Re: [題目] 簡諧運動
※ 引述《nerv3890 (阿淵)》之銘言：
: [領域] (題目相關領域)
: 簡諧
: [來源] (課本習題、考古題、參考書...)
: 考古
: [題目]
: 3.A pendulum of length L and mass M has a spring of force constant k
: connected to it at a distance h below its point of suspension as shown in a
: figure below. Find the period of vibration of the system if the amplitude of
: vibration is small. Assume the vertical suspension of length L is rigid and
: its mass is negligible.
: _____________________
: | |
: |h |
: | |
: L |__ /\ ____|
: | \/ \/ |
: | k |
: O
: M
: [瓶頸] (寫寫自己的想法，方便大家為你解答)
: 初步構想
: 是設力矩τ=-CΘ
: Θ為擺動角度，C為常數
: 之後就都沒想法了...Orz
: 有人能幫解答嗎
: 感恩
如下圖所示..以懸掛點作為支點..設系統角加速度為α..順時針為正..
http://www.wretch.cc/album/show.php?i=kramnik1&b=34&f=1825180178&p=11
因為"Assume the vertical suspension of length L is rigid"..
根據力矩守恆
M*g*sinθ*L + F(spring)*cosθ*h = -M*L^2*α ..............(a)
因為"the force constant of spring is k"
F(spring) =k*h*sinθ .....................................(b)
由(a)(b)知
M*g*sinθ*L + k*h^2*sinθ*cosθ = -M*L^2*α ..............(c)
因為"the amplitude of vibration is small"
對(c)做θ泰勒展開1階近似..可得
M*g*L*θ + k*h^2*θ = -M*L^2*α
θ = -M*L^2/(M*g*L + k*h^2)*α.............................(d)
(d)式之解為
θ = A^(i*w*t) ,其中 w = M*L^2/(M*g*L + k*h^2)
設The period of vibration of the system為T..則
T = 2*π/w
= 2*π*(M*g*L + k*h^2)/(m*L^2)
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