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Giải bài tập 2.9 trang 59 SGK Toán 12 tập 1 - Kết nối tri thức

Ba sợi dây không giãn với khối lượng không đáng kể được buộc chung một đầu và được kéo căng về ba hướng khác nhau (H.2.31). Nếu các lực kéo làm cho ba sợi dây ở trạng thái đứng yên thì khi đó ba sợi dây nằm trên cùng một mặt phẳng. Hãy giải thích vì sao.

Tổng hợp đề thi học kì 2 lớp 12 tất cả𒀰 các môn - Kết nối tri thức

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

Ba sợi dây không giãn với khối lượng không đáng kể được buộc chung một đầu và được kéo căng về ba hướng khác nhau (H.2.31). Nếu các lực kéo làm cho ba sợi dây ở trạng thái đứng yên thì khi đó ba sợi dây nằm trên cùng một mặt phẳng. Hãy giải thích vì sao.

Phương pháp giải - Xem chi tiết

Sử dụng quy tắc hình bình hành để giải thích: Nếu ABCD là hình bình hành thì \(\overrightarrow {AB}  + \overrightarrow {AD}  = \overrightarrow {AC} \)

Lời giải chi tiết

Biểu diễn lực các lực kéo của ba sợi dây bằng các vectơ, đặt tên các vectơ như hình vẽ:

 

Lấy điểm D sao cho tứ giác DCAE là hình bình hành (điểm D nằm khác phía với điểm B).

 

Do đó, giá của các vectơ \(\overrightarrow {AC} \) và \(\overrightarrow {AE} \) cùng nằm trên mặt phẳng (ACDE). (1) Vì DCAE là hình bình hành nên \(\overrightarrow {AC}  + \overrightarrow {AE}  = \overrightarrow {AD} \) (quy tắc hình bình hành) Vì các lực kéo làm cho ba sợi dây ở trạng thái đứng yên nên \(\overrightarrow {AD}  =  - \overrightarrow {AB} \), do đó hai vectơ \(\overrightarrow {AD} \) và \(\overrightarrow {AB} \) có giá cùng nằm trên một mặt phẳng (ACDE). (2) Từ (1) và (2) suy ra ba vectơ \(\overrightarrow {AC} \), \(\overrightarrow {AE} \) và \(\overrightarrow {AB} \) có giá cùng nằm trên mặt phẳng (ACDE). Vậy khi các lực kéo làm cho ba sợi dây ở trạng thái đứng yên thì khi đó ba sợi dây nằm trên cùng một mặt phẳng

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