課程名稱︰普物甲
課程性質︰
課程教師︰張寶棣
開課系所︰資工+@#$%^&*(
考試時間︰2005/12/16
試題 :
Midterm Exam
1. A small sphere of radius r = 1.5 cm rolls without slipping on the track
shown in Fig.1(抱歉沒圖) whose radius is R = 26.0 m. The sphere starts roll-
ing at a height R above a the bottom of the track. When it leaves the track
after passing through an angle of 135" as shown, (a)what will be its speed,
and (b)at what distance D from the base of the track will the sphere hit the
ground?(10%)
2. A string is wound around a uniform disk of radius R and mass M. The disk
is released from rest with the string vertical and its top end tied to a fixed
bar. Show that (a)the tension in the string is one third of the weight of the
disk, (b)the magnitude of the acceleration of the center of mass is 2 g/3, and
(c)the speed of the center of mass is (4gh)(power 1/2) after the disk has
descended through distance h. (15%)
3. Two disks of identical mass but different radii (r and 2r) are spinning on
frictionless bearings at the same angular speed w but in opposite directions.
The two disks are brought slowly together. The resulting frictional force
between the surfaces eventually brings them to a common angular velocity.
What is the magnitude of the final angular velocity in terms of w ?(10%)
(Hint: Angular momentum is conserved.)
4.A thin rod of mass M and length l rests on a frictionless table and is
struck at a point l/4 from its CM by a clay ball of mass m moving at speed v.
The ball sticks to the rod. Determine the translational and rotational motion
of the rod after the collision. (15%)(Hint: There are no external force and
torque in the whole system.)
5. There was an astronaut couple that took a spacecraft to a certain galax.
When they left the Earth with a constant speed of 0.95c, the female astronaut
found that she was pregnant. After 10 months spacecraft time, a male baby was
born and the male astronaut immediately sent out the message to the Earth
with speed of light to apply for a birth certificate. If we neglect the
acceleration time that the spacecraft took off, how far was the spacecraft
observed by the Earth people when the baby was born ? When did the application
signal reach the Earth according to the city goverment? If the date that the
couple left is March 2nd, 2004, what is the birth day of the baby in the
Earth time? (15%)
6. An electron (m=9.11*10 to the power -31 kg)has a kinetic energy that is
twice its rest energy. Determine its speed. (5%)
7. A certain galaxy has a Doppler shift given by F-f=0.797F. How fast was is
it moving away from us? (5%)
8. In class , we have obtained the relativistic velocity addition theorem for
one dimensional space and one dimensional time . Let's consider the case of
thee dimensional space. Assume that A frame moves relative to the B frame with
constant velocity v in the x direction. At tA=tB=0, frames A and B concide.
An object is moving with constant velocity u, with components ux, uy, uz. Write
down the lorentz transformation in the x-torx-ct frame. Since the relative
velocity is in the x direction , the y and z components are the same observed
in frames A and B, yA=yB and zA=zB. By definition , u(yA)=^yA (over) ^tA and
u(yB)=^yB(over)^tB . The same expression is for the z component . Aothough
^yA eqals ^yB, ^tA differs from ^tB, which results in u(yA)!=u(yB). Use the
relation between ^tA and ^tB to derive the transformation between u(yA)
and u(yB).(15%)
9. Choose the scale of Lorentz diagrams to be 1cm = 3*10 power 8m =1 sec.
(a) Construct four Lorentz diagrams in which a=30, 45, 60 and 75 degree.
and in each plot the event E whose B-coordinates are xB=6*10 power 8m and
tB=1sec.
(b). What happens to the position of event E as a approaches 90'.(10%)
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