Mechanics | Relativity Answers

Let’s get one more perspective on the twin paradox. It is always interesting
to ask what each twin sees during the trip. Now, note that what you
actually see has to do with light rays, and with when a bit of light happens
to reach your eye. So, to study this question, we should study light rays
sent from one twin to the other. We will again have Gaston go off to
Alpha Centauri (4 light-years away) and back at .8c while Alphonse stays
at home.
(a) Let’s first think about what Gaston (the traveling twin) sees. Start
by drawing a spacetime diagram for the trip in any inertial frame (itis easiest to use Alphonse’s frame of reference). Now, suppose that
Alphonse emits one light ray every year (according to his own proper
time). Draw these light rays on the diagram. How many of these
light rays does Gaston see on his way out?

Redraw everything using Alice’s frame of reference.
Once again, we and our friends, Alice and Bob, are inertial observers who
all meet at a single event. At this event, our clock, Alice’s clock, and Bob’s
clock all read zero and a firecracker explodes. Alice moves to our right at
c/2 and Bob moves to our left at c/2.

Draw a single spacetime diagram in our reference frame showing all of the
following:
(a) Alice, Bob, and the outgoing light from the explosion.
(b) The curve representing all events that are a proper time of one second
to the future of the explosion. Also draw in the curves representing
the events that are: i) one second of proper time to the past of the
explosion, ii) one light second of proper distance to the left of the
explosion, and iii) one light second of proper distance to the right of
the explosion.
(c) The events A, U, and B where Alice’s clock reads one second, where
our clock reads one second, and where Bob’s clock reads one second.
(d) Finally, suppose that we (but not Alice or Bob) are holding the middle
of a stick that is two light seconds long (and which is at rest relative
to us). Draw in the worldlines of both ends of that stick. Also mark
the events X and Y occupied by the ends of that stick on the line
tus = 0.

The figure here shows an overhead view of three horizontal forces acting on a cargo canister that was initially stationary but that now moves across a frictionless floor. The force magnitudes are F1 = 2.90 N, F2 = 3.50 N, and F3 = 10.0 N, and the indicated angles are θ2 = 52.0 ˚ and θ3 = 30.0 ˚. What is the net work done on the canister by the three forces during the first 4.20 m of displacement?

A block with mass m =6.5 kg is hung from a vertical spring. When the mass hangs in equilibrium, the spring stretches x = 0.21 m. While at this equilibrium position, the mass is then given an initial push downward at v = 4 m/s. The block oscillates on the spring without friction. After t = 0.37 s what is the speed of the block?

Consider an air-borne humanitarian mission, where food packages where dropped
with zero initial speed, from a helicopter hovering a height of h above the Earth
surface. Treating the package as particle of mass m, derive the horizontal deflection
from the particle caused by the Coriolis force acting on a particle falling freely in Earth’s
gravity field.

A small trailer has mass 400 kg. The wheels give friction, with coefficient of friction being 0.016. How much force must be exerted through the hitch to keep the trailer moving at steady speed? How much force in total must be exerted to accelerate the trailer at 1.5 m/s2?

A helicopter of mass 3500 kg is hovering. a) How much force must the rotors exert to do this? b) How much force in total must the rotors exert to make the helicopter accelerate upwards ate 1.5 m/s2 ? c) How much force in total must be exerted to allow the helicopter to accelerate downwards at -1.5 m/s2

A lump of metal of mass 5.0 kg is lying on a table top. It is tied to a string which runs over a pulley at the edge of the table. The other end of the string holds a 2.0 kg mass hanging down. Friction on the table is µ = 0.2 a) How much friction between the larger mass and the table? b) Is there any friction on the smaller mass? c) How much weight makes the system move? d) How much mass is moving? e) How much force is available to make the system move? f) How much acceleration can occur? G) How much force acts through the string to make the larger mass move

A rocket burns fuel at a rate of 251 kg/s and
exhausts the gas at a relative speed of 7 km/s.
Find the thrust of the rocket.
Answer in units of MN.
Don't round answer.