Physics for Scientists and Engineers, Technology Update (No access codes included)
9th Edition
ISBN: 9781305116399
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 6, Problem 6.24P
Review. A student, along with her backpack on the floor next to her, is in an elevator that is accelerating upward with acceleration a. The student gives her backpack a quick kick at t = 0, imparting to it speed v and causing it to slide across the elevator floor. At time t, the backpack hits the opposite wall a distance L away from the student. Find the coefficient of kinetic friction μk between the backpack and the elevator floor.
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Question B3
Consider the following FLRW spacetime:
t2
ds² = -dt² +
(dx²
+ dy²+ dz²),
t2
where t is a constant.
a)
State whether this universe is spatially open, closed or flat.
[2 marks]
b) Determine the Hubble factor H(t), and represent it in a (roughly drawn) plot as a function
of time t, starting at t = 0.
[3 marks]
c) Taking galaxy A to be located at (x, y, z) = (0,0,0), determine the proper distance to galaxy
B located at (x, y, z) = (L, 0, 0). Determine the recessional velocity of galaxy B with respect
to galaxy A.
d) The Friedmann equations are
2
k
8πG
а
4πG
+
a²
(p+3p).
3
a
3
[5 marks]
Use these equations to determine the energy density p(t) and the pressure p(t) for the
FLRW spacetime specified at the top of the page.
[5 marks]
e) Given the result of question B3.d, state whether the FLRW universe in question is (i)
radiation-dominated, (ii) matter-dominated, (iii) cosmological-constant-dominated, or (iv)
none of the previous. Justify your answer.
f)
[5 marks]
A conformally…
SECTION B
Answer ONLY TWO questions in Section B
[Expect to use one single-sided A4 page for each Section-B sub question.]
Question B1
Consider the line element
where w is a constant.
ds²=-dt²+e2wt dx²,
a) Determine the components of the metric and of the inverse metric.
[2 marks]
b) Determine the Christoffel symbols. [See the Appendix of this document.]
[10 marks]
c)
Write down the geodesic equations.
[5 marks]
d) Show that e2wt it is a constant of geodesic motion.
[4 marks]
e)
Solve the geodesic equations for null geodesics.
[4 marks]
Page 2
SECTION A
Answer ALL questions in Section A
[Expect to use one single-sided A4 page for each Section-A sub question.]
Question A1
SPA6308 (2024)
Consider Minkowski spacetime in Cartesian coordinates th
=
(t, x, y, z), such that
ds² = dt² + dx² + dy² + dz².
(a) Consider the vector with components V" = (1,-1,0,0). Determine V and V. V.
(b) Consider now the coordinate system x' (u, v, y, z) such that
u =t-x,
v=t+x.
[2 marks]
Write down the line element, the metric, the Christoffel symbols and the Riemann curvature
tensor in the new coordinates. [See the Appendix of this document.]
[5 marks]
(c) Determine V", that is, write the object in question A1.a in the coordinate system x'. Verify
explicitly that V. V is invariant under the coordinate transformation.
Question A2
[5 marks]
Suppose that A, is a covector field, and consider the object
Fv=AAμ.
(a) Show explicitly that F is a tensor, that is, show that it transforms appropriately under a
coordinate transformation.
[5 marks]
(b)…
Chapter 6 Solutions
Physics for Scientists and Engineers, Technology Update (No access codes included)
Ch. 6 - You are riding on a Ferris wheel that is rotating...Ch. 6 - A bead slides at constant speed along a curved...Ch. 6 - Consider the passenger in the car making a left...Ch. 6 - A basketball and a 2-inch-diameter steel ball,...Ch. 6 - A child is practicing for a BMX race. His speed...Ch. 6 - Consider a skydive r who has stepped from a...Ch. 6 - A door in a hospital has a pneumatic closer that...Ch. 6 - A pendulum consists of a small object called a bob...Ch. 6 - As a raindrop falls through the atmosphere, its...Ch. 6 - An office door is given a sharp push and swings...
Ch. 6 - Before takeoff on an airplane, an inquisitive...Ch. 6 - What forces cause (a) an automobile, (b) a...Ch. 6 - A falling skydiver reaches terminal speed with her...Ch. 6 - An object executes circular motion with constant...Ch. 6 - Describe the path of a moving body in the event...Ch. 6 - The observer in the accelerating elevator of...Ch. 6 - Prob. 6.6CQCh. 6 - It has been suggested dial rotating cylinders...Ch. 6 - Consider a small raindrop and a large raindrop...Ch. 6 - Why does a pilot lend to black out when pulling...Ch. 6 - Prob. 6.10CQCh. 6 - If the current position and velocity of every...Ch. 6 - A light string can support a stationary hanging...Ch. 6 - Whenever two Apollo astronauts were on the surface...Ch. 6 - In the Bohr model of the hydrogen atom, an...Ch. 6 - A curve in a road forms part of a horizontal...Ch. 6 - In a cyclotron (one type of particle accelerator),...Ch. 6 - A car initially traveling eastward turns north by...Ch. 6 - A space station, in the form of a wheel 120 m in...Ch. 6 - Consider a conical pendulum (Fig. P6.8) with a bob...Ch. 6 - A coin placed 30.0 cm from the center of a...Ch. 6 - Why is the following situation impossible? The...Ch. 6 - A crate of eggs is located in the middle of the...Ch. 6 - A pail of water is rotated in a vertical circle of...Ch. 6 - A hawk flies in a horizontal arc of radius 12.0 m...Ch. 6 - A 40.0-kg child swings in a swing supported by two...Ch. 6 - A child of mass m swings in a swing supported by...Ch. 6 - A roller-coaster car (Fig. P6.16) has a mass of...Ch. 6 - A roller coaster at the Six Flags Great America...Ch. 6 - One end of a cord is fixed and a small 0.500-kg...Ch. 6 - Prob. 6.19PCh. 6 - An object of mass m = 5.00 kg, attached to a...Ch. 6 - All object of mass m = 500 kg is suspended from...Ch. 6 - A child lying on her back experiences 55.0 N...Ch. 6 - A person stands on a scale in an elevator. As the...Ch. 6 - Review. A student, along with her backpack on the...Ch. 6 - A small container of water is placed on a...Ch. 6 - Review. (a) Estimate the terminal speed of a...Ch. 6 - The mass of a sports car is 1 200 kg. The shape of...Ch. 6 - A skydiver of mass 80.0 kg jumps from a...Ch. 6 - Calculate the force required to pull a copper ball...Ch. 6 - A small piece of Styrofoam packing material is...Ch. 6 - Prob. 6.31PCh. 6 - Prob. 6.32PCh. 6 - Assume the resistive force acting on a speed...Ch. 6 - Review. A window washer pulls a rubber squeegee...Ch. 6 - A motorboat cuts its engine when its speed is 10.0...Ch. 6 - You can feel a force of air drag on your hand if...Ch. 6 - A car travels clockwise at constant speed around a...Ch. 6 - The mass of a roller-coaster car, including its...Ch. 6 - A string under a tension of 50.0 N is used to...Ch. 6 - Disturbed by speeding cars outside his workplace,...Ch. 6 - A car of mass m passes over a hump in a road that...Ch. 6 - A childs toy consists of a small wedge that has an...Ch. 6 - A seaplane of total mass m lands on a lake with...Ch. 6 - An object of mass m1 = 4.00 kg is tied to an...Ch. 6 - A ball of mass m = 0.275 kg swings in a vertical...Ch. 6 - Why is the following situation impossible? A...Ch. 6 - (a) A luggage carousel at an airport has the form...Ch. 6 - In a home laundry dryer, a cylindrical tub...Ch. 6 - Prob. 6.49APCh. 6 - A basin surrounding a drain has the shape of a...Ch. 6 - A truck is moving with constant acceleration a up...Ch. 6 - The pilot of an airplane executes a loop-the-loop...Ch. 6 - Review. While learning to drive, you arc in a 1...Ch. 6 - A puck of mass m1 is tied to a string and allowed...Ch. 6 - Because the Earth rotates about its axis, a point...Ch. 6 - Galileo thought about whether acceleration should...Ch. 6 - Figure P6.57 shows a photo of a swing a ride at an...Ch. 6 - Review. A piece of putty is initially located at...Ch. 6 - An amusement park ride consists of a large...Ch. 6 - Members of a skydiving club were given the...Ch. 6 - A car rounds a banked curve as discussed in...Ch. 6 - In Example 6.5, we investigated the forces a child...Ch. 6 - A model airplane of mass 0.750 kg flies with a...Ch. 6 - A student builds and calibrates an accelerometer...Ch. 6 - A 9.00-kg object starting from rest falls through...Ch. 6 - For t 0, an object of mass m experiences no force...Ch. 6 - A golfer tees off from a location precisely at i =...Ch. 6 - A single bead can slide with negligible friction...Ch. 6 - Prob. 6.69CPCh. 6 - Because of the Earths rotation, a plumb bob does...
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