课程名称︰普通物理学甲 课程性质︰必修 课程教师︰梁启德 开课学院：工学院 开课系所︰机械系 考试日期（年月日）︰2010/11/11 考试时限（分钟）：110 mins 是否需发放奖励金：是 谢谢 （如未明确表示，则不予发放） 试题： 　　　　　　　 　　　　Mid-term examination(11/11/2010) 1.Three vectors (向量) are given by a=3.0i+3.0j-2.0k, b=-1.0i-4.0j+2.0k, and c=2.0i+2.0j+1.0k. Please find (a) a‧bxc(5%) (b) a‧(b+c)(5%) (c) ax(b+c)(5%) [Hint:right-hand rule, i, j, and k are the unit vectors in the x-, y- and z- directions, respectively. ixj=k, jxk=i, ixk=-j] 2.Please prove (证明) that the rotational inertia of a solid cylinder about 1 2　 1 2 central diameter (see Fig.1) is given by I=--- mR + --- mL , where m is 4 12 the mass of the cylinder.(10%) Axis │ Solid cylinder │ (or disk) about │ central diameter │ ┌──────┴──────┐┬ │ │ R │ │┴ │ │ └──────┬──────┘ ├ L │ ┤ │ │ Figure 1 │ 3.Figure 2 shows a stream of water flowing through a hole at depth h=10cm in a tank holding water to height H=40cm. (a) t what distance x does the stream strike the floor? (5%) (b) At what depth should a second hole be made to give the same value? (5%) (c) At what depth should a hole be made to maximize x? (5%) ║ ║ ┬ ╟────┬╢ │ ║ h║ v │ ║ │║ ----> │ ║ ┴ ─╮ H ║ ║ │ ║ ║ │ ║ ║ (抛物线) │ ║ ║ │ ║ ║ ┴ ╚═════╝ ╰─── ├───x───┤ 4.A conical pendulum consists of a bob of mass m in motion in a circular path in a horizontal plane as shown in Fig. 3. During the motion, the supporting wire of length L maintains the constant angle θ with the vertical. Show (证明) that the magnitude of the angular mometum of the bob about the 2 3 4 m gl sin θ 0.5 circle's center is given by L = (------------) ,where g is the free-fall cosθ acceleration.(10%) ████████ ▕＼ ▕θ＼ L ▕ ＼ ▕ ＼ ▕ ＼ ╭-----------┼---------＼╮ │ └-----------○m ╰-----------┴-----------╯ 5.The position of a particle is given by x(t)=4+ln(t)sin(2t), where t is the time in units of second (t>0). Please find the velocity v(t)(5%) and the acceleration a(t)(5%) [Hint: ln(t)=log (t), where e is a mathematical e constant, the base of the natural logarithm, which equals approximately x 2.718281828, and is also known as Euler's number.e is the exponential x de x function. ---- = e ] dx 6.As you read this page on paper, a cosmic ray proton passes along the left-right width of the page with a relative speed v and a total energy of -8 1.424 x 10 J. According to your measurements, that left-right width is 0.21m (rest length). (a) What is the width according to the proton's reference frame? (5%) How much time did the passage take according to (b) your reference frame (5%) and (c) the proton's frame? (5%) [Hint: Rest -10 (mass) energy of a proton is ~1.5 x 10 J]. 7.A 55kg woman runs up a flight of stairs with a height of 4.5 m in 3.5 s. What average power (平均功率) is required of her? (10%) [Hint: free-fall acceleration g=9.8m/s^2] 8.One end of a uniform 4.00-m-long rod of weight Fg is supported ba a cable. The other end rests against the wall, where it is held by friction, as shown in Figure 4. The cofficient of static friction between the wall and the rod is μs=0.500. Determine the minimum distance x from point A at which an additional object, also with the same weight Fg, can be hung without causing the rod to slip at point A.(15%) [sin(37)≒3/5 cos(37)≒4/5] ▉＼ ▉ ＼ ▉ ＼ ▉ ＼ ▉ ＼ ▉ ＼ ▉ ＼ Figure 4 ▉ ＼ ▉ ＼ ▉A 37.0 ( ＼ ▉═══╤══════ B ▉ ←x→│ ▉ │ ▉ ○ Fg --

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