Mechanics | Relativity Answers

Suppose that we a have tunnel of length 100m (measured in its rest frame)
and a train whose length is 100m (measured in its rest frame). The train
is moving along the track at .8c, and the robbers want to trap it using the
following scheme: one robber will sit at the entrance to the tunnel and
blow it up just after the back end of the train has entered, while the other
robber will sit at the exit of the tunnel and blow it up just before the front
end of the train gets there.
iii) Describe the events shown on your diagrams from the point of view
of the robbers (either one). Be sure to state the order in which the
events occur in their frame of reference, to discuss whether the train
can escape, and to explain why or why not.

Suppose that we a have tunnel of length 100m (measured in its rest frame)
and a train whose length is 100m (measured in its rest frame). The train
is moving along the track at .8c, and the robbers want to trap it using the
following scheme: one robber will sit at the entrance to the tunnel and
blow it up just after the back end of the train has entered, while the other
robber will sit at the exit of the tunnel and blow it up just before the front
end of the train gets there.
i) Draw a spacetime diagram of all of this in the (inertial) reference frame
of the tunnel. Be sure to include the explosions and the response of
the train. Use the graph paper to make your diagram accurate.
ii) Draw a spacetime diagram of all of this using the inertial frame that
matches the train while it is moving smoothly down the tracks. Be
sure to include the explosions and the response of the train. Use the
graph paper to make your diagram accurate.

For this problem, consider three inertial observers: You, Alice, and Bob.
All three of you meet at one event where your watches all read zero. Alice
recedes from you at 1
2
c on your left and Bob recedes from you at 1
2
c on
your right. Draw this situation on a spacetime diagram in your frame of
reference. Also draw in a light cone from the event where you all meet.
Recall that you measure distances using your own lines of simultaneity,
and note that you find each of the others to be ‘halfway between you and
the light cone’ along any of your lines of simultaneity.
Now, use the above observation to draw this situation on a spacetime
diagram in Alice’s frame of reference. Use this second diagram to estimate
the speed at which Alice finds Bob to be receding from her. What happens
if you draw in another observer (again meeting all of you at t = 0) and
traveling away from Bob on the right at 1
2
c as measured in Bob’s frame of
reference??

A muon is a particle that has a lifetime of about 10−6
seconds. In other
words, when you make a muon, it lives for that long (as measured in its
own rest frame) and then decays (disintegrates) into other particles.
a) The atmosphere is about 30km tall. If a muon is created in the upper
atmosphere moving (straight down) at 1
2
c, will it live long enough to
reach the ground?
b) Suppose that a muon is created at the top of the atmosphere moving
straight down at .999999c. Suppose that you want to catch this muon
at the surface and shoot it back up at .999999c so that it decays just
when it reaches the top of the atmosphere. How long should you hold
onto the muon?

Suppose that you are in an airplane and that you are watching another
airplane fly in the opposite direction. Your relative velocity is roughly
600m/s. Calculate the size of the time dilation effect you would observe if
you measured the ticking of a clock in the other airplane. If that clock ticksonce each second, how much time passes in your reference frame between
two of it’s ticks?
Hint: Your calculator may not be able to deal effectively with the tiny
numbers involved. As a result, if you just try to type formula (3.3) into
your calculator you may get the value 1. What I want to know is how
much the actual value is different from 1. In cases like this, it is helpful to
use Taylor series expansions. The expansions you need to complete this
problem were given in problem (2-2).

a) Draw the two worldlines on a spacetime diagram in your frame of refer-
ence, and draw and label (i) the event where your watch “ticks” t = 1
and (ii) the event where your friend’s watch “ticks” t = 1. At what
time is this second event in your reference frame? Did you draw it in
the right place?
b) On the same diagram, sketch the line of events which are at t = 1 second
as determined by your system of reference. This is your t = 1 line of
simultaneity.
c) On the same diagram, sketch the line of events which are at t = 1 second
as determined by your friend’s system of reference. (Hint: At what time
in your system of reference does his watch tick t = 1?) This is your
friend’s t = 1 line of simultaneity.
e) Consider the event where your t = 1 line of simultaneity crosses your
friend’s worldline. What time do you assign to this event? What time
does your friend assign to this event?

an object of mass 0.25kg moves at a height,h above the ground with a speed of 4m/s if its mechanical energy at this height 12J determine the value of h

A lump of gold is suspected to contain some quantity of aluminum.if the gold has a mass of 450g and is to have a relative density of 5.2, what is the mass of aluminium if the relative density of gold and aluminium are 19.3 and 2.6 respectively.

4. A 2m long fixed string has nodes at both ends when it vibrates in its fundamental mode. If the distance between this nodes is 12cm. what are the number of (1) nodes (b) anti-nodes?

A cylindrical tank with diameter d = 300 mm is subjected to internal gas pressure p = 2 MPa. The tank is constructed of steel sections that are welded circumferentially. The heads of the tank are hemispherical. The allowable tensile and shear stress are 60 MPa and 24 MPa, respectively. Also, the allowable tensile stress perpendicular to a weld is 40 MPa. Determine the thickness tmin of
(i) cylindrical part of the tank and
(ii) the hemispherical heads.