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103年 - 103 淡江大學 轉學考 理論力學#53325
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(b) [5%] Write down the Newton equation of motion for the particle.
其他申論題
【已刪除】2. An undamped harmonic oscillator satisfies the equation of motion m where m dt and k are the mass and spring constant of the oscillator, respectively. (a) [10%]When F(/)=0, show that x(t)=a sin(coo0+^ cos(coo/) is the solution of the equation of motion, where a, b and ©o are constants, and find coo (in terms of m and k).
#195388
(b) [10%]A driving force F(t)=Fo^m{(nf) is switched on at t=0, where Fo and co are constants. Find x(0 for t＞0 with the initial conditions x=0 and v=0 at 户0.
#195389
(c) [5%]Find x(t) for co=(0o by taking the limit co^coo in your result from part (b). Hint In part (b) you can find a particular solution of the form x=A sin(©0 and determine A.
#195390
3. An one-dimensional particle at rest is attracted toward a center by a force F=-2w^/jc3, where m is the mass of the particle, A: is a constant and x is the coordinate of the particle, (a) [10%] Find the potential energy resulted from the force.
#195391
(c) [10%] Show that the time required for the particle to reach x=0 from a distance d is
#195393
4. A system is composed of n particles, with each particle's mass described by where i=l, 2, n. The total mass of the system is denoted by M. Show that (a) [10%] The linear momentum of the system is the same as if a single particle of mass Mwere located at the position of the center of mass and moving in the manner the center of mass moves.
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(b) [10%] The time differential of the linear momentum of the system is equal to the sum of all the external forces, as long as the internal forces follow 广-fp, where fy is the force acted on particle i by particle j.
#195395
(c) [5%] The total linear momentum for a system free of external forces is constant and equal to the linear momentum of the center of mass.
#195396
5. Two particles of mass m and m2 move in a plane and interact with each other by a central forces with the potential energy U{r)=M2 k r2，where A: is a constant, r叫ri-r2|, ri and 1*2 are position vectors of the two particles. (a) [5%] Write down the Lagrangian of the system.
#195397
(b) [10%] Show that the center of mass of the system moves in a constant velocity.
#195398
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