thus the density of water relatively to the standard substance is known. 2. Find, without integration, the space traversed from rest under a uniform acceleration. Parkinson's Mechanics, Art. 68. 3. A particle slides down a rough inclined plane, and takes twice as long as if the plane were smooth; find the coefficient of friction. = If the plane is smooth, the space passed over and the time are connected by the relation sag sinať, where a is the inclination of the plane; but if the plane be rough (coefficient of friction = μ), this relation becomes s=1g (sina — μ cosa)ť"; therefore 4 (g sina-ug cosa) = g sina, Find the length of the line of quickest descent to an ellipse from its upper focus, the major axis being vertical. Let SP (fig. 29) be the line of quickest descent. Then SP is equally inclined to the vertical and normal at P, whence SG= GP; but SG=eSP=PG; therefore HP2 = SP2 + SH2 - 2SP.HS cos PSH, 4. Find the elevations which will make a projectile strike a given object on the same level with the point of projection, and shew how they are related. Parkinson's Mechanics, Art. 86. 5 State the velocities after direct impact of two spheres of masses m and m', whose velocities before impact are v and v', coefficient of elasticity e. Upon what physical assumption are they estimated, and what observation leads us to that assumption? Parkinson's Mechanics, Art. 57. Three perfectly elastic particles m, m', m" are placed in line, m impinges on m', and m' on m"; what is the effect of the interposition of m' on the velocity of m"? When one body m impinges on another m' at rest (the elasticity being perfect), the second body moves off at rest, m" will have communicated to it a velocity of but if m had impinged directly on m", the velocity communicated would have been 2mv m + m' The ratio of these two velocities is therefore 2m' (m+m") 6. Find the acceleration with which a particle will describe a given circle uniformly with given velocity. Parkinson's Mechanics, Art. 105. PLANE TRIGONOMETRY. THURSDAY, 21ST JUNE, 1876. 2 P.M. TO 5P.M. 1. Point out any advantages in Trigonometry that result (1) from considering the magnitude of an angle unlimited, (2) from considering that angles and lines may have negative as well as positive values. Define those of the elementary trigonometrical functions that may become infinite in value, and trace the signs and magnitudes of one of these as the angle varies from 0° to 360°. Todhunter's Trigonometry, Arts. 41, 42, 43, 49. The trigonometrical ratios which may become infinite are tangent, cotangent, secant, and cosecant. Prove that cos4= cos(-A) and sin A-sin(-A). 2. Find tan 60°, and prove that (sin 30°), (sin 45°), (sin 60°) are as the numbers 1, 2, 3. Express the values. of sin 660°, tan 660°, versed sine 660°. Trace the changes in the sign and value of (cos Asin A) as A varies from 0° to 360°. Todhunter's Trigonometry, Arts. 36, 37. sin 660°√3, tan 660° as A varies from 0° to 45° 45° to 90° 90° to 135° 135° to 180° 180° to 225° 225° to 270° = 3, versed sine 660°, cos A-sin A changes from 1 to 0, 0 to -1, – 1 to - √√2, -√/2 to -1, 0 to 1, 270° to 315° 315° to 360° 1 to √√/2, √/2 to 1. 3. Prove, by means of a geometrical figure, cos 24 = cos3A - sin2A, without assuming the formula for cos (A+B), and deduce from the above value of cos2A that of sin 2A. First part to be done as Todhunter, Art. 76, only two angles to be taken equal in the figure and each = A. sin 24 = √(1-cos2 24)=√(1-cos 2A) √(1 + cos24) = √(2) sin A √(2) cos A = 2 sin A cosA. 4. Find sin 18°, and thence determine sin 81° without reducing the surds that may be involved in the expression. Todhunter's Trigonometry, Arts. 107, 111. |