### My Journal

Maths Grinds
28 Nov 2018
##### Question 1

Show how to find the centre of gravity of a non-rectangular parallelogram.

A thin uniform square sheet of metal $ABCD$ of area $4$ square inches weighs $4$ ounces. At the corners $A$, $B$, $C$, $D$ weights of $1$ oz., $2$ oz., $3$ oz., and $4$ oz., respectively are placed. Locate the centre of gravity of the system.

##### Question 2

Establish the formulae $v=r\omega$ and $a=r\omega^2$ for the velocity and acceleration of a particle moving with uniform speed in a circle or radius $r$.

Assuming that the moon travels in a circular orbit whose radius is $3.8 \times 10^10$ cm. and whose centre ie the center of the earth, and completes a revolution every $2.4 \times 10^6$ seconds, calculate the ratio of its acceleration $g^1$ towards the centre of the earth to gravity $g$ at the earth’s surface.

##### Question 3

Two indentical cars $A$ and $B$ each $16$ feet in length travel with constant velocity $60$ feet per second along a straight level road, the front of $A$ being $44$ feet directly behind the rear of $B$. Immediately on reaching a line drawn across the road perpendicular to the direction of motion of the cars each car decelerates at $12$ feet per sec$^2$.

(i) Show that $A$ crashes into $B$.

(ii) At what distance from the line does the crash occur?

##### Question 4

An aeroplane $A$ about to drop a bomb on a small stationary target $T$. on the sea surface descends along a line which is inclined at an angle of $30^\circ$ to the horizontal and which passes directly over the target at a vertical height of $\frac{1}{2}$ mile. If the aeroplane travels at $600$ m.p.h. what duration of time must elapse between the release of the bomb and its impact on the target. (See diagram). ##### Question 5

Show that the work done by a body in coming to rest is the kinetic energy of the moving body. A body of mass $20$ lbs. sliding on a rough horizontal plane with initial velocity $22$ feet sec$^{-1}$. is brought to rest by friction in $15$ feet. Calculate the coefficient of friction.

At what average rate in horse-power was work done on the body in bringing it to rest?

##### Question 6

A particle executes simple harmonic motion with a frequency of $60$ vibrations per second. The amplitude is $1.5$ 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 $10$ grams and oscillates with simple harmonic motion at a frequency of $20$ vibrations per second, and with an amplitude of $1.5$ millimetres, calculate the maximum value of the driving force.

##### Question 7

A train is rounding a circular curve of radius $440$ 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 $4^\circ18’$ to the vertical. Estimate the speed of the train.

##### Question 8

A fire-engine raises $200$ gallons of water per minute from a well $16$ feet deep and then pumps it as a jet issuing with a nozzle velocity of $44$ feet per sec. At what horsepower is the engine working?

##### Question 9

The density of lead is $11.3$ gram cm$^{-3}$. and that of mercury is $13.6$ grams cm$^{-3}$. Explain why a block of lead will float on mercury.

A rectangular uniform block of wood floats on sea water with $\frac{1}{3}$ 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 $100$ lb. placed vertically above its centre of gravity?

(Take \$1.03 as the specific gravity of sea-water.)

##### Citation:

State Examinations Commission (2023). State Examination Commission. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru

Malone, D and Murray, H. (2023). Archive of Maths State Exams Papers. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/

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