課程名稱︰普通物理學甲上 課程教師︰蔡爾成 開課系所︰機械系 考試日期（年月日）︰2010/11/12 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. [10%] The only force acting on a particle is conservative force F. If the → particle is at point A, the potential energy of the system associated with F is 40J. If the particle moves from point A to point B, the work done on the → particle by F is +25J. What is the potential energy of the system with the particle at B? 2. [10%] After complete inelastic collision, two objects of the same mass and same initial speed move away together at half their initial speed. Find the angle between the initial velocities of the two objects. 3. [10%] A meter stick is held vertically with one end on the floor and is then allowed to fall. Find the speed of the other end just before it hits the floor, assuming that the end on the floor does not slip. (Hint: Consider the stick to be a thin rod and use the conservation of energy principle.) 4. [10%] The rotational inertia of a collapsing spinning star drops to 1/3 its initial value. What is the ratio of the new rotational kinetic energy to the initial rotational kinetic energy? → ^ ^ ^ 5. (a) [5%] Is the force field F(x,y,z) = xi + yj + zk conservative? Why? → ^ ^ ^ (b) [5%] Is the force field F(x,y,z) = -yi + xj + zk conservative? why? 6. For a thin spherical shell of mass M and radius R, (a) [7%] show that its rotational inertia about an axis passing through its 2 2 center is equal to ─mR ; 3 (b) [3%] show also that the rotational inertia for a thin spherical shell 5 2 about any axis that is tangent to the spherical shell is ─mR. 3 7. [10%] For the elastic collision of two particles observed in the center of mass frame, show that the initial speed is equal to the final speed for each of the two particles. 8. [10%] A body of radius R and mass m is rolling smoothly with speed ν on a horizontal surface It the rolls without sliding up a hill to a maximum height 2 3ν h. If h = ─, what is the body's rotational inertia about the rotational axis 4g through its center of mass? 9. [10%] Prove that for a general rigid body motion about a fixed point, the kinetic energy K is given by 1 → → K = ─L‧ω 2 → → where L is the total angular momentum of the rigid body and ω is the instantaneous angular velocity. 10. A particle of mass m is moving along the interior of a frictionless → hemispherical bowl of radius r, which is kept at rest. The force F acting on → → → → the particle is equal to N + mg where N is the normal force and mg is the gravitational force in the vertical direction. (a) [5%] Show that the component of the angular momentum along the vertical direction is conserved. (b) [5%] Show that ν sinθ is a constant where ν is the horizontal speed and h h θ, as indicated in the above figure, is the angle between the direction of → the particle and the direction of g. --

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