问题 计算题

如图所示,地面和半圆轨道面均光滑。质量M = 1kg、长L = 4m的小车放在地面上,其右端与墙壁的距离为S=3m,小车上表面与半圆轨道最低点P的切线相平。现有一质量m = 2kg的滑块(不计大小)以v0= 6m/s的初速度滑上小车左端,带动小车向右运动。小车与墙壁碰撞时即被粘在墙壁上,已知滑块与小车表面的滑动摩擦因数μ= 0.2,g取10m/s2

(1)求小车与墙壁碰撞时的速度;

(2)要滑块能沿圆轨道运动而不脱离圆轨道,求半圆轨道的半径R的取值。

答案

解:(1)滑块与小车的共同速度为v1,滑块与小车相对运动过程中动量守恒,有

mv0=(m+M)v1

代入数据解得v1=4m/s

设滑块与小车的相对位移为L1,由系统能量守恒定律,有

μmgL1 =

代入数据解得L1=3m

设与滑块相对静止时小车的位移为S1,根据动能定理,有

μmgS1=

代入数据解得S1=2m

L1LS1S,说明小车与墙壁碰撞前滑块与小车已具有共同速度,且共速时小车与墙壁还未发生碰撞,故小车与碰壁碰撞时的速度即v1=4m/s

(2)滑块将在小车上继续向右做初速度为v1=4m/s,位移为L2=LL1=1m的匀减速运动,然后滑上圆轨道的最低点P

若滑块恰能滑过圆的最高点,设滑至最高点的速度为v,临界条件为mg=

根据动能定理,有-μmgL2

①②联立并代入数据解得R = 0.24m

若滑块恰好滑至圆弧到达T点时就停止,则滑块也能沿圆轨道运动而不脱离圆轨道

根据动能定理,有-μmgL2

代入数据解得R = 0.6m

综上所述,滑块能沿圆轨道运动而不脱离圆轨道,半圆轨道的半径必须满足R≤0.24m或R≥0.6m

阅读理解

Getting rid of dirt, in the opinion of most people, is a good thing. However, the attitudes to dirt are always changing.

In the early 16th century, people thought that dirt on the skin was a means to block out disease, and washing off dirt with hot water could open up the skin and let ills in. A particular danger was thought to lie in public baths. By 1538, the French king had closed the bath houses in his kingdom. So did the king of England in 1546. Thus began a long time when the rich and the poor in Europe lived with dirt in a friendly way. Henry IV, King of France, was famously dirty. Upon learning that a nobleman had taken a bath, the king ordered that, to avoid the attack of disease, the nobleman should not go out.

Though the belief above was long­lived, dirt has no longer been regarded as a nice neighbor ever since the 18th century. Scientifically speaking, cleaning away dirt is good to health. Clean water supply and hand washing are practical means of preventing disease. Yet, it seems that standards of cleanliness have moved beyond science since World War Ⅱ.Advertisements repeatedly sell the idea; clothes need to be whiter than white, cloths ever softer, surfaces to shine. Has the hate for dirt, however, gone too far?

Attitudes to dirt still differ hugely nowadays. Many first­time parents nervously try to warn their children off touching dirt, which might be responsible for the spread of disease. On the contrary,Mary Ruebush,an American immunologist(免疫学家),encourages children to play in the dirt to build up a strong immune system. And the latter position is gaining some ground.

小题1:The kings of France and England in the 16th century closed bath houses because ________.

A.they believed disease could be spread in public baths

B.they thought bath houses were too dirty to stay in

C.they lived healthily in a dirty environment

D.they considered bathing as the cause of skin disease小题2:Which of the following best describes Henry IV's attitude to bathing?

A.Approving.

B.Afraid.

C.Curious.

D.Uninterested.小题3:How does the passage mainly develop?

A.By providing examples.

B.By making comparisons.

C.By following the order of importance.

D.By following the order of time.小题4: What is the author's purpose in writing the passage?

A.To stress the role of dirt.

B.To introduce the history of dirt.

C.To present the change of views on dirt.

D.To call attention to the danger of dirt.

判断题