The University of New Zealand. — Entrance Examination, 1885—Junior Scholarships. — Elementary Mechanics

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Elementary Mechanics.

1. If two forces P and Q are inclined at an angle ⍺, prove that the magnitude R of their resultant is given by
• R2 = P2 + Q2 + 2 PQ cos .

If the resultant of two equal forces inclined at an angle a is twice the magnitude of the resultant when the forces are inclined at an angle π−, prove that

2. Weights of 8lbs, 10lbs, 12lbs are placed respectively at the angular points A B C of a triangle; prove that the centre of gravity of the system lies on a line drawn parallel to A C through G, the centre of gravity of the triangle. Where in this line is the required centre of gravity situated?

3. Explain the meaning of the terms acceleration, momentum, work, energy.

State the principle of the conservation of energy.

A rifle bullet strikes a target. What becomes of the kinetic energy of the bullet?

4. A bucket is lowered into a well with uniform velocity and after it has decended 25 feet it is struck by a stone let fall from the top of the well one second after the bucket started. What is the velocity with which the bucket is lowered?

5. State and prove the proposition called the parallelogram of velocities.

6. How is an impulse measured?

A 64lb shot is fired from a 10 ton gun; the velocity of the shot being 2000 feet a second. Find the velocity with which the gun begins to recoil. If it be brought to rest in a distance of 8 feet by friction, asumed to be uniform throughout the motion, calculate the friction.

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7. Find the ratio of the power to the weight in that system of pulleys in which each string is vertical and each attached to the weight.

8. What is meant by the specific gravity of a body?

The specific gravities of gold and copper are 19·3 and 8·8 respectively, find the specific gravity of an alloy of gold and copper in the ratio of 11 to 1.

9. From a cubical tank whose edge is 4 ft. a pipe rises 10 feet high. If the tank and the pipe be filled with water, calculate the pressure on each face of the tank, assuming that a cubic foot of water weighs 1000 oz.

10. Investigate the conditions which must be satisfied if a body float freely in a liquid.