103年 - 103 淡江大學轉學考理論力學#53325

科目：轉學考-理論力學 | 年份：103年 | 選擇題數：0 | 申論題數：14

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所屬科目：轉學考-理論力學

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申論題 (14)

(a) [20%] Find the velocity v, the acceleration a and the speed of the particle.

(b) [5%] What is the angle between v and a at time

【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).

(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.

(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.

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.

(b) [5%] Write down the Newton equation of motion for the particle.

(c) [10%] Show that the time required for the particle to reach x=0 from a distance d is

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.

(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.

(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.

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.

(b) [10%] Show that the center of mass of the system moves in a constant velocity.

(c) [10%] Using Lagrangian dynamics to show that the motion of the system can be reduced to an equivalent one-body problem.