UC-NRLF
LIBRARY
OF THE
UNIVERSITY OF CALIFORNIA
GIFT OK
Received ) and "weight," (W). D. 56, 57.
2.* Distinguish three classes of levers, stating which of the
three forces is in the middle in each case. D. 56, 57.
3.* In which class of levers is the weight always greater
than the power ? always less ? sometimes greater and sometimes
less ?
4.* Which force in each class of levers is equal to the sum of
the other two ?
5.* State the law of levers. Q. XVI.
6.f Show that each of the three forces in equilibrium in the
lever is proportional to the distance between the lines of action
of the other two.
7.^ Describe one or more experiments with the arithmetical
lever, and state what points each illustrates. D. 25.
8.* What is meant by a bent lever? D. 57.
9. Distinguish the "arms " of a bent lever from the " arms "
of the couples which are in equilibrium. D. 57.
10* When are the forces acting on a bent lever in equili-
brium ? Q. XVI.
n.J Describe one or more experiments (with E. H. HALL'S
apparatus) illustrating the equality of moments in the bent lever.
23
XVIII.
COMPOSITION AND EQUILIBRIUM OF COUPLES IN
PARALLEL PLANES.
i.f Describe the condition of the rod of a torsion apparatus
during torsion.
2.f Show that a rod subject to torsion can offset the action of
a couple.
3. When a rod is twisted by a couple acting at one end of it,
what is the nature of its reaction against the agent through
which the couple is produced ? (Axiom).
4. Show (by considerations of symmetry) that a uniform
free rod, subject to torsion, must react equally at both ends
against agents producing the torsion. (Axiom).
5.t What is meant by a " shaft " or " axle," and when is it
in equilibrium ?
6.f Show that couples, like forces, exist under four aspects,
and obey the same laws of action and reaction.
7.f Prove that a couple is transmitted without loss at right-
angles to its plane, from one section of a body to another, and
hence from one end of the body to the other.
8. What is meant by the transmissibility of couples along
their axes ? Prob.
9.f Show that the plane or axis of a couple has no definite
position in space, but is determined only by parallelism to a given
plane or axis. Prob.
io.f Show that the resultant of any number of couples with
parallel planes is a single couple, with plane parallel to those of
the components, and with a moment equal to the algebraic sum
of the moments of the components. D. 27, Q. XIII. and XVIII.
1 1. 1 Describe one or more experiments (due to E. H. HALL)
illustrating the conditions of equilibrium between couples in
parallel planes.
i2.f Show that a couple is completely specified when its
moment and general direction of rotation are known. Prob.
24
XIX.
COMPOSITION AND RESOLUTION OF COUPLES IN
NON-PARALLEL PLANES.
i.f Find the locus of all antipodal points in a given spherical
surface at which tangential forces can be applied so as to produce
a couple equivalent to a given couple.
2.f Find two antipodal points in a given spherical surface
where tangential forces can be applied so as to produce two
couples equivalent to any two given couples in non-parallel
planes.
3.f Show that the parallelogram of forces can be applied to
the composition of couples in non-parallel planes.
4-f Prove that the resultant of two couples in non-parallel
planes is a single couple in some intermediate plane.
5. Apply the Pythagorean proposition to the relation between
the moments of two component couples in planes at right-angles,
and the moment of their resultant. Prob.
6.f Show that a couple can be resolved into components, the
first parallel to any plane, and the second making a right-angle
(or any other angle) with the first.
7. Find the trigonometric relation between a couple and its
(rectangular) component in any plane, making the angle A with
the plane of the couple. Prob.
8,f Find the component of a couple in any plane whose
normal makes an angle, A, with the axis of the couple. Prob.
9-t What is meant by the moment of a force about a given
axis? and what distinction is made between positive and negative
moments ?
io.f Show that the algebraic sum of the moments of two equal
and opposite forces about any axis is equal to the component of
the couple in a plane at right- angles to this axis.
n.f Show that, in a state of equilibrium, the algebraic sum of
the components of all the forces along any axis, and also the
algebraic sum of their moments about this axis must be zero.
XX.
WRENCHES.
1. Show that every force acting upon a body (whether part
of a couple or not) can be resolved into a force with line of action
passing through any given point, P, and a couple. D. 28, 29 ;
Q. XIV. i.
2. What is the nature of the resultant in the last question,
of all the forces passing through a single point, /*? Of all the
couples ? D. 29 ; Q. VII., XVIII., XIX.
3t Show that all possible forces acting on a body can be
resolved and recompounded into a single force with line of
action passing through a given point and a single couple.
D. 29; Q. 2.
4.f Show that the couple in the last question can be resolved
into two components, one at right-angles to the force, the other
parallel to it, and state the result of combining one of these
components with the force. D. 29, Q. XIX. 6, XIV. 4.
5.f Show that all possible forces acting on a body (whether
parts of couples or not), are equivalent to a single force with
definite line of application, and a single couple at right-angles
with this force. D. 29, Q.
6. What name is given to a force combined with a couple at
right-angles to it ? D. 29, 30.
7. Define the word "wrench" as used in (theoretical)
mechanics. D. 29, 30.
8. Show that all possible combinations of forces are reducible
to a wrench. D. 29, 30 ; Q.
9-f Distinguish right and left-handed wrenches, according to
the direction of rotation, as looked at in the direction of the
force.
26 WRENCHES. [XX.
io.f Show that the reaction against a right or left-handed
wrench, being equal and opposite in respect to both force and
couple, is a wrench of the same kind, as concerns the relation
between linear and rotatory displacement.
n.f Distinguish wrenches farther into two classes, according
to whether extension or compression is produced.
I2.f Discuss the wrenches exerted in certain familiar mechan-
ical processes (such as driving a screw).
13-f Show that extension in one part of an elastic system
implies compression in some other part.
14. In what respect are the wrenches exerted by two different
parts of an elastic system on each other similar, and in what
respect dissimilar ? Q.
i5.f Discuss the longitudinal and torsional effects of mutual
wrenches in the ordinary spring curtain-roller.
1 6. State the law of action and reaction as applied to
wrenches. D. 30.
2 7
XXI.
ELASTICITY OF SOLID BODIES.
i.f Name three effects of a wrench upon an elastic body.
2.* Distinguish stretching, bending, twisting, and com-
pression.
3. What is meant by elasticity ? by limits of (perfect)
elasticity? by a .set ? or permanent strain ? D. 126.
4. Define a stress, a strain, and a coefficient or modulus of
elasticity. D. 128.
5. 1 Describe one or more experiments illustrating effects of
longitudinal forces upon wires or rods.
6.f State the law connecting the stretching of a rod with the
force applied, with the length of the rod, and with its area of
cross-section.
7. What is meant by YOUNG'S modulus of elasticity?
D. 128.
8.f Give some idea of the relative magnitudes of the forces
necessary to produce a given stretch in rods of different materials,
but of the same dimensions.
g.f State the usual effect of stretching on the diameter of a
rod, and what is meant by " POISSON'S ratio."
io.f What connection (if any) exists between the relative
magnitudes of the forces necessary to produce a given amount of
stretching, and the relative magnitudes of those required to
produce a given amount of bending in bodies of given
dimensions ?
1 1. 1 Describe one or more experiments relating to the laws
of bending.
28 ELASTICITY OF SOLID BODIES. [XXI.
12. f State the laws connecting the bending of a. beam with
the load, and with the length, breadth, and thickness of a beam.
134 Describe one or more experiments relating to the laws of
torsion.
i4-t State the laws connecting the twisting of a rod with its
length, with its breadth, and with the magnitude of the couple
applied.
15. f Give some idea of the relative magnitudes of the couples
necessary to produce a given amount of twisting in bodies of the
same dimensions, but different materials.
i6.f What is meant by a modulus or coefficient of torsion ?
17. What is meant by elasticity of volume? by resistance to
compression? by compressibility ? D. 129.
i8.f Explain how the compressibility of solids (as well as
liquids) can be demonstrated by an (OERSTED'S) piezometer.
19. f What connection (if any) exists between YOUNG'S
modulus, the modulus of torsion, and the resistance to compres-
sion for a given material ?
20. Show that HOOKE'S law applies to cases of stretching,
bending, twisting, and compression.
XXII.
CENTRE OF GRAVITY.
i.f When is a force applied to a rigid body said to have a
definite point of application ? (D. 34).
2.f What is meant by a centre of force? by the centre of two
or more forces ?
3-t Distinguish between a true centre of force, as for instance
an ideal atom, and a virtual centre of force, as for instance the
centre of Saturn's rings.
4. Show that the centre of two parallel forces (not equal and
opposite) having definite points of application lies in the line
joining these points of application. D. 34.
5. Where is the point of application of the resultant of two
parallel forces (not equal and opposite) ? D. 34.
6. Show that if any number of parallel forces (whose
algebraic sum is not zero) have definite points of application,
they give rise to a resultant with a definite centre or point of
application. D. 34.
7. Show that the forces which the earth's gravity exerts
upon the molecules of a small body at the earth's surface, having
definite points of application, and being (practically) parallel,
must have a definite centre or point of application. D. 33, 34.
8. What name is given to the centie or ] cint of i f pl.'catic n
of the (parallel) forces which gia\ ity exerts i yon the different
elementary particles of a body ? D. 34.
9. Define the term " centre of gravity," and state whether
it is applicable to solids only, or also to liquids and gases.
D- 34. 35-
io.f Show that if masses are measured by the forces which
gravity exerts upon them, the centre of gravity coincides (by
definition) with the "centre of mass."
30 CENTRE OF GRAVITY. [XXII.
n.f Would the statement in the last question hold if the
masses were not measured by the forces which gravity exerts
upon them ? or if the forces in question were not parallel ?
12. Under what circumstances does the centre of (the forces
due to) attraction coincide with the centre of mass ? Q.
13. Under what conditions does the resultant of two or more
forces have a definite centre or point of application ? Q.
14. Where is the centre of gravity of two equal masses? of
two unequal masses ? D. 34.
15. f Show that the centre of gravity of two, three, or four equal
masses is at the geometrical centre of this line, triangle, or
pyramid which they define. (D. 37-39).
i6.f Show that three or more unequal masses must have a
fixed centre of gravity, independent of the order in which they
are considered. (D. 37).
17. Show (by geometry) that in a system of mass, M,
consisting of two masses, m^ and m 2 , at the heights, /i, and // 2 ,
the height, H, of the centre of the gravity is such that
MH=m^ h^ -j- m 2 h 2 . D. 22, 45.
1 8. Extend the principle of the last question to the case of
any number of masses. D. 23, 45.
19. Show that the principle of the last two questions applies
not only to the heights, H, h^ h z , etc., of the centre of gravity
and separate masses composing a system, but also to their
distances, D, d^ d 2 , etc., from any plane. D. 22, 23.
XXIII.
CENTRE OF GRAVITY OF GEOMETRICAL FIGURES.
1. What assumption (as to the distribution of weight) is
made in calculating the position of the centre of gravity of bodies
approximating to geometrical figures ? D. 35.
2. What is meant by the centre of gravity of a geometrical
solid? surface? or line ? D. 35.
3.* Show that the centre of gravity of a unifoim thin rod (or
line) is at its centre, and that it will balance about any axis
passing through this centre. D. 36.
4.f Show that the centre of gravity of two equal triangles or
pyramids, symmetrically situated with respect to their common
apex, is at this apex.
5. Prove that if a figure can be cut up into strips, slices,
pairs of triangles, pairs of pyramids, etc., each having its centre
of gravity at or in a given point or axis, the centre of gravity of
the whole figure must be at or in this point or axis. Prob.
6. Show that if any figure is symmetrical with respect to a
centre (so that it can be cut up into equal and opposite pairs of
pyramids, with common apex at this centre) its centre of gravity
must lie at this centre. Prob. D. 36. Q. XVIII.
7. Show that if any figure is symmetrical with respect to -an
axis, (so that it can be cut up into equal and opposite pairs of
triangles with common apex on this axis) its centre of gravity
must lie on this axis. Prob. (Q. XXIII.)
8. Show that if any figure is symmetrical with respect to a
plane (so that it can be cut up into uniform thin strips, each with
its centre of gravity in this plane), the centre of gravity of the
figure must lie in this plane. Prob. (Q. XXIII.)
9. Where is the centre of gravity of a figure symmetrical
with respect to two axes ? an axis and a plane ? two planes ?
three planes ? Prob.
32 CENTRE OF GRAVITY OF GEOMETRICAL FIGURES. [XXIII.
10. Find the centre of gravity of a circle, square, ellipse,
parallelogram, cube, block, parallelepiped, prism, cylinder,
sphere, ellipsoid, etc. Prob. D. 36.
11. Show (by cutting up a triangle into strips parallel to the
base, and considering the centre of gravity of each strip) that
the centre of gravity of a triangle must lie in its medial
line. D. 37.
12. Show that the centre of gravity of a triangle must lie at
the point of intersection of its medial lines, and find (by
geometry) the relative altitudes of the apex and centre of gravity.
D. 37, Q. ii.
13. Show (by cutting up a pyramid into sections parallel to
the base) that if a line be drawn from the apex to the centre of
gravity of the base (so as to pass through the centre of gravity
of each section), this line will contain the centre of gravity of
the pyramid. D. 38.
14. Show that the centre of gravity of a pyramid must lie at
the point of intersection of two or more lines drawn as in the
last question, and find (by geometry) the relative altitudes of
the centre of gravity and apex of a triangular (or any other)
pyramid. D. 38.
15. Find the centre of gravity of a cone by the same method
as in the last question. D. 38.
33
XXIV.
CENTRE OF GRAVITY OF SUSPENDED BODIES.
i.f Find the moment of the force exerted by gravity on a
body of weight, W, about a horizontal axis in a vertical plane at
the distance, D, from the centre of gravity. (D. 58).
2.f A body of weight, W, and centre of gravity at the
horizontal distance, D, from a fulcrum (or axis) is balanced by a
force, P, acting on a horizontal arm, A. Find the equal and
opposite moments. (D. 58).
3. In the last question, express each of the four quantities,
W, Z>, P, and A, in terms of the other three. Prob. Q. 3.
4.J How can you find the weight of a body by balancing it
on an axis by means of a known weight ? Q.
5-J How can you locate the centre of gravity of a body of
known weight when balanced on an axis with another body of
known weight ? Q.
6.J Show how the weight of any object, great or small, can
be found by a graduated lever of known weight. Q.
7-1 Show how the position of a bullet of known weight in a
gun-barrel of known weight can be found from the displacement
of the centre of gravity. Q.
8. * Describe one or more experiments with levers showing
that their weight acts always as if concentrated at their centre of
gravity.
9.* Explain and describe one cr mure experiments illustrating
the behavior of bodies suspended at their centre of gravity.
10. Show that a body is in equilibrium if suspended at any
point in a vertical line passing through the centre of gravity.
D. 41.
IT.* What is the result of suspending a body at a point not
contained in the vertical line passing through the centre of
gravity ?
34 CENTRE OF GRAVITY OF SUSPENDED BODIES. [XXIV.
12. State the condition of equilibrium in a body suspended
at a point. D. 41.
13.* Describe the experimental location of the centre of
gravity of a body by the method of suspension. D. 43, 44.
14. J Describe and explain one or more experiments in which
a body is suspended upon a horizontal axis in a vertical plane
containining the centre of gravity, and state how the centre of
gravity may be located by such experiments.
15.* What is the result of suspending a body upon a horizontal
axis not containing the centre of gravity ?
1 6. What is the result of suspending a body upon an oblique
axis containing the centre of gravity ? not containing it? Prob.
17.* Describe the condition of a body suspended about a
vertical axis.
1 8. State the conditions of equilibrium of a body suspended
upon an axis, in terms of the position of the centre of gravity
with respect to this axis. D. 41.
19. Show that in all the cases above, where a body is not in
equilibrium, the action of gravity is such as to cause the centre
of gravity to descend. D. 16.
20. Show that a heavy body is in equilibrium when, and only
when, the nature of its restraints is such as to prevent the centre
of gravity from descending. Prob.
2 1. 1 Explain the action of a double-pointed cone in rolling up
two inclined, but diverging rails.
22. Explain the action of a ball in rolling into the deepest
portion of a hollow. D. 46.
23. Show that, in general, the centre of gravity tends to
descend. D. 46.
35
XXV.
STABILITY, AND ITS RELATIONS TO CENTRE OF
GRAVITY.
1. Show that a body, in which two points are fixed, is
equivalent to one suspended upon an axis passing through these
points. Prob.
2. How many points in a body must be fixed to prevent
movement of any kind ? Prob.
3.f Three points in the same horizontal plane are fixed. Find
the direction of the force on each, according to the position of
the centre of gravity.
4. Describe the forces by which a tripod is usually held in
place, stating the direction of these forces, and whether they
vary uniformly or " per saltum " under displacement. D. 40, 52.
5. State the conditions under which a body may be fixed by
the upward reaction of three forces. Q.
6.f What name is given to the area of a horizontal section
included between the forces in the last question ?
7.* Define, in general, the "area of support" of a bod) T . Q.
8.f Show that a body is stable if its centre of gravity lies
above its area of support. (D. 40).
9. What is meant by the limit of stability in a body
supported at three or more points ? D. 54.
10. Show that a body, displaced within its limits of stability,
tends to return to its original position. Prob.
11. When is a body said to be " practically stable ?" D. 54.
12. What, in general, is meant by stability? D. 42.
13. Distinguish stable, unstable, and neutral equilibrium,
according to the tendency of a body when displaced to return to
its original position of equilibrium. D. 42.
36 STABILITY AND CENTRE OF GRAVITY. [XXV.
14. When is a heavy body, suspended by a point or by an
axis, in stable equilibrium ? in unstable equilibrium ? in neutral
equilibrium? State the position of the centre of gravity
relatively to the point or axis of suspension in each case.
D. 41, 42.
15. Explain the toy called " the balancer." D. 53.
1 6. Show that, in a condition of stability, the height of the
centre of gravity is at a minimum, in a condition of instability
at a maximum. D. 46.
17. State the conditions which determine whether equilibrium
exists, and whether it is stable or unstable, in terms of the path
of the centre of gravity consequent upon a given displacement
of the body. D. 46.
1 8. Find the state of equilibrium in a sphere or cylinder
rolling on a flat, on a concave, and on a convex surface. Prob.
(D. 46).
19. Describe and explain the action of a toy known as " the
tumbler," or any other similar device. D. 53.
37 XXVI.
SENSITIVENESS OF A BALANCE AS RELATED TO
CENTRE OF GRAVITY.
i. Name the essential parts of a balance. D. 70.
2.J Describe the beam of an ordinary balance, the knife-
edges and their bearings. D. 75.
3.J Describe the pointer or index of a balance, the scale-pans,
and the means for arresting the loads, or lifting them from the
knife-edges.
4. Explain the importance of preserving the sharpness of
the knife-edges, and the ordinary precautions taken to this end.
(D. 75).
5. Where is the centre of gravity of a balance-beam, and how
is the position of the centre of gravity adjusted ? D. 73.
6. Show that the weights act as if concentrated at the knife-
edges. Prob. Q. XXII. i.
7. Show that if the three knife-edges be in the same straight
line, the centre of gravity is not disturbed by loading the scale-
pans so as to produce equilibrium. Prob. Q. XXII. 5.
8. Show that in a straight-arm balance, we have for the
angle, a, of deflection due to excess of weight, p, on aim of
length, /, of the beam of weight, w, with its centre of gravity at
a distance, d, from the axis,
tan a pi -j- wd. D. 73.
9-f What additional terms must be considered if the three
knife-edges are not in the same straight line ? D. 74.
10. Show that if the outer knife-edge be above the middle
knife-edge, the equilibrium of the balance may become unstable
from loading. D. 74.
11. Show that if the outer knife-edges be below the central
knife-edge, the equilibrium of the balance will become more
stable on loading. Prob. D. 74.
12. Distinguish three types of balance, according to align-
ment of the knife-edges. Q.
13. Show that a balance beam may belong, successively, to
the three several types named in the last question, through
bending under an increasing load. Prob.
i4.f What is meant by the sensitiveness of a balance ?
38 SENSITIVENESS OF A BALANCE. [XXVI.
15. State certain conveniences in the use of a balance of
known sensitiveness. D. 74.
i6.f Describe the method of weighing under a constant load,
and state some of the advantages and disadvantages of this
method. D. 74.
ly.J How is the sensitiveness of a balance under a given load
ordinarily determined?
i8.f Explain the use and construction of a table showing the
sensitiveness of a balance under different loads.
ig.J Describe the use of a " rider," and show that in weighing
with a rider, the sensitiveness of a balance need not be kngwn.
20. J Explain the method of weighing by "oscillations" (or
by "vibrations "), and state some of its advantages.
2i.f Show that friction, within certain limits, does not
necessarily affect the sensitiveness of a balance, if the method of
oscillations is employed.
22. f State what advantages are to be gained by using a balance
with rapid oscillations, to what extent sensitiveness should be
sacrificed to this end, and to what extent sensitiveness is
consistent with rapidity. D. 71, 74.
23. f Show that it is convenient, but not necessary, that the
pointer should indicate how much the loads differ.
24. Correct DESCHANEL'S statement (73) that the centre of
gravity and axis of a balance " must not coincide." Q.
25. What qualities are sought for in a balance? D. 71, 73.
Q. XXVI. 1-26.
26.* Describe the ordinary "steelyard," andshowthat equality
of the arms of balance is not necessary in order that the true
weight of a body may be found. D. 76.
27. Describe (BORDA'S method of) weighing by substitution,
(or double weighing, according to DESCHANEL and other French
writers). D. 72.
28. J Describe (what is meant by most English writers by) the
method of "double weighing," (or GAUSS'S method, involving
an interchange of the loads).
29. f Show that the method of double weighing (by inter-
change) is twice as accurate as weighing by substitution.
30. J Describe, in detail, the processes actually employed in
very accurate weighings.
"-.'
Qg THE
39
XXVII.
WORK, AND ITS RELATIONS TO CENTRE OF
GRAVITY.
i.* What name is given to the product of a force (when
constant in magnitude and in direction) and the displacement of
its point of application (in the same direction) ?
2.* Define work, and name some of the units in which it is
measured.
3.* Define the foot-pound and kilogram-metre.
4.f Define the foot-poundal, and erg (or dyne-centimetre).
5.* Find the work, W, necessary to raise a weight, w,
through the vertical distance, d. D. 45, 47.
6.* Distinguish between the work done by a force upon a
weight or against gravity, and that done by gravity or by the
weight against the force. D. 47.
7. How is work measured when the directions of the force
and displacement are nut the <-ame ? D. 45, 48.
8. The point of application, A, of a force, AB, moves to C.
What kind of work is done by the force if BAC is acute? if
BAG is obtuse ? if BAC is a right-angle ? D. 48.
9. Find the work done by a force, AB, in producing the
displacement, AC, in terms of AB, AC, and a function of the
angle, BAC. D. 48.
10. Show that the work done by a force in producing a given
displacement may be measured (i) by the product of the displace-
ment and the component of the force in the direction of the
displacement, or (2) by the product of the force and the
component of the displacement in the direction of the force.
Prob. Q.
40 WORK AND CENTRE OF GRAVITY. [XXVII.
11. Show that the work necessary to move a weight in any
direction is the product of the weight and the vertical component
of the displacement. D. 48.
12. A body of weight, W, remains fixed, while another body,
of weight, w, moves through the distance, D. Find (by
geometry) the displacement, d y of the centre of gravity of the
two weights. Prob.
13. Compare, in the last question, the components of the
two displacements in the vertical (or any other) direction. Prob.
i4.f Show that the product of a weight, forming part of a
system, by its vertical displacement is equal to the product of the
weight of the whole system by the vertical displacement of its
centre of gravity. Q.
i5-t What is meant by the work done upon the centre of
gravity of a system ?
i6.f Show that the algebraic sum of the quantities of work
done upon the separate weights of a system (by considering these
one by one) is equal to the work done upon the centre of
gravity of the system. Prob.
17. State the "principle of work " as applied to the centre of
gravity of a system of weights. D. 45.
1 8. Bxplain how the calculation of work done upon a body
can be simplified by the principle of the last question. D. 45.
19. Find the work, W, necessary to pile a given weight, w,
of fragments into a rectangular, into a triangular, or into a
pyramidal heap of height, h, above the (mean) level of the
fragments. D. 45.
20. Modify the last question so as to apply to the case of
filling cisterns ot different shapes with liquids, or exhausting
wells of different depths and shapes. Prob.
XXVIII.
WORK, AS A CRITERION OF STABILITY.
1. Find the work, W, necessary to turn a block of length, /,
breadth, b, thickness, /, and weight, w, from its position of
maximum stability into its position of medium or minimum
stability. Prob. D. 45.
2. Modify the last question so as to apply to an ellipsoid.
Prob.
3. A barrel of weight, w, rests upon a floor in stable
equilibrium, because its centre of gravity is at a distance, d,
below the axis : find the work, W, necessary to start the barrel
rolling. Prob. Q.
4.* State some of the most important properties of the centre
of gravity. Q. XXII.-XXVII.
5. Show that work must be done upon a body to displace it
from a position of stable equilibrium. D. 50.
6. Show that work is done by a body when it is displaced
from a position of unstable equilibrium. D. 50.
7. Show that no work is done upon or by a body when it is
displaced from a position of neutral equilibrium. D. 50.
8. State the criterion of the stability of equilibrium in terms
of the total work done upon or by a body when displaced from
its position of equilibrium. D. 50.
9f What condition of equilibrium is assumed to exist in an
ideal lever ?
10. t State, in general, what kind of equilibrium exists in
various ideal machines or " mechanical powers."
XXIX.
PRINCIPLE OF WORK APPLIED TO MECHANICAL
POWERS.
, by geometry, that the work spent by a force acting
upon an ideal lever is equal to the work utilized in the weight
raised.
. 2.* State the " principle of work " (or " virtual velocities ")
as applied to the lever. D. 49.
3.* What assumptions (in regard to loss of work by friction,
etc.) are made in the investigation of ideal machines? (D. 49).
4. What distinction is made between the force known as
the "power" and the "weight" in various machines? D. 59.
5. What is meant by mechanical advantage ? D. 59.
6. Name some of the most important " mechanical powers."
D. 55-
7. Show that the wheel and axle may be regarded as an
endless lever. D. 60.
8. Find (by the principle of work) the relation between the
power, weight, and radii of a wheel and axle. Prob.
9. How would the results obtainable with a wheel and axle
be modified by the thickness of the ropes, if these were perfectly
flexible ? D. 60.
ro.f In what way would the stiffness of the ropes of a wheel
and axle modify the result if the ropes were perfectly elastic, so
that no work need be spent in bending them ?
ii. Show that a pulley may be regarded as an endless lever.
D. 61.
i2.f What assumption as to the tension of the cords passing
round one or more pulley-wheels is made in calculating the
mechanical advantage of ideal pulleys ?
43 MECHANICAL POWERS. [XXIX.
13.* What mechanical advantage is gained by a single fixed
pulley ? D. 61.
i4.f What allowance must be made for the weight of a
movable pulley ? for the weight of the cords ?
154 How is the influence of the weight of pulleys and cords
eliminated in experimental demonstrations ?
1 6.* State the mechanical advantage of a single movable
pulley, to which the weight is attached, assuming the cords
parallel. D. 61.
17.* State in general the mechanical advantage of a pulley
niOv'ed by x cords with uniform tension in the same direction.
Prob. D. 62.
1 8.* Show that the principle of work applies to the case of a
pulley with x cords, as in the last question. Prob. D. 61.
19. Find the mechanical advantage of a series of x single
pulleys, each attached to the power cord of the next. Prob.
D. 63.
20. f What is meant by an ideal smooth surface or plane? and
what limitations exist in the direction of the force which such a
surface is capable of exerting upon a body ?
21% Describe one or more experiments with inclined planes
approximating to the ideal conditions.
22. Explain the resolution of forces in a body resting on a
perfectly smooth inclined plane. D. 64.
23. A body of weight, w, rests on an inclined plane of length,
AC, base, AB, and height, BC: find the force by which it is held
in place, (i) if this force is tangential, (2) if the force is
horizontal. Prob. Q.
24. Show that the principle of work applies to the forces
known as the " power " and the " weight " in an inclined plane.
D. 65, 66.
25. Show that a wedge may be regarded as a special case of
inclined plane. D. 67.
44 MECHANICAL POWERS. [XXIX.
26. Find the mechanical advantage of an ideally smooth
wedge of length, /, and thickness, /. Prob. Q.
27.* Show that a screw can be treated as a special case of
inclined plane. D. 68.
28.* Find the mechanical advantage of a screw of circum-
ference, c, and distance, d, between threads measured parallel to
the axis of the screw, if the force is applied at the circumference,
at right-angles to the axis. Prob. Q.
29.* By what means is the mechanical advantage of a screw
greatly increased in ordinary screw presses ? D. 69.
30.* Find the mechanical advantage of a screw press with
arms of length, /, and threads, at the distance, d. D. 69.
3i.*fKstimate, approximately, the force which can be brought
to bear in an ordinary screw press, neglecting friction.
32.* Show that the principle of work applies to the screw
press.
33. Show that all the mechanical powers are reducible to two
types (the lever and the inclined plane).
34.* State, in general, the application of the principle of
work to mechanical powers. D. 49.
35.* What is meant by the statement that " what is gained in
power is lost in speed ?" Prob. Q.
36.* An ideal machine is actuated by a force, /% acting through
a distance, D, and produces a force,/", acting through a distance,
d ; express each of the four quantities, /% /" D, and d, in terms
of the other three.
37.* In the last question, state what must be, and what need
not be known about the construction of the machine, in order
that the principle of work may be applied. Q.
45
XXX.
CONSERVATION OF WORK.
i.f A weight, w, is moved through the distances, AB, BC,
. . . YZ, all lying in the same vertical line. Show that the work
done is the same (w X AZ}, whatever may be the position of
the intermediate points, B, C, etc.
2. Prove that if a weight, w, is moved along any broken
path, A" B" C" . . . Z" , the work done is the same as along the
projection of this path, A' B' C' . . . Z 1 ', on a vertical line, and
hence, equal to w X A'Z'. Prob.
3. Show that in returning from Z" to A", the weight gives
out the same amount of work that it received between A" and
Z". Prob. D. 123.
4. Show that work done against constant forces, like those
exerted by gravity, depends only upon the initial and final
positions of a body, and is independent of the path of the body.
Q. (D. 123).
5. Show that when a body returns by any " closed path " to
its original position, the total work done upon it, or by it, is
zero. Q. (D. 123).
6.f Discuss, from the point of view of the last question, the
possibility of various proposed forms of "perpetual motion."
(D. 49)-
7. State the relation between the work received by a body
in one part of a closed path, and that given out by it in the
remainder of its path. Q.
8.*fWhat is meant by the "conservation of work" in
-mechanics ?
XXXI.
LOSS OF WORK BY FRICTION.
i.* What element, entering into the working of all practical
machines, causes a departure from ideal conditions ? D. 49,
125, 131.
2.* Define (kinetical) friction (D. 131), and state the
peculiarity of sign in the work to which it gives rise.
3-t Describe and explain one or more experiments showing a
great discrepancy between the force required to start a body
sliding (no matter how slowly) over a surface, and that required
to maintain the motion. (D. 131, 132).
4. | State the results of one or more experiments showing
whether the force required to drag a body over a surface is or is
.not affected by the velocity. (D. 131).
1 5.f Why is it necessary, in measuring forces due to friction,
that bodies should be moved with uniform velocity ?
6. 1 State the results of one or more experiments showing the
relation between the force required to drag a body over a surface,
and the (normal) force urging the body and the surface together.
D. 131.
7. J What is the result of varying the ?rea of rubbing
surfaces, without varying either their nature or the force urging
them together ?
8. What is meant by a ' ' coefficient of friction ?" D. 131, 132.
9-J Describe one or more experiments in which the coefficient
of friction between two horizontal surfaces is determined.
10. A body of weight, W, requires a force, w, to drag it with
uniform velocity, along a horizontal surface ; find the coefficient
of friction between the body and the surface. Prob. Q.
47 LOSS OF WORK BY FRICTION. [XXXI.
ii. What is meant by the "limiting angle of friction " for
two surf aces? D. 132.
1 2. J Explain the experimental determination of coefficents of
friction by means of an inclined plane. D. 133.
13. A body slides with uniform velocity down an inclined
plane with base, AB, and height, BC: find the coefficient of
friction. D. 133.
14. f Give some idea of the magnitude of coefficients of friction
between wooden or metallic surfaces, and the effect of grease or
oil upon these coefficients.
15. t Show that the mechanical advantage of a wedge cr screw,
no matter how fine the pitch, cannot exceed the reciprocal of the
coefficient of friction.
i6.f Find a relation between the weight, W, of a truck, the
diameters, D, and d, of the wheel and axle, the coefficient of
friction,/, of the (loose) bearing upon the axle, and the force, F,
required to pull the truck. Illustrate by a numerical example.
ry.f Find the difference between the forces, F' and F" , trans-
mitted by parallel cords to and from a pulley of diameter, Z?,
with axle of diameter, d, and coefficient of friction, /, on its
loose bearing, (calling the force borne by the pulley F' -\- F").
i8.f Trace out the effect in a double block (with forr cords)
of a loss of 10 per cent., 20 per cent., etc., in the tension of each
cord passing round each wheel, on the force required to lift a
given weight by the block, remembering that the weight raised
is given by the sum of the tensions on the cords.
ig.f Find, conversely, the weight on the block necessary to
overcome a given force at one end of the cord.
20. f Explain the enormous loss of work in long trains of
clock-work, and the surprising effects of oil upon these.
XXXII.
EFFICIENCY.
i. J Describe one or more experiments showing that the work
spent upon the cord of a tackle in raising a weight is greater
than that utilized through the pull upon this cord exerted by the
descending weight.
2.t What name is given to the ratio of the work spent to the
work utilized in a given case ?
3. Define "efficiency," as used in mechanics. Q. 2.
4.f Distinguish between the efficiency of a tackle as a
machine for raising weights, its efficiency as a machine for
utilizing work done by the descent of a weight, and its efficiency
as a machine for storing useful work.
5.f State a certain necessary relation between the three
efficiencies in the last question.
6. A man slides a box up into a wagon along a board inclined
at the limiting angle of friction : (Q. XXXI.) find the efficiency
of his combination. Prob.
7. Show that the efficiency of a screw, irreversible through
friction, cannot exceed 50 per cent. Prob.
8.f Show, from a point of view of efficiency, the disadvantage
of .a screw with too fine a thread (or of a differential screw).
49
XXXIII.
POWER.
i. J Describe one or more experiments with an ergometer (or
friction-brake), showing how it is possible to measure the work
spent by a given agent in a given time.
2.J Describe a "transmission dynamometer," and its uses.
3.f What name is given to the quotient of work by time? to
the product of force and velocity ?
4.f Distinguish power, in its technical sense, from power as a
name for one of the forces in a machine.
5f Define pow r er in two different ways, and show that these
definitions are identical. Q. 3.
6.f Define "horse-power" (English or French), the erg per
second, and the "watt."
7-t Show that the horse-power varies slightly in different
latitudes, but that the erg and the watt are constant.
8.f Find the horse-power of a locomotive with two cylinders
100 square inches in section, and double two-foot stroke, making
125 revolutions per minute under a mean pressure of 33 Ibs. of
steam per square inch, in both the forward and backward stroke,
making no allowance for friction.
g.f Find the power, P, spent upon a water-motor with piston
of area, a (sq. cm.), making n double strokes of length, /, (cm.)
in every second, under a pressure, p (dynes per sq. cm.), and
state in what units the result is expressed.
io.f Show that the power, P, spent upon a motor (by an
incompressible liquid) is equal to the product of the pressure, p,
per unit of area, and current, c, in units of volume per unit of
time.
ii.t Show that (as a consequence of the principle of the last
question), if pressure is transmitted by an incompressible fluid
without loss, power must also be transmitted without loss.
XXXIV.
SOLIDS AND FLUIDS, DISTINGUISHED.
1. 1 Describe an experiment (with OERSTED'S piezometer),
showing that liquids are compressible. D. 130.
2.* Distinguish fluids from solids (in respect to their relative
resistance to, and limits of recovery from, changes of form or
shape unaccompanied by changes of bulk or volume). (D. 129).
3.f Distinguish fluids from solids in respect to their rates of
yielding to tangential or transverse forces, and state whether this
distinction is one of kind or of degree.
4.f What name is given to that property in solids which is
connected with their slow yielding under transverse stresses ?
5.f When is a solid, and when is a fluid said to be especially
viscous? (D. 135).
6.f Show that viscosity is used in two opposite senses in its
applications to solids and fluids.
y.f Distinguish frictional forces within the body of a fluid
(whether due to viscosity or not) from forces due to the rubbing
of two solid surfaces, in respect to their dependence upon (i) the
compression, (2) the area, and (3) the relative velocity of the
moving parts.
8.* State some of the characteristic distinctions between
solids and fluids. Q.
XXXV.
LIQUIDS AND GASES DISTINGUISHED.
1. 1 Describe an experiment (with a gas-bag and air-pump)
illustrating the indefinite expansibility of a gas. D. 194.
2.* Distinguish fluids into two classes, according to their
tendency toward uniform distribution throughout any space
within which they are confined. D. 194.
3.* How do liquids compare, as a class, with gases, in respect
to density ?
4.* How do liquids compare, as a class, with gases, in respect
to compressibility ?
5.f What is meant by the " free surface " of a fluid, and what
kind of fluids alone present such surfaces under ordinary
conditions ?
6.J Describe one or more experiments (e.g., with floating
needles), showing that the surfaces of liquids resist deformation
to a slight extent. D. 159.
7.J Describe one or more experiments (with camphor and
water, oil films on water, and the effects of local heat or alcohol)
showing that slight tangential forces are exerted by the surfaces
of liquids. D. 192.
8. J Describe one or more experiments (with PLATEAU'S films)
showing that the surface of a film is a minimum consistent with
its boundary. (D. 186).
9. Explain the spherical shape of rain-drops or bubbles.
D. 189.
10. J Describe one or more experiments showing that films of
liquid, when free to do so, cause their boundaries to contract.
ii. State the influence of the length, the breadth, and the
thickness of a film on the force with which it contracts
longitudinally. D. 185.
52 LIQUIDS AND GASES DISTINGUISHED. [XXXV.
12. What is meant by ".surface tension ?" D. 185.
13. How many surfaces is a (soap-bubble) film considered to
possess ? and how is the surface tension of such a film measured ?
D. 188.
14.* Describe one or more experiments illustrating adhesion
between a liquid and a solid which it wets.
15.* Describe the surfaces of liquids of two different kinds,
near the edges of the vessels containing them. D. 182.
1 6.* What is meant by a capillary tube ? by capillary forces?
and by capillarity in general ?
17.* Describe the rise and fall of liquids caused by capillary
tubes. D. 182.
1 8.* When will a liquid rise in a capillary tube, and when will
it be depressed by it ? D. 182.
19. Describe the " meniscus " (or curved surface) of a liquid
accompanying (i) capillary ascensions and (2) capillary depres-
sions. D. 182, 190.
20. Kxplain the rise of liquids in capillary tubes which they
wet, and their depression by capillary tubes which they do not
wet. D. 1 86.
21. State some of the conditions which influence the amount
of the rise or fall of a liquid in a capillary tube. D. 183, 184.
22.f What is meant by the height of the meniscus (or
sagitta) ?
23. What is meant by the angle of contact between a liquid
and a solid? D. 184, 185.
24. Find the height, /i, of a column of liquid of density, d,
sustained by a surface tension (in gravitation units) t, in a tube
of radius, r, assuming that the tube is wet by a film of liquid
tangent to its surface. D. 186.
25. State the law of diameters governing capillary ascensions
and depressions. D. 184.
26. J Describe one or more experiments (e.g., with inclined
plates) illustrating the law of diameters.
53 LIQUIDS AND GASES DISTINGUISHED. [XXXV.
27. f How high are liquids known to rise by capillary action in
certain vegetable structures ?
28. f Give reasons for supposing that a practical limit exists in
the height attainable by capillary action.
29. J Describe an experiment with an air-pump, showing that
capillary ascensions and depressions are not due to atmospheric
pressure. (D. 185).
30.J Describe one or more experiments illustrating the tensile
strength under certain specified conditions, of columns of liquid
of considerable cross section.
31.* In what class of fluids, alone, are phenomena due to
capillarity, cohesion, or surface tension perceptible ? Q.
32.* State some of the characteristic distinctions between
liquids and gases. Q.
33. 1 Describe one or more experiments illustrating effects of
endosmose (or diffusion through a diaphragm) and endosmotic
pressure. D. 193.
34. Distinguish solutions into two classes with respect to the
facility with which they pass through a diaphragm. D. 193.
35. Describe the process of separating colloids and crystalloids
by "dialysis." D. 193.
36. Describe one or more experiments illustrating the diffusion
of gases, and the pressures to which such diffusion may give
rise. D. 193, 227.
54
XXXVI.
FUNDAMENTAL ASSUMPTIONS IN HYDROSTATICS.
i.f What assumptions are usually made in the solution of
problems in hydrostatics, with respect to viscosity, capillarity,
osmosis, etc. ?
2. Why does not viscosity enter into problems is hydrostatics?
D. 135-
3-t State reasons for assuming that capillary forces may be
neglected in the treatment of bodies of liquid of considerable
size, or in the case of solids of considerable size immersed in
these liquids.
4.J Give experimental grounds for the assumption that forces
or pressures due to osmosis or to diffusion are not perceptible
in the wide channels usually employed in hydrostatics.
5-t Show that, in the absence of viscosity and capillarity,
normal forces and pressures are the only ones which need be
taken into account. (D. 135).
6. Distinguish hydrostatics from hydrokinetics, treating
both as branches of hydrodynamics. D. 134.
7. Distinguish pneumatics from hydrostatics. D. 3.
8.f To what extent are the properties of gases studied under
the head of hydrostatics ?
Q.f State certain obvious experimental evidence of the fact
that fluids, when undisturbed for a long time, fall (practically)
into a state of equilibrium.
io.f What general principle would lead to the anticipation of
the result stated in the last question ?
u.f What hypothesis (as to the equilibrium of fluids) lies at
the foundation of hydrostatics ?
55
XXXVII.
CONDITIONS OF EQUILIBRIUM IN FLUIDS.
i.f Show (by considerations relating to the centre of gravity)
that a bubble, contained in a liquid within any enclosure, seeks
the highest possible point.
2. Describe the construction and use of an ordinary spirit-
level, its attachment to telescopes with cross-hairs, the conditions
of, and one or more processes of testing its sensitiveness, and the
method of eliminating errors of adjustment. D. 180, 181.
3. State the condition of equilibrium in the free surface of a
liquid. D. 140.
4. Show, by the resolution of forces, that if the free surface
of a fluid is not horizontal, it is not in equilibrium. D. 140.
5.f Show, by considerations relating to the centre of gravity,
that the bounding surface between any two fluids of unequal
density is in equilibrium only when horizontal. (D. 145).
6. 1 Describe one or more experiments illustrating the equili-
brium of bounding surfaces between different fluids. D. 145.
7. Show that the equilibrium of a fluid would not be
disturbed if any portion of it should become rigid, or should be
replaced by a fixed rigid body. D. 153.
8.f Show that it is possible, by imagining rigid supports
substituted for certain portions of a body of fluid, to treat the
remainder as if contained in a /-tube, or in communicating
vessels of any shape, without any change of equilibrium.
9.* State the condition of equilibrium of a liquid contained in
communicating vessels. D. 178.
i o.*J Describe one or more experiments illustrating the fact
that a liquid stands at the same level in communicating vessels.
D. 178.
56 CONDITIONS OF EQUILIBRIUM IN FLUIDS. [xXXVII.
ii. Explain the construction and use of a water-level.
D. 179.
i2.f Show (by the principle of action and reaction) that the
resultant of all the forces exerted by a fluid upon the walls of its
enclosure must be equal and opposite to the resultant of the forces
exerted by the walls of the enclosure upon the fluid, and hence
(by the fundamental principles of equilibrium) equal to the
weight of the fluid. (D. 148).
i3-t Show that the weight of any portion of a fluid, of
whatever shape, is equal and opposite to the resultant of the
forces exerted upon this portion by the body of fluid which
surrounds it.
14. Show that the difference between the forces exerted upon
the top and bottom of a column of fluid with vertical sides (from
which the column is supposed to derive no support) must be equal
to the weight of the column. D. 139.
15. Show that the resultant forces upon the two ends of a
fluid prism with horizontal axis must be equal and opposite.
D. 138.
1 6. Find by the triangle of forces, the relation between the
resultant forces upon the three rectangular faces of a triangular
fluid prism of negligible weight, and show (by similar triangles)
that the three forces are proportional to the areas of the three
faces in question. D. 137.
57
XXXVIII.
PRESSURE.
i.f What is meant by intensity of pressure? and in what
units is it expressed ? D. 136.
2.f In what sense is the word " pressure," when unqualified,
used by most modern writers ? and how 7 does this differ from
another sense in w 7 hich it was sometimes employed by earlier
writers? D. 136.
3-f Find the (intensity of) pressure, p, (expressed in gravi-
tation units) exerted by a weight, w, upon a horizontal surface
of area, a.
4. A weight, w, of fluid is contained in a tube with vertical
sides, and with an area of cross-section, a. Show that if the
upper surface is free from force, the weight of the fluid is borne
by the bottom of the tube, alone ; and find the pressure due to the
weight of the fluid upon the bottom of the tube, supposing it to
be horizontal. Prob. Q. XXXVII.-XXXVIII.
5. Find the weight, w, of fluid of density, d, standing at a
height, h, in a tube with vertical sides and horizontal base of
area, a ; find also the pressure upon the bottom of the tube.
Prob. Q. XXXVII.-XXXVIII.
6. Find, as in the last question, the pressure, p, at the
bottom of a vertical column of fluid of density, d, and depth, h,
when in equilibrium with the surrounding fluid. Prob. Q.
7. Show (by the principle of action and reaction) that the
upward pressure beneath any horizontal area, a, in the body of a
fluid, must be equal to the downward pressure on this area, due
to the fluid above it. Prob.
8. \ Describe one more experiments showing that the upward
pressure at a given depth in a liquid is equal to the downward
pressure due to a column of liquid of the same depth and density.
D. 144.
5 8 PRESSURE. [XXXVIII.
9. Prove (by the conditions of equilibrium of the pressures
on the faces of a small triangular prism, with axis horizontal)
that the pressure upon any surface of given area at a given depth
in a fluid is the same as upon a horizontal surface of the same
area, and at the same depth. D. 137. Q. XXXVII. 15.
10.* Explain the statement that the pressure of a fluid is the
same in all directions. D. 137.
n. Prove (by considering the conditions of equilibrium of
the forces on the ends of a prism with horizontal axis) that the
pressure at two points at the same level in a fluid must be the
same. D. 138.
12. Prove (by considering the conditions of equilibrium of a
column of fluid with vertical sides) that in descending through
the vertical distance, h, in a liquid of density, d, the pressure
(measured in gravitation units) increases by the amount, hd,
whether the starting point is in the free surface of the liquid or
not. D. 139.
I3.| Describe one or more experiments (with communicating
tubes) illustrating the equality of pressure between columns of
liquid of given vertical height in tubes having different
inclinations.
14. Show (by a zigzag of horizontal and vertical prisms) that
the pressure at two points on the same le\el in communicating
vessels must be the same. Prob. D. 139, 147.
15. Show (as in the last question) that the pressure of a fluid
of density, d, at a depth, 7z, below a given surface is greater than
at this surface by the amount, hd, regardless of the shape of the
vessel. D. 139, 147.
i6.*J Describe one or more experiments illustrating the fact
that the force exerted on a given area by a liquid of given depth
and density is independent of the shape of the vessel containing
the liquid. D. 147.
I7.J Describe one or more experiments {e.g. , with E. H.
HAUL'S pressure-gauge) illustrating the fact that the pressure
59 PRESSURE. [XXXVIII.
in a liquid increases with the depth, and is the same in all
directions.
1 8. Discuss the applicability of the principles of hydrostatic
pressure to the case of liquids in capillary tubes. D. 190.
19. Show that the pressure of a liquid just within its curved
surface in a capillary tube must differ considerably from that just
outside of it, and show that the difference is explainable by
surface tension. D. 190, 191.
20. Find the sign of the pressure of a liquid raised in vacuo by
a capillary tube, on the assumption that its pressure outside of
this tube, is zero. D. 190.
21. Explain the attraction between two parallel plates, or
between two floating bodies of which (i) both are wet, or (2)
neither is wet by the liquid. D. 190.
6o
XXXIX.
BALANCING COLUMNS.
i. *J Describe an experiment in which water is introduced into
one branch of a /-tube containing some mercury. (D. 146).
2.* If the water in the last question reaches from the level, a,
to the level, b, and the mercury reaches from the level, b, down
to the bottom of the /-tube and up again to the level,