Một sợi dây đàn hồi dài 30 cm có hai đầu cố định. Trên dây đang có sóng dừng. Biết sóng truyền trên dây với bước sóng 20 cm và biên độ dao động của điểm bụng là 2 cm. Số điểm trên dây mà phần tử tại đó dao động với biên độ 6 mm là bao nhiêu?
Đáp án:
Một sợi dây đàn hồi dài 30 cm có hai đầu cố định. Trên dây đang có sóng dừng. Biết sóng truyền trên dây với bước sóng 20 cm và biên độ dao động của điểm bụng là 2 cm. Số điểm trên dây mà phần tử tại đó dao động với biên độ 6 mm là bao nhiêu?
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Hãy suy nghĩ và trả lời câu hỏi trước khi xem đáp án
Bước sóng $\lambda = 20$ cm, chiều dài dây $L = 30$ cm.
Số bụng sóng: $k = \frac{2L}{\lambda} = \frac{2*30}{20} = 3$ bụng sóng (vì hai đầu cố định).
Biên độ tại bụng sóng: $A = 2$ cm.
Ta có biên độ dao động của một điểm trên dây là $A' = A \left| \sin(\frac{2\pi x}{\lambda}) \right|$, với $x$ là khoảng cách từ điểm đó đến nút sóng gần nhất.
Để $A' = 0.6$ cm, ta có:
$0.6 = 2 \left| \sin(\frac{2\pi x}{20}) \right|$
$\left| \sin(\frac{\pi x}{10}) \right| = 0.3$
$\frac{\pi x}{10} = \arcsin(0.3)$ hoặc $\frac{\pi x}{10} = \pi - \arcsin(0.3)$
$x = \frac{10}{\pi} \arcsin(0.3)$ hoặc $x = 10 - \frac{10}{\pi} \arcsin(0.3)$
Mỗi bó sóng có 2 vị trí có cùng biên độ. Do có 3 bó sóng, và mỗi bó sóng có 2 điểm thỏa mãn nên sẽ có $3 \times 2 = 6$ điểm.
Tuy nhiên, ta cần xem xét các bó sóng đối xứng qua nút. Trong mỗi bó sóng, có hai điểm có cùng biên độ 0.6 cm. Vì có 3 bó sóng nên có $2 \times 3 = 6$ điểm trong 3 bó sóng. Nhưng do tính đối xứng qua các nút nên mỗi khoảng giữa 2 nút liền kề (1 bó sóng) sẽ có 2 điểm thỏa mãn. Do đó, trên toàn sợi dây có $2 imes 4 = 8$ điểm thỏa mãn. (bốn khoảng $\lambda/2$).
Số điểm trên dây dao động với biên độ 0.6 cm là 8.
Câu hỏi liên quan
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