Analytical View of Sir Isaac Newton's Principia

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1855 edition. Excerpt: ...had been undisturbed by/, the same equation would have given the arc measured from C. If then the particle when resisted by f be at P at any time, and if another particle not resisted by/ be at Q at the same time, then DP bears to PQ the constant ratio DO to DC. The effect of any constant resistance is to diminish the arc continually in an arithmetical progression, but not to affect the time of oscillation. B2 A particle constrained to move in a cycloid whose axis is vertical, is acted on by gravity and resisted by the medium in which it moves in the ratio of the velocity. To determine the motion. Newt. xxvi. Let / be twice the radius of the generating circle. Let be the distance of the particle at any time t from the lowest point of the cycloid, and v be the velocity, and m the mass of the particle. Let x be the coefficient of resistance. Then the moving force along the tangent will be m a y-. s--x v, the particle being supposed ascending its arc. But this is Ps dl ds x rf, .9 _ft dP m d t I is the equation to find s in terms of t. To integrate this equation, put s = . e where y. is some constant at our disposal; on substitution, the equation becomes Let us then choose y so that 2 + i, = m and the above equation is reduced to the standard form du fg x1 Putn= %, n'2 = 9-u = L cos n'(t--X). Supposing the time to be counted from the epoch when the pendulum began its first complete oscillation from its highest point, then = 0. _ x t.'. S = Le 2m cos n' t Hence we learn, 1. That the oscillations will be isochronal, and the time of oscillation ' fg x_ V I 4m2 This differs from the time of oscillation in a vacuum; if x be small, and T the time in vacuo, the difference will be very nearly x/ If we neglect terms so small as...show more

Product details

• Paperback | 138 pages
• 189 x 246 x 8mm | 259g
• United States
• English
• black & white illustrations
• 1236993411
• 9781236993410