《2016年泛珠三角物理竞赛综合试卷一题》由会员分享，可在线阅读，更多相关《2016年泛珠三角物理竞赛综合试卷一题（5页珍藏版）》请在金锄头文库上搜索。
1、1 Pan Pearl River Delta Physics Olympiad 2016 2016 年泛珠三角及中华名校物理奥林匹克邀请赛 Sponsored by Institute for Advanced Study, HKUST 香港科技大学高等研究院赞助 Simplified Chinese Part-1 (Total 5 Problems) 简体简体版版卷卷-1（共（共5题）题） (9:00 am 12:00 pm, 18 February, 2016) Please fill in your final answers to all problems on the summar
2、y sheet. 请在总答案纸总答案纸上填上各题的最后答案。 There are 5 problems. Please answer each problem using a new sheet. 卷一 5题，每答 1 题，须采用新一张纸新一张纸。 Please answer on each sheet using a single column. Do not use two columns on a single sheet. 每页纸请用单一直列单一直列的方式答题。不可以在一页纸上以双直列方式答题。 Please answer on only one side of each sheet.
3、 Do not use both sides of the sheet. 每张纸单面单面作答。不可以双面作答。 At the end of the competition, please arrange the summary sheet as the first page, followed by the answers to each problem in sequential order of the problems. If your answer to a problem requires more than one sheet, please arrange the sheets
4、of the same problem in sequential order of the parts. 比赛结束时，请将总答案纸放在首页总答案纸放在首页，随后把答题纸按题目次序排好。若答一题超过一 页，请按分部次序排好答题纸。 1. Electrostatic Force (4 marks) 静电力静电力 （4 分）分） Consider a 2017-side regular polygon. There are 2016 point charges, each with charge q and located at a vertex of the polygon. Another p
5、oint charge Q is located at the center of the polygon. The distance from the center of the regular polygon to its vertices is a. Find the force experienced by Q. 考虑一 2017-边正多边形。其中 2016个角上各有一点电荷 q。 另有一个点电荷 Q位于 多边形的中心。 中心到每一个角的距离为 a。 求 Q所受的力。 2. Capacitors (13 marks) (a-c) Consider two clusters of ele
6、ctric charges. Cluster A consists of N charges q1, q2, , qN, located at positions respectively. Cluster B consists of M charges , located at positions respectively. (a-c) 考虑两组电荷。组 A 由 N 个电荷 q1, q2, , qN组成, 并分别位于位置 。组 B 由 M个电荷 组成, 并分别位于位置 。 (a) Write the electric potential at position due to the char
7、ges in cluster A. (1 mark) 写下于位置 由组 A电荷形成的电势 。（1 分） (b) Write the electric potential energy of cluster B due to the electric potential . (1 mark) 写下组 B电荷因电势 产生的电势能 。 (1 分) (c) What is the relation between and ? (1 mark) 2 和 有何关系? (1 分) (d) Consider two large conducting plates as shown in Fig. 1a. Th
8、e upper plate carries a uniform surface charge density and the lower plate is grounded. Find the surface charge density of the lower plate and the potential , where is the height of an arbitrary location from the lower plate. (5 marks) 考虑如图 1a 所示两块很大的电导板。上板带有均匀面电荷密度 , 而下板则接地。求 下板的面电荷密度和电势 , 其中 为任意一点
9、距离下板的高度。（5 分） Figure 1a 图 1 a Figure 1b 图 1b (e) A point charge q is placed between two very large grounded parallel conducting plates. If z0 is the distance between q and the lower plate, find the total charge induced on the upper plate in terms of q, z0, and l, where l is the distance between the
10、plates, as shown in Fig. 1b. (5 marks) 如图 1b 所示, 在相距为 l 的两块平行大电导板间放置电荷 q, 其到下板的距离为 z0。求 上板的总感应电荷。以 q、z0和 l 表示你的答案。（5 分） 3. Cannonballs and Bombs (10 marks) 砲彈和炸彈砲彈和炸彈 （1010 分）分） (a) Envelope of safety: A ground based cannon can fire a cannonball at a fixed speed of in any direction. The envelope of
11、safety is the curve inside which a target can be hit by the cannonball, and outside which there is no possibility of a target getting hit by the cannonball. Find the equation of the envelope of safety in space. (3 marks) 安全区域边界: 一门位于地面的大炮能以固定速率 u 向任何方向发射炮弹。 若目标在 安全区域边界内, 则有可能被炮弹打中。 若在其外, 则不可能被炮弹打中。
12、求在空中的 安全区域边界的方程式。 （3 分） (b) A bomb explodes at a height of into many small fragments. It is given that after the explosion the fragments have the same speed and a uniform angular distribution in all directions. After some time all fragments hit the ground and the collisions with the ground are perfe
13、ctly inelastic. Find the radius R of the distribution of the debris. (2 marks) z 0 l z 0 l q z0 3 一个炸弹在高度 H处爆炸成很多小碎片。 已知刚爆炸后各碎片以同样的速率 u和均匀 的角分布向各个方向散开。 其后各碎片都坠到地面上。 假设所有碎片与地面的碰撞皆 为完全非弹性碰撞。 求炸弹残骸的分布半径 R。 (2分) (c) A bomb explodes on the ground. Its fragments are projected at the same speed , and the a
14、ngular distribution is uniform within a narrow angle with the upward vertical direction. After some time all fragments hit the ground. Let the mass of the bomb be M. Find the radius R of the distribution of the debris up to order . Calculate the radial density distribution within radius R up to orde
15、r , where is the mass of the debris located at a distance to from the centre of the distribution. (5 marks) Remark: and for 一个炸弹在地面爆炸。爆炸后各碎片以同样速率 u 射出，角度分布则限在与垂直向上 方向的狭小夹角 内，而在这范围内角度分布均匀。其后各碎片都坠到地面上。设炸 弹的质量为 M。求炸弹残骸的分布半径 R, 准确至 的第一阶。定义径密度分布 , 使得 为距离残骸中心 r 至 r+dr 范围内的残骸质量。求半径 R 内的 , 准确 至 r 的第二阶。（5分）
16、注: 当 1时, 及 。 4. Collisions (14 marks) 碰撞碰撞 (14分分) A thin rod with length L, mass m and uniform density lies on the y-axis with its midpoint at the origin. A point object A with mass m travels with velocity u in the positive x direction hits the rod with impact parameter h, where , as shown in Fig. 2a. The collision is perfectly inelastic. 如图2a所示, 一根长度为L、质量为m、密度均匀
