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武汉大学2008-2009学年《理论力学》期末考试试卷及参考答案详解

武汉大学2008-2009学年《理论力学》期末考试试卷及参考答案详解
武汉大学2008-2009学年《理论力学》期末考试试卷及参考答案详解

解:(1)分别选择整体和ABC 为研究对象 (2分)

(2)分别进行受力分析(两图每图各3分)

(3) 分别列平衡方程

整体:0=∑x

F

,0=Ex F

0)

(=∑F D M

02

3=?+?a

qa a F Ey

2

3qa

F Ey -= (4分) ABC 杆:

=0)(F C M ,02

45sin 0=?

+?-a

qa a F BD qa F BD 22

=

分)

四、均质杆AD 重P ,与长为2l 的铅直杆BE 的中心D 铰接,如图所示。柔绳的下端

吊有重为G 的物体M 。假设杆BE 、滑轮和柔绳的重量都忽略不计,连线AB 以及柔绳的CH 段都处于水平位置,求固定铰链支座A 的约束反力。(15分)

解:(1)分别选整体和杆AD 为研究对象(2分)

(2)分别画出它们的受力图(5分) (3)分别列平衡方程 整体: 由

()0B

M

=∑F ,有

o o 2cos30(2)cos300Ay HC F l G r F l r P l -?-?--+?= (3分)

杆AD :

()0D

M

=∑F ,有

o o o 2sin302cos30cos300Ax Ay F l F l P l -?-?+?= (3分) 其中HC F G =。联立求解,可得

2Ax F G =,2Ay P F =

(2分)

五、如图所示,曲柄OA 长20cm ,绕轴O 以匀角速度010/rad s ω=转动。此曲柄借助

F Bx Dx

BD

q

连杆AB 带动滑块B 沿铅垂方向运动,连杆长100cm 。求当曲柄与连杆相互垂直并与

水平线各成o

45α=与o

45β=时,连杆的角速度、角加速度和滑块B 的加速度。 (15

分)

解:(1)由A v 和B v 的速度方向可知P 点为杆AB 的速度瞬心。故连杆的角速度为

0o 2010

2(/)tan45100

A AB

OA v rad s PA AB ωω??==== (4分) (2)由n

n B A BA

BA τ

=++a a a a 作B 点的加速度合成图。 (4分)

列投影方程,n

BA

a 方向的投影方程,有

o cos45n

B BA

a a -= (3分) 而2

222100400(/)n BA AB a AB cm s ω=?=?=,故有

o 2/cos45566(/)n

B BA a a cm s =-=-=- (1分)

BA τa 方向的投影方程,有

o sin 45n

B BA A

a a a τ

=- (3分)

而222010202000(/)n A

a OA cm s ω=?=?=,故有 o 2

cos451600(/)n BA B A a a a cm s τ=+= (1分)

连杆的角加速度为

21600

/16(/)100

BA BA a BA rad s τε==

= (1分)

六、跨过定滑轮B 的绳索,两端分别系在滚子A 的中心和物块C 上。滚子A 和定滑轮

B 都是半径为r 的均质圆盘,各重G ,物块

C 重1G 。滚子沿倾角是α的斜面向上作纯

滚动(见图)。绳的倾斜段与斜面平行,绳与轮B 不打滑,不计绳重和轴承摩擦。试

求:(1)滚子A 的质心加速度;(2)绳索AB 段的拉力;(3)轴承O 的反力。(15分)

解:(1)分别选滚子A 、滑轮B 和物块C 为研究对象(1分) (2)受力分析和运动分析如图所示(5分)

(3)列动力学方程

滚子A :sin AB sA A G G

F F

G a r g g

αε--=

=? 2

12sA G r F r g

ε?=? (3分) 滑轮B :cos 0Ox BA F F α-=

sin 0Oy BA BC F F G F α---=

2

12BC BA G F r F r r g

ε?-?=

? 物块C :111CB G G

G F a r g g

ε-==? (4分) 联立求解,可得

B

P

n

A a

NA

sA

AB C

1

Ox BA F

11sin 2A G G a r g G G αε-=?=

+,1113(2)

2(2)

AB G G G F G G G ++=+

111cos [3(2)sin ]2(2)

ox G F G G G G G α

α=

+++,

1111cos {46[5(2)sin ]sin }2(2)

Oy G F G G G G G G G α

αα=

+++++ (2分)

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