College Physics
1st Edition
ISBN: 9781938168000
Author: Paul Peter Urone, Roger Hinrichs
Publisher: OpenStax College
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Chapter 11, Problem 25PE
What force must be exerted on the master cylinder of a hydraulic lift to support the weight of a 2000-kg car (a large car) resting on the slave cylinder? The master cylinder has a 2.00-cm diameter and the slave has a 24.0-cm diameter.
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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 11 Solutions
College Physics
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How much force is exerted on one side of an 8.50...Ch. 11 - What pressure is exerted on the bottom of a...Ch. 11 - Calculate the average pressure exerted on the palm...Ch. 11 - The left side of the heart creates a pressure of...Ch. 11 - Show that the total force on a rectangular dam due...Ch. 11 - How much pressure is transmitted in the hydraulic...Ch. 11 - What force must be exerted on the master cylinder...Ch. 11 - A crass host pours the remnants of several bottles...Ch. 11 - A certain hydraulic system is designed to exert a...Ch. 11 - (a) Verify that work input equals work output for...Ch. 11 - Find the gauge and absolute pressures in the...Ch. 11 - (a) Convert normal blood pressure readings of 120...Ch. 11 - How tall must a water-filled manometer be to...Ch. 11 - Pressure cookers have been around for more than...Ch. 11 - Suppose you measure a standing person's blood...Ch. 11 - A submarine is stranded on the bottom of the ocean...Ch. 11 - Assuming bicycle tires are perfectly flexible and...Ch. 11 - What fraction of ice is submerged when it floats...Ch. 11 - Logs sometimes float vertically in a lake because...Ch. 11 - Find the density of a fluid in which a hydrometer...Ch. 11 - If your body has a density of 995 kg/m3, what...Ch. 11 - Bird bones have air pockets in them to reduce...Ch. 11 - A rock with a mass of 540 g in air is found to...Ch. 11 - Archimedes' principle can be used to calculate the...Ch. 11 - In an immersion measurement of a woman's density,...Ch. 11 - Some fish have a density slightly less than that...Ch. 11 - (a) Calculate the buoyant force on a 2.00-L helium...Ch. 11 - (a) What is the density of a woman who floats in...Ch. 11 - A certain man has a mass of 80 kg and a density of...Ch. 11 - A simple compass can be made by placing a small...Ch. 11 - What fraction of an iron anchor's weight will be...Ch. 11 - Scurrilous con artists have been known to...Ch. 11 - A twin-sized air mattress used for camping has...Ch. 11 - Referring to Figure 11.21, prove that the buoyant...Ch. 11 - (a) A 75.0-kg man floats in freshwater with 3.00%...Ch. 11 - What is the pressure inside an alveolus having a...Ch. 11 - (a) The pressure inside an alveolus with a 2.00104...Ch. 11 - What is the gauge pressure in millimeters of...Ch. 11 - Calculate the force on the slide wire in Figure...Ch. 11 - Figure 11.35(a) shows the effect of tube radius on...Ch. 11 - We stated in Example 11.12 that a xylem tube is of...Ch. 11 - What fluid is in the device shown in Figure 11.29...Ch. 11 - If the gauge pressure inside a rubber balloon with...Ch. 11 - Calculate the gauge pressures inside...Ch. 11 - Suppose water is raised by capillary action to a...Ch. 11 - Calculate the contact angle for olive oil if...Ch. 11 - When two soap bubbles touch, the larger is...Ch. 11 - Calculate the ratio of the heights to which water...Ch. 11 - What is the ratio of heights to which ethyl...Ch. 11 - During forced exhalation, such as when blowing up...Ch. 11 - You can chew through very tough objects with your...Ch. 11 - One way to force air into an unconscious person's...Ch. 11 - Heroes in movies hide beneath water and breathe...Ch. 11 - Gauge pressure in the fluid surrounding an...Ch. 11 - A full-term fetus typically has a mass of 3.50 kg....Ch. 11 - If the pressure in the esophagus is -2.00 mm Hg...Ch. 11 - Pressure in the spinal fluid is measured as shown...Ch. 11 - Calculate the maximum force in newtons exerted by...Ch. 11 - During heavy lifting, a disk between spinal...Ch. 11 - When a person sits erect, increasing the vertical...Ch. 11 - (a) How high will water rise in a glass capillary...Ch. 11 - A negative pressure of 25.0 atm can sometimes be...Ch. 11 - Suppose you hit a steel nail with a 0.500-kg...Ch. 11 - Calculate the pressure due to the ocean at the...Ch. 11 - The hydraulic system of a backhoe is used to lift...Ch. 11 - Some miners wish to remove water from a mine...Ch. 11 - You are pumping up a bicycle tire with a hand...Ch. 11 - Consider a group of people trying to stay afloat...Ch. 11 - The alveoli in emphysema victims are damaged and...
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