課程名稱︰普通物理學甲 課程性質︰系訂必修 課程教師︰梁啟德 開課學院：工學院 開課系所︰機械系 考試日期（年月日）︰98/11/12 考試時限（分鐘）：110(13:20-15:10) 是否需發放獎勵金：是，謝謝 （如未明確表示，則不予發放） 試題 : 1.Please explain, in your own words, what is an inertial reference frame (10%)? 2.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‧b×c (5%) (b) a‧(b+c) (5%) (c) a×(b+c) (5%) [Hint right-hand rule, i, j, and k are the unit vectors in the x-, y- and z- directions, respectively.] 3.A supertrain (rest length 100.0 m) travels at a speen of 0.9500c as it passes through a tunnel (proper length 50.00m). As seen by a trackside observer, is the train ever completely within the tunnel? (5%) If so, how much space is there to spare (tunnel length minus the train length)? (5%) [c=3*10^8 m/s] 4.Please prove that the relativistic expression for the kinetic energy of a particle is given by 1 K=(--------------- -1)mc^2 √[1-(v/c)^2] where v is the speed of the particle, c is the speen of light and m is the rest mass (靜止質量) of the particle, respectively. (5%) Please show that in the low speed limit v<<c, the kinetic energy is given by K=1/2 mv^2 (5%). 5.Figure 1 represents the total acceleration of a particle moving clockwise in a circle of radius 2.50m at a certain instant of time. At this instant, find (a) the radial acceleration (徑向加速度), (5%) (b) the speed of the particle, (5%) and (c) its tangential acceleration (切線加速度) (5%). 　　　　　　　　　　　　　　　　 　︵　　 . * 　* . a=15.0m/s^2 .* ／＼ θ=30.0度 　　　　　　　　　　　　 .`　 　　　／ /　＼ 　　　　　　　　　　　 　: 　　 　／θ/ 　 :↘v向量 (←2.50m→。 a向量 ) : : `. .` 。 。 `‧. .‧` ︶ 6.A particle of mass 1.18 kg is attached between two identical springs on a horizontal, frictionless tabletop. Both springs have spring constant k and are initially unstressed. (a) The particle is pulled a distance x along a direction perpendicular to the initial configuration of the springs as shown in Fugure 2. Show that the force exerted by the springs on the particle is L F=-2kx(1- -------------)i (5%) √(x^2+L^2) (b) Show that the potential energy of the system is given by U(x) = kx^2 + 2kL(L-√(x^2+L^2) (5%) ┌──────┬────────────┐ │ │ ＼k │ │ L│x ＼ │ ├──────┼───m───────→x┤ │ │ ／ │ │ L│ ／ │ └──────┴────────────┘ 7.A 5.00-kg block is set into motion up an inclined plane with an initial speed of 8.00 m/s (fig. 3). The block comes to rest after traveling 3.00 m along the plane, which is inclined at an angle of 30.0度 to the horizontal. For this motion, determine (a) the change in the block's kinetic energy, (5%) (b) the change in the potential energy of the block-Earth system, (5%), (c) the friction force exerted on the block (assumed to be constant). (5%) (d) What is the coefficient of kinetic friction? (5%) v=8.00m/s ╲ ╱▏ ↗ 3.00m ◇╱ ▏ ╲ ╱ ▏ ◇╱ ▏ ╱ ▏ 　　　　　　　　　　　　　　╱ 30度 ▏ ￣￣￣￣￣￣ 8.What horizontal force must be applied to the cart shown in Figure 4 so that blocks remain stationary relative to the cart? Assume all surfaces, wheels, and pulley are frictionless. Notice that the force exerted by the string accelerates m1.(10%) ┌─┐ │m1├─────╮ ┌──┴─┴────◎│ │ ├┴┐ F →│ M │m2│ │ ├─┘ └─⊙─────⊙─┘ --

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