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14 Nov 2018

Leaving Certificate Examination 1957 Honours Applied Mathematics

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Question 1

If three co-planar non-parallel forces acting on a rigid body are in equilibrium, prove that their lines of action are concurrent and that the forces may be represented in magnitude and direction by the sides of a triangle taken in order.

A uniform bar quicklatex.com-bb11fdb4c82e898ea8147471416d775f_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds which weighs quicklatex.com-2b18fa15a2f284d990fd81b9cae524ce_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds lb. is supported by two strings quicklatex.com-371fe833e9a89e2790fc66adb618b944_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds, quicklatex.com-3675ba693016b80ac93fbf734e15678b_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds attached to a fixed peg at quicklatex.com-e86ae6aeacc8696c140ee1ca29d51f96_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds. If quicklatex.com-7b330dfb209da651a53c38169c2108e5_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds, quicklatex.com-2c27280c41528d19602df95964a65370_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds, quicklatex.com-a8856cf92524d2ce25911ac60741ad48_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds, find the tension in each of the strings.

Question 2

State the main laws of friction. Explain the term “coefficient of friction,” and describe briefly how its value for two given surfaces may be found by experiment.

A box, lying on the floor of a railway carriage, just begins to slide back along the floor as the carriage is descending an incline of quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds in quicklatex.com-2b18fa15a2f284d990fd81b9cae524ce_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds with a uniform acceleration of quicklatex.com-ed5decd492fdc6648538222379ceaf27_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds ft. per sec.quicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds. Show by diagram the forces acting on the box, and find the coefficient of friction between the box and the floor.

Question 3

Show that the moment of a couple is the same about all points in the plane of the couple. Explain how couples may be compounded, and show that a force and a couple acting in the same plane are equivalent to a single force.

Prove that four forces represented in magnitude, direction and position by the sides of a quadrilateral taken in order are equivalent to couple.

Question 4

(i) A block is sliding freely down the line of greatest slope of a smooth inclined plane. If the velocity of the block increases from quicklatex.com-1ee1c2caec46fc384f9e0e830a2e8b84_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds ft. per sec. to quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds ft. per sec. in a distance of quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds feet, find the slope of the plane.

(ii) A car which weigh quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds ton is descending an incline of quicklatex.com-d39c7a34226b6a7fc7b5d3ae51b6b5e4_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds in quicklatex.com-5291abebe16995c90201b72e133d15d8_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds ; the velocity of the car is quicklatex.com-2dda138cd42bede9cb97214fc5c8a205_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds m.p.h. and it is accelerating at the rate of quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds ft. per sec.quicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds. If the frictional resistances to motion are equivalent to quicklatex.com-1b5ffe190011734769d91db377610058_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds lb. wt., find the horse-power at which the car is working.

Question 5

A bullet weighing quicklatex.com-afd1938e804f165b249d5afbf9c5cefd_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds lb. is fired horizontally with a velocity of quicklatex.com-3c29056d21e2c67ab8d951fa0b157cf7_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds ft. per sec. from a gun which is free to recoil. If the gun weighs quicklatex.com-2dda138cd42bede9cb97214fc5c8a205_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds lb., find the velocity with which the gun begins to recoil, and find the total kinetic energy, in foot-lbs., of the gun and the bullet.

If the same bullet were fired from a gun weighing quicklatex.com-dc2e717421b16a9f080dba4c8c0e3437_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds lb., and the total kinetic energy was the same as above, show that the velocity of the bullet would be less by about quicklatex.com-1ee1c2caec46fc384f9e0e830a2e8b84_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds ft. per sec.

Question 6

A particle is describing a circle of radius quicklatex.com-95d7c225dd31ad7d6dd5fb9f0980b531_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds, with constant angular velocity quicklatex.com-d66dbaeb0eebc210a255b3fd356eacd0_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds: show that is acceleration is quicklatex.com-21b1f5d26e2a330299275bfe78d4329f_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds directed towards the centre of the circle.

A quicklatex.com-23b0ff706f1ddcdb8e00b5b72339e7a0_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grindsounce mass supported from a fixed point by a light inextensible string quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds feet long is describing a horizontal circle at a uniform rate of quicklatex.com-ce939b244df556d77c12e315c6165e7d_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds revolutions per second. Find the tension in the string in lbs. wt., and find the vertical distance from the fixed point to the plane of the circle.

Question 7

quicklatex.com-d613350484b582bc83e2a6f9473437f8_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds,quicklatex.com-6ce7c226153d55058c10348c6739a4c5_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds are two points at ground level quicklatex.com-2f45e5fc001351ff8b3c7ebace7c1802_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds feet apart. An aeroplane is flying at a steady height of quicklatex.com-63bf388232c3aa08847f9559026724e7_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds feet above the ground in the direction quicklatex.com-44ff489534c25536d47781ec27ee92fa_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds, with a uniform velocity of quicklatex.com-c5e1fff724fe36afee16a8302b93e856_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds ft. per sec. When the aeroplane is directly over quicklatex.com-6ce7c226153d55058c10348c6739a4c5_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds a shell is fired, at an angle of projection quicklatex.com-0dce58ae1ea930bb9c39e85ab5e4c977_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds, from a gun at quicklatex.com-d613350484b582bc83e2a6f9473437f8_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds and strikes the aeroplane quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds seconds later. If quicklatex.com-b97f42f845482bb5cff804f74afebb7f_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds, find the value of quicklatex.com-db2144d676b0c010fa61116d07db1982_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds and the initial velocity of the shell.

Question 8

A point quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds is describing a circle, centre quicklatex.com-5b090119e833ac90162777d07e3fe243_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds, at constant speed. If quicklatex.com-4635d0bb2c5d9acbe6771e05c8a643ef_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds is the foot of the perpendicular from quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds to a fixed diameter, show that quicklatex.com-4635d0bb2c5d9acbe6771e05c8a643ef_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds is moving with an acceleration which is directly proportional to is distance from quicklatex.com-5b090119e833ac90162777d07e3fe243_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds.

When quicklatex.com-4635d0bb2c5d9acbe6771e05c8a643ef_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds is quicklatex.com-fd6542ae3d49d38a0f73fefe83b285af_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds feet from quicklatex.com-5b090119e833ac90162777d07e3fe243_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds its velocity is quicklatex.com-ee873c946a42a316d07b2d164e2b7d67_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds ft. per sec. away from quicklatex.com-5b090119e833ac90162777d07e3fe243_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds, and its acceleration is quicklatex.com-3c4de7b9ce82b47f9788dc2416e4c750_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds ft. per sec.quicklatex.com-025303222356a63c2332a429c33d35a9_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds towards quicklatex.com-5b090119e833ac90162777d07e3fe243_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds. Find the amplitude and the period. How many seconds later does quicklatex.com-4635d0bb2c5d9acbe6771e05c8a643ef_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds reach quicklatex.com-5b090119e833ac90162777d07e3fe243_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds from that position? (Give your answer correct to the nearest tenth of a second.)

Question 9

If a plane lamina is immersed vertically in a liquid at rest, prove that the total thrust on it due to the liquid is equal to the area of the immersed surface multi[lied by the pressure at its centre of gravity.

A quadrilateral lamina quicklatex.com-0bc2407dfe67fc49000363238a97afd0_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds in which quicklatex.com-2bc8b92466fb8df4de567451d7b7b774_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds and quicklatex.com-01b08e0c92406c2e91d632ae5d546817_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds is immersed in water, with the vertex quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds at the surface and the vertex quicklatex.com-e86ae6aeacc8696c140ee1ca29d51f96_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds vertically below quicklatex.com-cba4133f939edee31a8ce51d790a9cd0_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds. Find the total thrust of the water on the lamina in lbs. wt. correct to one decimal place.

[A cubic foot of water weighs quicklatex.com-2a62b4fd8e04c54613a6683d3f1636ef_l3 | Leaving Certificate Examination 1957 Honours Applied Mathematics | Maths Grinds lb.]


Citation:

State Examinations Commission (2018). State Examination Commission. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru

Malone, D and Murray, H. (2016). Archive of Maths State Exams Papers. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/

Licence:

“Contains Irish Public Sector Information licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) licence”.

The EU Directive 2003/98/EC on the re-use of public sector information, its amendment EU Directive 2013/37/EC, its transposed Irish Statutory Instruments S.I. No. 279/2005, S.I No. 103/2008, and S.I. No. 525/2015, and related Circulars issued by the Department of Finance (Circular 32/05), and Department of Public Expenditure and Reform (Circular 16/15 and Circular 12/16).

Note. Circular 12/2016: Licence for Re-Use of Public Sector Information adopts CC-BY as the standard PSI licence, and notes that the open standard licence identified in this Circular supersedes PSI General Licence No: 2005/08/01.

Links:

https://circulars.gov.ie/pdf/circular/per/2016/12.pdf

https://creativecommons.org/licenses/by/4.0/legalcod

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