Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

the magnitude and law of force tending to its centre round which it would revolve in the same manner if the cohesion of the particles were destroyed.

6. Compare the centrifugal forces at similar points of similar curves, round centres of force similarly situated.

7. If a body describe a semicircle by a force tending to a point so remote that all lines drawn from the circle to it may be considered parallel, shew that the force at any point P varies reciprocally as PM3. [Section II. Prop. 8.]

8. If the lines drawn to the circle from the centre of force, in the same proposition, be only very nearly parallel, shew that the force at any point P must be increased by a quantity which varies

[blocks in formation]

9. The area which a body describes round a centre of force in the time tptv, p being the perpendicular corresponding to the velocity v.

10. The force in any conic section tending to the centre of force

CG3 ; C being the centre, and CG parallel to the distance RP RP2 meeting the tangent at P in G; [Scholium, Section III.] Prove

this, and shew that if R be the vertex of an ellipse, the force oc

RP

RN

PN being perpendicular to the major axis.

11.

There are three equal bodies, two of which describe circles round the third fixed, in such a manner that the bodies are always in the same right line: compare the radii of the circles, with the

1

three bodies attracting each other with forces which vary as dist.

12. If the central force of a body moving in a parabola round the focus, were to cease acting at the vertex, and continue interrupted till the body had described an angle of 60° above the focus; find the eccentricity of the orbit it would afterwards describe.

ST. JOHN'S COLLEGE, 1826.

1. A SEGMENT of a sphere: its inscribed cone :: 3 : 2 ultimately.

2. Shew how to find when the curves whose equations are obtained by altering one constant in a given equation are similar, and find whether the curves formed by giving different values to a in the equation y — ax + ab = 0 are similar.

3. If the Earth be struck by a comet of a given mass, which moves in the plane of the ecliptic, and the perihelion distance of which is equal to the radius of the Earth's circular orbit, find the alteration produced in the length of the year.

4. If the least distance of the comet be not quite equal to the radius of the Earth's orbit, find how the length of the day is altered supposing the ecliptic and equator to coincide.

5. Find the equation to the curve in which the centrifugal and centripetal forces are in a given ratio.

6. Compare the forces at similar points of similar curves; the force acting in parallel lines and the velocities in a direction perpendicular to that in which the force acts being the same in the different curves.

7. Compare the periods in two equal circles round the same centre of force which is situated in the centre of one circle, and in the circumference of the other.

8. If a body move from one point to another acted on by any centripetal force, shew that the velocity is independent of the curve in which it moves.

9. The velocity is also independent of the curve if the body be acted on by any number of centripetal forces. Does the proposition hold for all forces not centripetal ?

10. Explain Newton's method of finding the angle between the apsides in an orbit nearly circular. Find the angle between the

apsides in the curve whose equation is

1

r

= C cos.yv + C' sin.yv, r

being the radius vector, v the angle, and C, C', y constants.

11. Find the disturbing force of the Sun perpendicular to the plane of the lunar orbit.

12. If a body move freely on the surface of a solid of revolution acted on by forces tending to all points of the axis, shew that the angular velocity round the axis varies inversely as the square of the distance from the axis.

13. Explain why the Sun acting on the protuberant matter of the Earth's equator causes a precession of the equinoxes.

ST. JOHN'S COLLEGE, DEC. 1826.

1. THE ultimate ratio of evanescent quantities is a ratio of equality, when their difference is an evanescent of an higher order than themselves; explain this phrase, and shew that the rule is contained in the words of the first Lemma; the term data differentia being understood a quantity of finite magnitude.

2. Shew from the preceding question that the axiom, "If equals be taken from equals, the remainders are equal," does not necessarily hold when applied to the Geometry of the first Section, and give an example.

3. Define similar curves when referred to polar co-ordinates;

r

prove that curves represented by the equation 0 = 92

a

are similar, and hence determine on what condition different conchoids will be similar.

4. In a curve of finite curvature, the chord, arc, and tangent, are ultimately in a ratio of equality.

5. AB is the chord of an arc ACB of finite curvature, take AC: CB always as mn; join AC, BC, and prove that the

[blocks in formation]

6. Given three points in an orbit and the three corresponding angular velocities; to find the centre of force.

7. Find the law of force, that the velocities in different circles about the same centre may be the same.

8. A body revolving in a sets out from A; required to draw the chord AF, along which a body descending by the force in the circumference acting uniformly parallel to the radius CA, may meet the revolving body again in F.

9. Prove fully that the force in any curve varies inversely as SY2. PV; when SY is the on the tangent, and PV the chord of curvature through the centre of force. And apply the expression to find the law of force in the reciprocal spiral.

10. If a body be projected from a given point with the same velocity in different directions, and be acted on by a force tending to a given point, which varies inversely as D°; find the curve, which is the locus of the centres of all the ellipses described.

11. Prove that when the velocity in any curve equals velocity in the e.d, the angular velocities of the radius-vector and on the tangent are equal, and find when this takes place in a O, force not in the centre.

12. Force to C varies as distance. A body is projected from P along the line zPy, with velocity = m. times velocity in rad. CP. Draw PK = m. CP, making the ZzPK = CPy and join CK. Shew that the axis-major of the ellipse described bisects the LPCK, and complete the construction, so as to determine the magnitude of the axes.

13. A body is revolving in an ellipse; the centre of force is suddenly transferred from the focus G to H. Shew that if at the same instant, the body be so situated, that the normal divides SH in extreme and mean ratio, it will describe a parabola, and find its latus-rectum.

14. If a comet describing a parabolic orbit inclined to the plane of the ecliptic, be seen to pass over the Sun's disc, and of a year after, to strike the planet Mars, whose distance from the Sun is that of the Earth; required the distance of the comet from the Earth at the first observation in parts of the radius of the Earth's orbit, supposed circular, and its plane coincident with that of Mars.

ST. JOHN'S COLLEGE, MAY 1827.

1. PROVE that the force in any orbit varies as limit

[blocks in formation]

QR is the subtense of an arc drawn parallel to the force, and T is the time of moving through it.

2. Find the law of force parallel to the base, by which a body may move in a cycloid, and find when the force is least.

3. Required the position of a plane, against which a corpuscle revolving in a circle, at a distance equal twice the Earth's radius must impinge, that after reflection it may just pass round the Earth, elasticity being perfect; and find the time it remains within its former orbit.

4. Determine what Newton's trochoid in the sixth section becomes, when the axis minor of the ellipse vanishes; and reconcile the construction with that given in the seventh section, for the time in a descent towards a centre of force, which varies inversely as the square of the distance.

1 D3

5. oc Force A body is projected with a velocity which is to

that in a circle at the same distance as √2:1, and in a direction making an angle of 30° with the distance; find the equation to the orbit reckoned from the asymptote, draw the asymptote and find the time of descent to the centre.

6. If the fixed orbit, which is a parabola, force in focus, be projected in antecedentia with an angular velocity equal to half that of the body in the fixed orbit; required to trace the curve formed; find its point of contrary flexure, and the point where the forces in the two orbits are as 2: 1.

7. Equal absolute forces are placed in A and B which vary as the distance. Required the direction and velocity with which a body must be projected from a given point, so as to describe an ellipse having A and B in the foci.

8. Investigate an approximate expression for the ablatitious force in Prop. 66, and shew that it equals three times the addititious force × sine of the angular distance of P from quadratures.

« ΠροηγούμενηΣυνέχεια »