Show how to find the centre of gravity of a non-rectangular parallelogram.
A thin uniform square sheet of metal of area square inches weighs ounces. At the corners , , , weights of oz., oz., oz., and oz., respectively are placed. Locate the centre of gravity of the system.
Establish the formulae and for the velocity and acceleration of a particle moving with uniform speed in a circle or radius .
Assuming that the moon travels in a circular orbit whose radius is cm. and whose centre ie the center of the earth, and completes a revolution every seconds, calculate the ratio of its acceleration towards the centre of the earth to gravity at the earth’s surface.
Two indentical cars and each feet in length travel with constant velocity feet per second along a straight level road, the front of being feet directly behind the rear of . Immediately on reaching a line drawn across the road perpendicular to the direction of motion of the cars each car decelerates at feet per sec.
(i) Show that crashes into .
(ii) At what distance from the line does the crash occur?
An aeroplane about to drop a bomb on a small stationary target . on the sea surface descends along a line which is inclined at an angle of to the horizontal and which passes directly over the target at a vertical height of mile. If the aeroplane travels at m.p.h. what duration of time must elapse between the release of the bomb and its impact on the target. (See diagram).
Show that the work done by a body in coming to rest is the kinetic energy of the moving body. A body of mass lbs. sliding on a rough horizontal plane with initial velocity feet sec. is brought to rest by friction in feet. Calculate the coefficient of friction.
At what average rate in horse-power was work done on the body in bringing it to rest?
A particle executes simple harmonic motion with a frequency of vibrations per second. The amplitude is millimetres.
Find (i) the maximum speed of the particle,
(ii) the maximum acceleration of the particle.
If the cutting-head of an electric razor has mass grams and oscillates with simple harmonic motion at a frequency of vibrations per second, and with an amplitude of millimetres, calculate the maximum value of the driving force.
A train is rounding a circular curve of radius yards. A bob hangs at one end of a light string which is fixed in the ceiling of a carriage and which is inclined at an angle of to the vertical. Estimate the speed of the train.
A fire-engine raises gallons of water per minute from a well feet deep and then pumps it as a jet issuing with a nozzle velocity of feet per sec. At what horsepower is the engine working?
The density of lead is gram cm. and that of mercury is grams cm. Explain why a block of lead will float on mercury.
A rectangular uniform block of wood floats on sea water with of its volume above the water-line. What is the volume of the smallest rectangular block of the wood which will float without being totally submerged while supporting a weight of lb. placed vertically above its centre of gravity?
(Take $1.03 as the specific gravity of sea-water.)
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/
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