[普物] 蔡爾成老師老筆記與考古題
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老筆記
2002
１力場 F=Yi-Xj+Zk是否為保守力場？
２在質心靜止的座標觀察兩物體彈性碰撞，證明碰撞前後兩物體運動速度大小不變
３一多質點系統動能可以因在不同慣性座標而有不同，證明質心座標中所觀察到的動能為
最小值
４證明一多質點系統所受總力矩為其角動量對時間變化率，且總力矩不含內部相互作用所
產生之力矩
５一空心球殼，半徑Ｒ，質量Ｍ，求通過其質心知轉動慣量
６若重力加速度大小為ｇ，一圓柱體，質量Ｍ半徑Ｒ，由距地面ｈ處順著一斜坡
向下滾動但不滑動，求此圓柱體到地面時的速度大小
７舉一反例，說明角動量和角速度方向可能不平行
８A meter stick is held vertically with one end on the floor and is then
allowed to fall.
Find the speed of the other end when it hit the floor, assuming that the end
on the floor does not slip.
９Two cylinders having radius R1 and R2 and rotational inertias I1 and I2
about their central axis are supported by axles perpendicular to the plane of
the figure on the right. The larger cylinder is initially rotating clockwise
with angular ω0 and. The small cylinder is moved to the right until it
touches the larger one and is caused to rotate by the frictional force
between the two. Eventually slipping ceases and the two cylinders rotate in
the opposite direction. (a)Is angular momentum or kinetic energy conserved?
(b)Find the final angular velocityω2 of the smaller one in terms of I1 I2
R1 R2 and ω0.
１０一靜止半球碗內面半徑R，一質點沿其無摩擦之碗內面水平射出，如此質點剛好可爬
到碗上邊緣，求質點初速Ｖ與與起始角度ψ的關係（起始角度：質點與球心連線與垂直線
的夾角）
2003
１三度空間運動之單一質點，證明外力對質點所作之功等於該質點增加的動能（須是三度
空間證明）
２一多質點系統動能可以因在不同慣性座標而有不同，證明質心座標中所觀察到的動能為
最小值
３在保守力場運動之質點如何定義其位能
４一圓柱體在一平面上滾動但不滑動，圓柱體和平面接觸之摩擦力是否作功
５一在向心力場F=f(r) r內運動之質點為何(a)其角動量為一常數向量(b)其軌道在一平面
上
６一均勻實心圓球半徑Ｒ質量Ｍ，證明通過質心之轉動慣量I=(2/5)(MR^2)
７對一轉動之剛體，證明K=(1/2)L．ω（注意兩者可能不平行）
８A rigid rod of length L and negligible mass has a ball with mass m attached
to one end and its other end fixed, to form a pendulum. The pendulum is
released at rest from a horizontal position. At what angle from the vertical
does the tension in the rod equal to the weight of the ball?
９A girl of mass M stands on a rim of a frictional merry-go-round of radius R
and rotational inertia I that us not moving. She throws a rock of mass m
horizontally in a direction that is tangent to the outer edge of the
merry-go-round. The speed of the rock, relative to the ground, is v.
Afterwards, what is the angular speed of the merry-go-round?
１０The rotational inertia of a collapsing spinning star change to 1/2 its
initial value. What is the ratio of the new rotational kinetic energy to the
initial rotational kinetic energy.
2004
1.The potential energy of a diatomic molcule(a two atom system like H2O or O2)
is given by U= A/r^12 - B/r^6 (A,B >0)
where r is the separation of the two aToms of the molecule and A and B are
constants.This potential energy is associated with the force that binds the
two atoms together.
(a)[5%]Find the equilibrium separation-that is , the distance between the
atoms at which the force on each atom is zero.
(b)[5%]Is the above equilibrium configuration stable?
2.[10%]Consider a rocket that is in deep space and at rest initially relative
to an inertial reference frame.The rocket's engine is to be fired for a
certain interval. Derive the rocket equation to evaluate the rocket's mass
ratio (ration of initial to final mass) if the rocket's final speed relative
to the inertial frame is to be equal to the exhaust speed(speed of the
exhaust products relative to the rocket).
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 the end on the floor does not slip.
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?
5.[10%]If the two particals are of the same mass and one of them is at rest
before collision,show that these two particals move in directions that are
perpendicular to each other after an elastic collision.
6.[10%]Show that the rotational inertia for a thin spherical shell about any
axis that is tangent to the spherical shell is 5/3mR^2.
7.[10%]Prove that for a general rigid body motion about a fixed point,the
kinetic energy K is given by
→ →
→ K=1/2L‧ω →
where L is the total angular momentum of the rigid body and ω is the
instantaneous angular velocity.
8.[10%]To get a billard ball to roll without slipping from the start,the cue
must hit the ball at a hight 2/5R above the center.Prove the result.The
rotational inertia for a homogeneous shere is 2/5mR^2.
9.[10%]For a particle in a central force field
→ ^ ^ ^
F (x,y,z)=g(x,y,z)(xi+yj+zk),
prove that the orbit of the particle must be on a plane.
10.[10%]A particleis projected horizontally along the interior of a
frictionless hemisherical bowl of radius r,which is kept at rest. We wish
to find the initial speed required for the partical to just reach the top
of the bowl.Find as a function of θ,the initial angular position of the
partical.
2005
1.A block of mass M is pulled along a horizontal (*frictionless) surface by a
rope of mass m. A horizontal force F(->) acts on the rope.
(a) show that the rope must sag, even if only by an imperceptible amount.
Then, assuming the sag is negligible, find (b) the tension in the rope at its
midpoint.
2.A student want to determine the coefficients of static friction and kinetic
frinction between a box and a plank. She places the box on the plank and
gradually raises one end of the plank. When the angle of inclination with
horizontal reaches θ, the box starts to slip, and it then slides distance s
down the plank in time T at constant acceleration. What are the coeffcient of
static friction and the coefficient of kinetic friction between the box and
the plank.
3.After a completely inelastic collision, two objects of the same mass and
same initial speed are found to move away together at 1/2 their initial
speed. Find the angle between the initial velocities of the objects.
4.A rigid rod of length L and negligible mass has a ball with mass m attached
to one end and its other end fixed, to form a pendulum. The pendulum is
released at rest from horizontal position. At what angle from the vertical
does the tension in the rod is equal to weight of the ball?
5.How do we define the potential energy for a conservative force field such
that the sum of the kinetic energy and potential energy is a constant?
6.A paticle moves from (0.0.0) to (1,2,3) on a straight line. What's the work
done by the force field F(->) =xy i -z^2 j +7x^2 k on this particle?
(*i,j,k means unit vectors <1,0,0> <0,1,0> <0,0,1>)
7.What's the force field due to the potential energy field U(x,y,z)=xyz?
8.The total kinetic energy of a system of particles depends on the inertial
reference frame. Prove that the total kinetic energy is minimal in the
center of mass frame.
9.For the elastic collision of two particles observed in the center of mass
frame, show that the initial and final speed for each of two particles
remain same.
10.Two particles are of the same mass and one of them is at rest before
collision. Show that these two particles move in directions that are
perpendicular to each other after an elastic collision.
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