. Differential and integral calculus. PC = g sin $ = acceleration in direction of motion. Mechanical Applications 375 But if we let PB = ds, then PA = dy; hence -^ = sin 0. ds Hence, § 82 d2s a¥ ~~ dsd2s df- Jy- = gds = gdy. Integrating between limits y andy we have ds2 2dt2^ g(y- ?/) (0 .-. § 17, v*= 2g(y-/). Comparing the last equation with (c) § 233, Cor., we seethat the velocity acquired by a body in rolling down a curve isthe same as it would acquire in falling freely through the verticalheight. Cor. From (1) we have ds ds dy dt ^2g(y- /) dy ^2g(y- y^f> ds dy .. /= I -r , • (2) J ) W W
. Differential and integral calculus. PC = g sin $ = acceleration in direction of motion. Mechanical Applications 375 But if we let PB = ds, then PA = dy; hence -^ = sin 0. ds Hence, § 82 d2s a¥ ~~ dsd2s df- Jy- = gds = gdy. Integrating between limits y andy we have ds2 2dt2^ g(y- ?/) (0 .-. § 17, v*= 2g(y-/). Comparing the last equation with (c) § 233, Cor., we seethat the velocity acquired by a body in rolling down a curve isthe same as it would acquire in falling freely through the verticalheight. Cor. From (1) we have ds ds dy dt ^2g(y- /) dy ^2g(y- y^f> ds dy .. /= I -r , • (2) J ) W We are to find what this expression becomes when appliedto the cycloid. 376 Integral Calculus From the equation of the cycloid, x — a vers-1- — V2 ay — y*, we obtain dx y. This value in (a) gives t = y- 1 dy gj V(2^-j)(7-/)Let y — y = z; then dy = dfe, and 2 a — y = 2a — y — z. dz Hence, -41 V(2 a - y)z - z2 2 z 2 a — y -, + C. If we suppose the body to fall from C to B we have 2=oatC, and z = 2a — yf at B. Hence between these limits of zwe have Mechanical Applications 377 is the time it takes the body to fall from the position C to thelowest point B of the curve. Since y is any value of y, thepoint C is any point of the cycloid; hence the time required for abody to fall from any point of an inverted cycloid to its lowestpoint is constant. Theoretically, therefore, the cycloidal arc isthe path of a pendulum which vibrates in equal times. 240. A projectile is thrown obliquely upward with a velocityv ; find (i), the equation of its path; (2), the coordinates of itshighest point; (3), the angle of projection in order that its rangemay be a maximum.
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