University Physics Volume 1
18th Edition
ISBN: 9781938168277
Author: William Moebs, Samuel J. Ling, Jeff Sanny
Publisher: OpenStax - Rice University
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Textbook Question
Chapter 10, Problem 36P
The angular position of a rod varies as
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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 10 Solutions
University Physics Volume 1
Ch. 10 - Check Your Understanding The fan blades on a...Ch. 10 - Check Your Understanding A centrifuge used in DNA...Ch. 10 - Check Your Understanding A boy jumps on a...Ch. 10 - Check Your Understanding A nuclear submarine...Ch. 10 - Check Your Understanding What is the moment of...Ch. 10 - Check Your Understanding A large ocean-going ship...Ch. 10 - Check Your Understanding The fan blades on a jet...Ch. 10 - Check Your Understanding A constant torque of...Ch. 10 - A clock is mounted on the wall. As you look at it,...Ch. 10 - What is the value of the angular acceleration of...
Ch. 10 - A baseball bat is swung. Do all points on the bat...Ch. 10 - The blades of a blender on a counter are rotating...Ch. 10 - If a rigid body has a constant angular...Ch. 10 - If a rigid body has a constant angular...Ch. 10 - If the angular acceleration of a rigid body is...Ch. 10 - A massless tether with a masses tied to both ends...Ch. 10 - Explain why centripetal acceleration changes the...Ch. 10 - In circular motion, a tangential acceleration can...Ch. 10 - Suppose a piece of food is on the edge of a...Ch. 10 - What if another planet the same size as Earth were...Ch. 10 - A solid sphere is rotating about an axis through...Ch. 10 - If a child walks toward the center of a...Ch. 10 - A discus thrower rotates with a discus in his hand...Ch. 10 - Does increasing the number of blades on a...Ch. 10 - The moment of inertia of a long rod spun around an...Ch. 10 - Why is the moment of inertia of a hoop that has a...Ch. 10 - What three factors affect the torque created by a...Ch. 10 - Give an example in which a small force exerts a...Ch. 10 - When reducing the mass of a racing bike, the...Ch. 10 - Can a single force produce a zero torque?Ch. 10 - Can a set of forces have a net torque that is zero...Ch. 10 - Can a set of forces have a net force that is zero...Ch. 10 - Prob. 25CQCh. 10 - If you were to stop a spinning wheel with a...Ch. 10 - Prob. 27CQCh. 10 - Calculate the angular velocity of Earth.Ch. 10 - A track star runs a 400-m race on a 400-m circular...Ch. 10 - A wheel rotates at a constant rate of...Ch. 10 - A particle moves 3.0 m along a circle of radius...Ch. 10 - A compact disc rotates at 500 rev/min. If the...Ch. 10 - Unreasonable results. The propeller of an aircraft...Ch. 10 - A gyroscope slows from an initial rate of 32.0...Ch. 10 - On takeoff, the propellers on a UAV (unmanned...Ch. 10 - The angular position of a rod varies as...Ch. 10 - A wheel has a constant angular acceleration of...Ch. 10 - During a 6.0-s time interval, a fly-wheel with a...Ch. 10 - The angular velocity of a rotating rigid body...Ch. 10 - A flywheel slows from 600 to 400 rev/min while...Ch. 10 - A wheel 1.0 m in diameter rotates with an angular...Ch. 10 - A vertical wheel with a diameter of 50 cm starts...Ch. 10 - A circular disk of radius 10 cm has a contant...Ch. 10 - The angular velocity vs. time for a fan on a...Ch. 10 - A rod of length 20 cm has two beads attached to...Ch. 10 - At its peak, a tornado is 60.0 m in diameter and...Ch. 10 - A man stands on a merry-go-round that is rotating...Ch. 10 - An ultracentrifuge accelerates from to 100,000 rpm...Ch. 10 - A wind turbine is rotating conterclockwise at 0.5...Ch. 10 - What is (a) the angular speed and (b) the linear...Ch. 10 - A child with mass 40 kg sits on the edge of a...Ch. 10 - A bicycle wheel with radius 0.3 m rotates from...Ch. 10 - The angular velocity of a flywheel with radius 1.0...Ch. 10 - A system of point particles is shown in the...Ch. 10 - (a) Calculate the rotational kinetic energy of...Ch. 10 - Calculate the rotational kinetic energy of a 12-kg...Ch. 10 - A baseball pitcher throws the ball in a motion...Ch. 10 - A diver goes into a somersault during a dive by...Ch. 10 - An aircraft is coming in for a landing at 300...Ch. 10 - If air resistance is present in the preceding...Ch. 10 - A neutron star of mass 21030kg and radius 10 km...Ch. 10 - An electric sander consisting of a rotating disk...Ch. 10 - A system consists of a disk of mass 2.0 kg and...Ch. 10 - While punting a football, a kicker rotates his leg...Ch. 10 - Using the parallel axis theorem, what is the...Ch. 10 - Find the moment of inertia of the rod in the...Ch. 10 - A uniform rod of mass 1.0 kg and length 2.0 m is...Ch. 10 - A pendulum consists of a rod of mass 2 kg and...Ch. 10 - A solid sphere of radius 10 cm is allowed to...Ch. 10 - Calculate the moment of inertia by direct...Ch. 10 - Two flywheel fo negligible mass and different...Ch. 10 - The cylindrical head bolts on a car are to be...Ch. 10 - (a) When opening a door, you push on it...Ch. 10 - When tightening a bolt, you push perpendicularly...Ch. 10 - What hanging mass must be placed on the cord to...Ch. 10 - A simple pendulum consists of a massless tether 50...Ch. 10 - Calculate the torque about the z-axis that is out...Ch. 10 - A seesaw has length 10.0 m and uniform mass 10.0...Ch. 10 - A pendulum consists ofa rod of mass 1 kg and...Ch. 10 - A torque of 5.00103Nm is required to raise a...Ch. 10 - A horizontal beam of length 3 m and mass 2.0 kg...Ch. 10 - What force must be applied to end of a rod along...Ch. 10 - What is the torque abot the origin of the force if...Ch. 10 - You have a grindstone (a disk) that is 90.0 kg,...Ch. 10 - Suppose you exert a force of 180 N tangential to a...Ch. 10 - A flywheel (l=50kgm2) starting from rest acquires...Ch. 10 - A constant torque is applied to a rigid body whose...Ch. 10 - A torque of 50.0Nm is applied to a grinding wheel...Ch. 10 - A flywheel (I=100.0kgm2) rotating at 500.0 rev/min...Ch. 10 - A uniform cylindrical grinding wheel of mass 50.0...Ch. 10 - Suppose when Earth was created, it was not...Ch. 10 - A pulley of moment of inertia 2.0kgm2 is mounted...Ch. 10 - A block of mass 3 kg slides down an inclined plane...Ch. 10 - The cart shown below moves across the table top as...Ch. 10 - A uniform rod of mass and length is held...Ch. 10 - A thin stick of mass 0.2 kg and length L=0.5m is...Ch. 10 - A wind turbine rotates at 20 rev/min. If its power...Ch. 10 - A clay cylinder of radius 20 cm on a potter’s...Ch. 10 - A uniform cylindrical grindstion has a mass of 10...Ch. 10 - A uniform disk of mass 500 kg and radius 0.25 m is...Ch. 10 - A propeller is accelerated from rest to an angular...Ch. 10 - A sphere of mass 1.0 kg and radius 0.5 m is...Ch. 10 - A uniform rod of length L and mass M is held...Ch. 10 - An athlete in a gym applies a constant force of 50...Ch. 10 - A 2-kg block on a frictionless inclined plane at...Ch. 10 - Small bodies of mass m1 and m2 are attached to...Ch. 10 - A cyclist is riding such that the wheels of the...Ch. 10 - Calculate the angular velocity of the orbital...Ch. 10 - A phonograph turntable rotating at 331/3rev/min...Ch. 10 - With the aid of a string, a gyroscope is...Ch. 10 - Suppose a piece of dust has fallen on a CD. If the...Ch. 10 - A system of point particles is rotating about a...Ch. 10 - Calculate the moment of inertia of a skater given...Ch. 10 - A stick of length 1.0 m and mass 6.0 kg is free to...Ch. 10 - A pendulum consists of a rod of length 2 m and...Ch. 10 - Calculate the torque of the 40-N force around the...Ch. 10 - Two children push on opposite sides of a door...Ch. 10 - Prob. 118APCh. 10 - An automobile engine can produce 200Nm of torque....Ch. 10 - A grindstone with a mass of 50 kg and radius 0.8 m...Ch. 10 - The angular acceleration of a rotating rigid body...Ch. 10 - Earth’s day has increased by 0.002 s in the last...Ch. 10 - A disk of mass m , radius R , and area A has a...Ch. 10 - Zorch, an archenemy of Rotation Man, decides to...Ch. 10 - A cord is wrapped around the rim of a solid...
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