

Pertanyaan

Resultan dari penjumlahan dua buah vektor yang sama besarnya 3 satuan. Jika kedua vektor tersebut membentuk sudut $60 to the power of ring operator$ , nilai masing-masing vektor tersebut adalah …. $space space$

5 satuan
4 satuan
3 satuan
2 satuan
1 satuan

Pembahasan Video:

Pembahasan Soal:

$open vertical bar straight a with rightwards harpoon with barb upwards on top plus straight b with rightwards harpoon with barb upwards on top close vertical bar equals square root of open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar squared plus vertical line straight b with rightwards harpoon with barb upwards on top vertical line squared plus 2 open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar open vertical bar straight b with rightwards harpoon with barb upwards on top close vertical bar cos open parentheses straight theta close parentheses end root 3 equals square root of open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar squared plus open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar squared plus 2 open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar squared cos open parentheses 60 to the power of straight o close parentheses end root 9 equals open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar squared plus open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar squared plus 2 open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar squared left parenthesis 1 half right parenthesis 9 equals 3 open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar squared open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar squared equals 3 open vertical bar straight a with rightwards harpoon with barb upwards on top close vertical bar equals square root of 3$

Jadi, vektor tersebut masing-masing besarnya adalah $square root of 3$ satuan

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Puspita

Terakhir diupdate 30 Agustus 2021

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Sebuah balok ditarik oleh dua gaya seperti pada gambar berikut. Tentukan resultan gaya tersebut!

Pembahasan Soal:

Resultan gaya vektor dihitung menggunakan metode analitis adalah:

Hitung dahulu resultan gaya komponen x

$table attributes columnalign right center left columnspacing 0px end attributes row cell begin inline style sum for blank of end style F subscript x end cell equals cell F subscript 1 times cos space 120 degree plus F subscript 2 end cell row blank equals cell 100 times open parentheses negative 1 half close parentheses plus 100 end cell row blank equals cell 50 space straight N end cell end table$

Hitung resultan gaya komponen y

$table attributes columnalign right center left columnspacing 0px end attributes row cell begin inline style sum for blank of end style F subscript y end cell equals cell space F subscript space 1 end subscript sin space 30 degree plus F subscript 2 end cell row blank equals cell 100 times 1 half plus 0 end cell row blank equals cell 50 space straight N end cell end table$

Resultan gaya nya adalah:

$table attributes columnalign right center left columnspacing 0px end attributes row cell begin inline style sum for blank of end style F end cell equals cell square root of open parentheses 50 close parentheses squared plus open parentheses 50 close parentheses squared end root end cell row blank equals cell square root of 2500 plus 2500 end root end cell row blank equals cell 70 comma 7 space straight N end cell end table$

Sehingga besar resultan gaya nya dalah 70,7 N

Roboguru

Dua gaya masing-masing memiliki besar 30 N dan 40 N. Berapa resultan gaya jika kedua gaya saling tegak lurus. . .

Pembahasan Soal:

Diketahui :

$F_{1} = 30 N F_{2} = 40 N$

Ditanya : resultan gaya?

Penyelesaian :

Untuk menentukan resultan vektor yang saling tegak lurus dapat menggunakan persamaan berikut

$F_{R} = F_{1}^{2} + F_{2}^{2} F_{R} = 3 0^{2} + 4 0^{2} F_{R} = 50 N$

Dengan demikian, besar resultan vektor adalah 50 N.

Roboguru

Jika diketahui vektor-vektor , , dan . Jika vektor tegak lurus terhadap vektor b dan vektor c tegak lurus terhadap vektor d, maka - d adalah ....

Pembahasan Soal:

Diketahui :

$a with rightwards arrow on top equals 2 i with hat on top minus 4 j with hat on top plus k with hat on top$
$b with rightwards arrow on top equals x i with hat on top minus z j with hat on top plus 4 k with hat on top$
$c with rightwards arrow on top equals 5 i with hat on top minus 3 j with hat on top plus 2 k with hat on top$
$d with rightwards arrow on top equals 2 i with hat on top minus z j with hat on top plus x k with hat on top$
$a perpendicular b c perpendicular d$

Ditanya : $a$ - d

Jawab :

Dua buah vektor dikatakan saling tegak lurus jika hasil perkalian kedua vektor adalah 0. Pada soal di atas, nilai variabel x dan z belum diketahui sehingga yang harus kita lakukan adalah mencari nilai x dan nilai z.

Saat $a perpendicular b$ , maka
$table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top end cell equals cell 2 i with hat on top minus 4 j with hat on top plus k with hat on top equals open parentheses 2 comma negative 4 comma 1 close parentheses end cell row cell b with rightwards arrow on top end cell equals cell x i with hat on top minus z j with hat on top plus 4 k with hat on top equals open parentheses x comma negative z comma 4 close parentheses end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top times b with rightwards arrow on top end cell equals 0 row cell open parentheses 2 comma negative 4 comma 1 close parentheses times open parentheses x comma negative z comma 4 close parentheses end cell equals 0 row cell 2 x plus 4 z plus 4 end cell equals 0 end table$

Dari perhitungan di atas, kita jadikan $table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x plus 4 z plus 4 end cell equals 0 end table$ menjadi persamaan 1.

Saat $c perpendicular d$ , maka
$table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top end cell equals cell 5 i with hat on top minus 3 j with hat on top plus 2 k with hat on top equals open parentheses 5 comma negative 3 comma 2 close parentheses end cell row cell d with rightwards arrow on top end cell equals cell 2 i with hat on top minus z j with hat on top plus x k with hat on top equals open parentheses 2 comma negative z comma x close parentheses end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell c with rightwards arrow on top times d with rightwards arrow on top end cell equals 0 row cell open parentheses 5 comma negative 3 comma 2 close parentheses times open parentheses 2 comma negative z comma x close parentheses end cell equals 0 row cell 10 plus 3 z plus 2 x end cell equals 0 end table$

Dari perhitungan di atas, kita jadikan $table attributes columnalign right center left columnspacing 0px end attributes row cell 10 plus 3 z plus 2 x end cell equals 0 end table$ menjadi persamaan 2.

Dari kedua persamaan di atas, dapat kita eliminasi menjadi

$table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x plus 4 z plus 4 end cell equals cell 0 space dikurangi space 10 plus 3 z plus 2 x equals 0 end cell row blank blank cell stack attributes charalign center stackalign right end attributes row 2 x plus 4 z plus 4 equals 0 end row row minus 2 x minus 3 z minus 10 equals 0 end row horizontal line row z minus 6 equals 0 end row row z equals 6 end row end stack end cell end table$

saat z = 6 maka nilai x adalah

$table attributes columnalign right center left columnspacing 0px end attributes row cell 10 plus 3 z plus 2 x end cell equals 0 row cell 10 plus 3 times left parenthesis 6 right parenthesis plus 2 x end cell equals 0 row cell 10 plus 18 plus 2 x end cell equals 0 row 28 equals cell negative 2 x end cell row x equals cell negative 14 end cell end table$

Berdasarkan perhitungan di atas, diperoleh nilai $x equals negative 14$ dan $z equals 6$ , sehingga hasil $a$ - d adalah

$table attributes columnalign right center left columnspacing 0px end attributes row cell a with rightwards arrow on top minus d with rightwards arrow on top end cell equals cell open parentheses 2 i with hat on top minus 4 j with hat on top plus k with hat on top close parentheses minus open parentheses 2 i with hat on top minus z j with hat on top plus x k with hat on top close parentheses end cell row blank equals cell open parentheses 2 i with hat on top minus 4 j with hat on top plus k with hat on top close parentheses minus open parentheses 2 i with hat on top minus 6 j with hat on top plus open parentheses negative 14 close parentheses k with hat on top close parentheses end cell row blank equals cell 2 i with hat on top minus 4 j with hat on top plus k with hat on top minus 2 i with hat on top plus 6 j with hat on top plus 14 k with hat on top end cell row blank equals cell 2 j with hat on top plus 15 k with hat on top end cell end table$

Dengan demikian nilai $a$ - d adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 2 j with hat on top plus 15 k with hat on top end cell end table$ .

Sepertinya terdapat kesalahan pada pilihan jawaban, karena jawaban yang benar adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 2 j with hat on top plus 15 k with hat on top end cell end table$ .

Roboguru

Jika terdapat 3 buah vektor , dan seperti gambar berikut. Maka, resutan ketiga vektor adalah ....

Pembahasan Soal:

Diketahui :

Ditanya : R

Jawab :

Untuk menyelesaikan soal di atas perlu digunakan metode analisis.

Gambar di atas, menunjukkan penguraian gaya pada masing-masing sumbu.

Gaya F₁
$table attributes columnalign right center left columnspacing 0px end attributes row cell F subscript 1 end cell equals cell 10 space straight N end cell row cell F subscript 1 x end subscript end cell equals cell F subscript 1 times cos space 37 end cell row blank equals cell 10 cross times 0 comma 8 end cell row blank equals cell 8 space straight N end cell row cell F subscript 1 y end subscript end cell equals cell F subscript 1 times sin space 37 end cell row blank equals cell 10 cross times 0 comma 6 end cell row blank equals cell 6 space straight N end cell end table$
Gaya F₂
$table attributes columnalign right center left columnspacing 0px end attributes row cell F subscript 2 end cell equals cell space 25 space straight N end cell row cell F subscript 2 x end subscript end cell equals 0 row cell F subscript 2 y end subscript end cell equals cell F 2 end cell row blank equals cell 25 space straight N end cell end table$
Gaya F₃
$table attributes columnalign right center left columnspacing 0px end attributes row cell F subscript 3 end cell equals cell 15 space straight N end cell row cell F subscript 3 x end subscript end cell equals cell F subscript 3 times cos space 53 end cell row blank equals cell 15 cross times 0 comma 6 end cell row blank equals cell 9 space straight N end cell row cell F subscript 3 y end subscript end cell equals cell F subscript 3 times sin space 53 end cell row blank equals cell 15 cross times 0 comma 8 end cell row blank equals cell 12 space straight N end cell end table$

Berdasarkan gambar di atas, kita harus mengetahui dulu besar vektor searah sumbu x dan y, yaitu F_x dan Fy. Ingat bahwa arah vektor yang ke atas dan ke kiri dari kita anggap negatif. Sehingga:

$table attributes columnalign right center left columnspacing 0px end attributes row cell F subscript x end cell equals cell F subscript 1 x end subscript plus F subscript 2 x end subscript plus F subscript 3 x end subscript end cell row blank equals cell 8 plus 0 plus open parentheses negative 9 close parentheses end cell row blank equals cell negative 1 space straight N end cell row cell F subscript y end cell equals cell F subscript 1 y end subscript plus F subscript 2 y end subscript plus F subscript 3 y end subscript end cell row blank equals cell open parentheses negative 6 close parentheses plus 25 plus open parentheses negative 12 close parentheses end cell row blank equals cell 7 space straight N end cell end table$

Selanjutnya untuk mencari nilai resultan kedua vektor gaya bisa dicari dengan rumus Phytagoras, yaitu :

$table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar F close vertical bar end cell equals cell square root of F subscript x squared plus F subscript y squared end root end cell row blank equals cell square root of open parentheses negative 1 close parentheses squared plus 7 squared end root end cell row blank equals cell square root of 1 plus 49 end root end cell row blank equals cell square root of 50 space straight N end cell end table$

Berdasarkan perhitungan di atas, maka besar resultan vektor adalah $square root of 50$ N.

Jadi, jawaban yang tepat adalah A.

Roboguru

Sebuah perahu menyebrangi sungai yang memiliki lebar 250 m dengan kecepatan alirannya 3 m/s. Perahu bergerak dengan arah menyilang dengan kecepatan 2 m/s. Lintasan yang ditempuh perahu sepanjang...

Pembahasan Soal:

Kecepatan aliran air $begin mathsize 14px style v subscript 1 equals 3 space bevelled straight m over straight s end style$

Kecepatan menyilang $begin mathsize 14px style v subscript 2 space equals space 2 space bevelled m over s end style$

Lebar sungai = 250 m

karena arah kedua gerak tegak lurus, maka kecepatan akhir adalah:

$begin mathsize 14px style v equals square root of left parenthesis v subscript 1 squared plus v subscript 2 squared right parenthesis end root v equals 3 comma 6 space bevelled straight m over straight s end style$

panjang lintasan:

$begin mathsize 14px style s equals left parenthesis L over v subscript 1 right parenthesis space v s equals 300 space straight m end style$

Jadi, jawaban yang tepat adalah B.

Roboguru