課程名稱︰普通物理上
課程性質︰大一必修
課程教師︰蔡爾成
開課系所︰電機系
考試時間︰2005/1/13
試題 :
1.A physical pendulum has two possible pivot points A and B; point A has a
fixed position, and point B is adjustable along the length of the pendulum.
The period of the pendulum when suspended from A is found to be T. The
pendulum is then reversed and suspended from B, which is moved until the
pendulum again has the period T. Show that the free-fall acceleration g at
the pendulum's location is given by g = 4*π^2*L/T^2 in which L is the
distance between A and B for equal periods.
2.Consider a homogeneous ring of mass M and radius R.
(a) What gravitational attraction does it exert on a particle of mass m
located on the ring's axis a distance x from the ring center?
(b) Suppose the particle falls from reset as a result of the attraction of the
ring of matter. What is the speed of the ring with which it passes through
the center of the ring?
3.The orbit for a planet moving around the Sun is an ellipse whose semi-major
axis (one half the major axis) and semi-minor axis (one half the minor axis)
are a and b. If the speed of the planet is Vο at the point of closest
approach. Express the period of the motion in terms ofa, b, Vο.
4.The damped oscillator satisfies the equation mx" + bx' + kx = 0. Assume
b^2 < 4mk, find the solution x(t) that satisfies the initial condition
x(0) = Xοand x'(0) = 0.
5.The equation of motion for a damped oscillator under an external driven force
is mx" + bx' + kx = Fοcos( w*t + θ) and the steady state solution is
x(t) = A*cos( w*t + ψ). Prove that A = Fο/((m*w^2 - k)^2 + b^2*w^2)^0.5.
6.The amplitude S(x, y, z, t) for a spherical wave travelling out uniformly in
all direction from a point source at the origin is described by expression
S(x, y, z, t) = (b/r)*sin(k(r - vt)) where r = (x^2 + y^2 + z^2)^0.5. Show
that it satisfies the three dimensional wave equation
(1/v^2)*(δ^2S/δt^2) － δ^2S/δx^2 － δ^2S/δy^2 － δ^2S/δz^2 = 0
【δ^2S/δt^2 表示S對t偏微分兩次】
7.It is known that the Maxwell probability distribution function of molecular
speeds in a gas at temperature T is
P(v) = 4π(M/(2*π*R*T))^1.5*v^2*e^(－M/(2RT)*v^2). What is the average of the
cubic power of the speed ﹤v^3 ﹥? You may need to make use of the integral
∞
∫ x^2*e^-x dx = 2
0
8.For a thermodynamic system, let dF = TdQ where dQ is the energy transferred
as heat to or from the system and T is the temperature. Prove that F cannot
be a state function by showing that ∫TdQ is path dependent for an ideal gas.
9.For an ideal gas, prove that the relaton between the pressure and the volume
during an adiabatic process is PV^γ = C in which C is a constant and
γ= Cp/Cv.
10.The second law of thermodynamics may be formulated in terms of entropy
change. i.e., in any thermodynamics process that proceeds from one
equilibrium state to another, the entropy of the system + environment
remains unchanged or increases. Show that the following two alternative
statements for the second law may be deduced from this entropy statement of
the second law.
(c)Kelvin-Planck form of the second law: There are no perfect heat engines.
(c)Clausius form of the second law: There are no perfect refrigerators.
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