问题 计算题

如图所示的“S” 形玩具轨道,该轨道是用内壁光滑的薄壁细圆管弯成,放置在竖直平面内,轨道弯曲部分是由两个半径相等的半圆对接而成,圆半径比细管内径大得多,轨道底端与水平地面相切,轨道在水平方向不可移动。弹射装置将一个小球(可视为质点)从a点水平弹射向b点并进入轨道,经过轨道后从最高点d水平抛出(抛出后小球不会再碰轨道),已知小球与地面ab段间的动摩擦因数μ=0.2,不计其他机械能损失,ab段长L=1.25m,圆的半径R=0.1 m,小球质量m=0.01 kg,轨道质量为M=0.26 kg,g=10 m/s2,求:

(1)若v0=5 m/s,小球从最高点d抛出后的水平射程;

(2)若v0=5 m/s,小球经过轨道的最高点d时,管道对小球作用力的大小和方向;

(3)设小球进入轨道之前,轨道对地面的压力大小等于轨道自身的重力,当v0至少为多少时,小球经过两半圆的对接处c点时,轨道对地面的压力为零。

答案

解:(1)设小球到达d点处速度为v,由动能定理,得

-μmgL-mg·4R=

小球由d点做平抛运动,有

4R=

s=vt ③

联立①②③并代入数值,解得小球从最高点d抛出后的水平射程:m≈0.98 m ④

(2)当小球通过d点时,由牛顿第二定律得

代入数值解得管道对小球作用力FN=1.1 N,方向竖直向下 ⑥

(3)设小球到达c点处速度为vc,由动能定理,得

-μmgL-mg·2R=

当小球通过c点时,由牛顿第二定律得 ⑧ 

要使轨道对地面的压力为零,则有F'N=Mg ⑨

联立①②③并代入数值,解得小球的最小速度:v0=6 m/s ⑩

单项选择题
单项选择题

The questions in this group are based on the content of a passage. After reading the passage, choose the best answer to each question. Answer all questions following the passage on the basis of what is stated or implied in the passage.

Although hard statistics are difficult to come by, there is substantial anecdotal evidence that use of performance-enhancing drugs, or doping, is rampant in professional sports. Of perhaps greater significance to society are the estimated 1.5 million amateur athletes who use steroids, either to improve their appearance or to emulate the performance of their favorite professional athletes. This chemical epidemic is a pernicious threat to both the nation’s health and our collective sense of "fair play."

Nonprescription anabolic steroids have been illegal in the United States since 1991, and most professional sports leagues have banned them since the 1980s. These bans are partly a matter of fairness--a talented athlete trained to the peak of her ability simply cannot compete with an equivalent athlete using steroids--but also based on issues of health. Anabolic androgenic steroids ("anabolic" means that they build tissues; "androgenic" means that they increase masculine traits) have been linked to liver damage, kidney tumors, high blood pressure, balding, and acne. They function by increasing the body’s level of testosterone, the primary male sex hormone. In men, this dramatic increase in testosterone can lead to the shrinking of testicles, infertility, and the development of breasts; in women, it can lead to the growth of facial hair and permanent damage to the reproductive system. Steroids have also been linked to a range of psychological problems, including depression and psychotic rage.

The punishments for getting caught using steroids are severe, and the serious health consequences are well documented. Despite this, millions of professional and amateur athletes continue to use performance-enhancing drugs. Why is this

One clear pattern is that many athletes will do whatever it takes to get an edge on the competition. Since the 1950s, Olympic athletes have played a cat-and-mouse game with Olympic Committee officials to get away with doping, because the drugs really do work. Athletes who dope are simply per and faster than their competitors who play fair. Professional athletes in football and baseball have found that steroids and human growth hormone can give them the edge to score that extra touch-down or home run, and in the modern sports market, those results can translate into millions of dollars in salary. For the millions of less talented athletes in gyms and playing fields across the country, drugs seem like the only way to approach the abilities of their heroes in professional sports.

The other clear pattern, unfortunately, is that it has been all too easy for abusers to get away with it. Steroid abuse is often regarded as a "victimless crime." One of the favored ways to trick the testers is to use "designer" steroids. There are thousands of permutations of testosterone, such as THG, that can be produced in a lab. Chemists have discovered that they can create new drugs that produce androgenic effects but do not set off the standard doping tests. Other methods have been to use the steroids but stop a few weeks before testing, to use other chemicals to mask the traces of steroids, or to switch in a "clean" sample of urine at the testing site. Other athletes use steroid precursors, such as androstene-dione, that have androgenic effects similar to those of steroids but are not illegal because they are not technically steroids. The sad fact is that unless the government and professional sports organizations are willing to get tough on the steroid problem, the use of performance-enhancing dugs in sports is not going to end.

According to the passage, all of the following are known potential consequences of steroid use except for which of the following ?()

A. Damage to reproductive organs

B. Decreased blood pressure

C. Increases in the user’s strength and speed

D. Kidney tumors

E. Increased risk of depression