题目内容
如图,两个圆形飞轮通过皮带传动,大飞轮O1的半径为2r(r为常数),小飞轮O2的半径为r,O1O2=4r.在大飞轮的边缘上有两个点A,B,满足∠BO1A=,在小飞轮的边缘上有点C.设大飞轮逆时针旋转,传动开始时,点B,C在水平直线O1O2上.
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(1)求点A到达最高点时A,C间的距离;
(2)求点B,C在传动过程中高度差的最大值.
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(1)求点A到达最高点时A,C间的距离;
(2)求点B,C在传动过程中高度差的最大值.
(1)
·r. (2)
r.
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解:(1)以O1为坐标系的原点,O1O2所在直线为x轴,建立如图所示的直角坐标系.当点A到达最高点时,点A绕O1转过
,则点C绕O2转过
.
此时A(0,2r),C(
r,
r).
∴AC=
=
·r.
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(2)由题意,设大飞轮转过的角度为θ,
则小飞轮转过的角度为2θ,其中θ∈[0,2π].
此时B(2rcos θ,2rsin θ),C(4r+rcos 2θ,rsin 2θ).
记点B,C的高度差为d,则d=|2rsin θ-rsin 2θ|,
即d=2r|sin θ-sin θcos θ|.
设f(θ)=sin θ-sin θcos θ,θ∈[0,2π],
则f′(θ)=(1-cos θ)(2cos θ+1).
令f′(θ)=(1-cos θ)(2cos θ+1)=0,得cos θ=-
或1,则θ=
,
,0或2π.
f(θ)和f′(θ)随θ的变化情况如下表:
∴当θ=
时,f(θ)取得极大值
;当θ=
时,f(θ)取得极小值-
.
综上所述,点B,C在传动过程中高度差的最大值dmax=
r.
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此时A(0,2r),C(
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∴AC=
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(2)由题意,设大飞轮转过的角度为θ,
则小飞轮转过的角度为2θ,其中θ∈[0,2π].
此时B(2rcos θ,2rsin θ),C(4r+rcos 2θ,rsin 2θ).
记点B,C的高度差为d,则d=|2rsin θ-rsin 2θ|,
即d=2r|sin θ-sin θcos θ|.
设f(θ)=sin θ-sin θcos θ,θ∈[0,2π],
则f′(θ)=(1-cos θ)(2cos θ+1).
令f′(θ)=(1-cos θ)(2cos θ+1)=0,得cos θ=-
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f(θ)和f′(θ)随θ的变化情况如下表:
θ | 0 | ![]() | ![]() | ![]() | ![]() | ![]() | 2π |
f′(θ) | | + | 0 | - | 0 | + | |
f(θ) | 0 | ? | 极大值 f ![]() | ? | 极小值 f ![]() | ? | 0 |
∴当θ=
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综上所述,点B,C在传动过程中高度差的最大值dmax=
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