Ren R

01 Maret 2024 02:59

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Ren R

01 Maret 2024 02:59

Pertanyaan

Two waves that set up a standing wave in a long string are given by the wave function y1=A sin (kx - wt + 𝞱) and y2 = A sin (kx + wt).Determine the position of the nodes of the resulting standing wave. (n = 0, 1, 2, 3, ...)

Two waves that set up a standing wave in a long string are given by the wave function y1=A sin (kx - wt + 𝞱) and y2 = A sin (kx + wt).Determine the position of the nodes of the resulting standing wave. (n = 0, 1, 2, 3, ...)

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Kevin L

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21 Agustus 2024 04:54

Soal ini meminta kita untuk: * Menentukan posisi simpul: Titik-titik pada tali yang selalu diam (amplitudonya nol) dalam gelombang berdiri. * Dari dua persamaan gelombang: y1 = A sin(kx - ωt + θ) dan y2 = A sin(kx + ωt). Konsep Gelombang Berdiri: * Terbentuk dari interferensi dua gelombang berjalan yang memiliki amplitudo, frekuensi, dan panjang gelombang yang sama, tetapi arah rambatnya berlawanan. * Simpul adalah titik-titik di mana amplitudo total gelombang selalu nol. Penyelesaian Soal 1. Menentukan Persamaan Gelombang Berdiri: Gelombang berdiri yang dihasilkan dari interferensi kedua gelombang tersebut dapat dinyatakan sebagai: y = y1 + y2 = A sin(kx - ωt + θ) + A sin(kx + ωt) = 2A cos(θ/2) sin(kx + θ/2) 2. Menentukan Posisi Simpul: Simpul terjadi ketika amplitudo total y sama dengan nol. Artinya: sin(kx + θ/2) = 0 Ini berlaku ketika: kx + θ/2 = nπ di mana n adalah bilangan bulat (0, 1, 2, 3, ...). 3. Mencari Persamaan Posisi Simpul: Dari persamaan di atas, kita dapat mengisolasi x untuk mendapatkan persamaan posisi simpul: x = (nπ - θ/2) / k Jadi, posisi simpul dari gelombang berdiri yang terbentuk adalah: x_n = (nπ - θ/2) / k di mana: * x_n: Posisi simpul ke-n * n: Bilangan bulat (0, 1, 2, 3, ...) * k: Bilangan gelombang * θ: Fase awal gelombang pertama Interpretasi Hasil * Posisi simpul: Bergantung pada nilai n, k, dan θ. * Jarak antara dua simpul berurutan: Selalu konstan dan sama dengan setengah panjang gelombang. * Simpul pertama: Terjadi pada n = 0. * Simpul berikutnya: Dapatkan dengan mensubstitusikan nilai n yang berbeda ke dalam persamaan. Kesimpulan: Persamaan x_n = (nπ - θ/2) / k memberikan kita formula umum untuk mencari posisi simpul dalam gelombang berdiri yang dibentuk oleh dua gelombang sinusoidal. Dengan mengetahui nilai-nilai konstanta yang terlibat, kita dapat menentukan posisi semua simpul pada tali. Catatan: * Nilai θ: Jika θ tidak diketahui, biasanya dianggap nol untuk menyederhanakan perhitungan. * Bilangan gelombang k: Berhubungan dengan panjang gelombang (λ) melalui persamaan k = 2π/λ.


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