課程名稱︰普通物理甲上 課程性質︰選修 課程教師︰張寶棣 開課學院：電資學院 開課系所︰資工系 考試日期（年月日）︰99.12.03 考試時限（分鐘）：120分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題： 1. Two objects are attached to ropes that are attached to wheels on a common axle as shown in Fig. 1. The two wheels glued together so that they form a single object. The total moment of inertial of the object is 40 Kg·m^2. The radii of the wheels are R1 = 1.2 m and R2 = 0.4 m. (a) If m1 = 24 Kg, find m2 such that there is no angular acceleration of wheels. (b) If 12 Kg is gently added to the rope of m1, find the angular acceleration of the wheels and the tensions in the ropes. (10%) 2. A spool of thread consists of a cylinder of radius R1 with end caps of radius R2 as in the end view shown in Fig. 2. The mass of the spool, including the thread, is m and its moment of inertial about an axis through its center is I. The spool is placed on a rough horizontal surface so that it rolls without slipping when a force T acting to the right is applied to the free end of the thread. Find the friction force exerted by the surface of the spool and its direction. (10%) 3. In Fig. 3, a uniform beam with a weight of 60 N and a length of 3.2 m is hinged at its lower end, and a horizontal force P of magnitude 50 N acts at its upper end. The beam is held vertical by a cable that makes angle θ = 25° with the ground and its attached to the beam at height h = 2.0 m. What are (a) the tension the cable and (b) the force on the beam from the hinge in unit-vector notation? (15%) 4. The fresh water behind a reservoir dam is 15 m deep. A horizontal pipe 4.0 cm in diameter passes through the dam 6.0 m below the water surface, as shown in Fig. 4. A plug secures the pipe opening. (a) Find the magnitude of the frictional force between plug arid pipe wall. (b) The plug is removed, What volume of water flows out of the pipe in three hours (assume that the water height does not change during flowing. 1 atm = 1.01 x 10^5 Pa)? (15%) 5. Two concentric shells of uniform density having masses M1 and M2 are situated as shown in Fig. 5. Find the magnitude of the net gravitational force on a particle of the mass m, due to the shells, when particle is located at (a) point A, at the distance r = a from the center, (b) point B at r = b, and (c) point C at r = c. The distance r is measured from the center of the shells. (Note that you don't need to do the integral to obtain the gravitational force when the mass distribution is spherical and uniform.) (15%) 6. A cup of water, only half full, is on top of a scale. You put your index finger inside the cup without touching its bottom and no water flow over the cup. Does the scale reading increase, decrease or remain the same after inserting your finger? Explain your answer. (5%) 7. The pendulum in Fig. 6 consists of a uniform disk with radius 10.0 cm and mass 500 g attached to a uniform rod with length 500 mm and mass 270 g. (a) Calculate the rotational inertial of the pendulum about the pivot. (b) What is the distance between the pivot point and the center of mass of the pendulum? (c) Calculate the period of oscillation. (15%) 8. A solid sphere (radius = R) with uniform density rolls without slipping in a cylindrical trough (radius = 5R) as shown in Fig. 7. Show that for small displacements from equilibrium perpendicular to the length of the trough, the sphere execute simple harmonic oscillation and fine the period. (Hint: Write down the total kinetic energy in terms of the angular velocity of the sphere (ω) in it's center of mass frame. Express the kinetic energy by converting ω to dθ/dt. Find the total energy by including the potential energe at the angle θ. Since the total energy is conserved, dE/dt = 0. Therefore, you should be able to obtain an equation and prove that the sphere moves as simple harmonic oscillation when θ is small.) (10%) 9. What is the frequency of a simple pendulum 2.0 m long in an elevator accelerating upward at a rate of 2.5 m/s^2 in the surface of Earth? How about free fall? (10%) Appendix： Moments of inertial for homogeneous rigid bodies: I = (1/2)MR^2 for a disk/cylinder, (2/5)MR^2 for a sphere, and (1/12)ML^2 for a long rod. --

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