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Derek F. Lawden's A Course in Applied Mathematics, Vol. 1 and 2 PDF

By Derek F. Lawden

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Extra resources for A Course in Applied Mathematics, Vol. 1 and 2

Example text

But this force is, by definition, one pound weight. 32 ·2 lbals. 2) 1 lb wt acting upon 1 = m x 32·2 m = 1/32·2. 2 32 ·2 lb. 3) Throughout the subsequent work, we shall always suppose that forces, masses and accelerations are expressed in terms of any set of consistent units. 1) is true for any such consistent set. Experiment reveals that, provided air resistance is eliminated or can be neglected, all particles near the earth's surface fall with the same acceleration 32·2 ftjsec2 or 981 cmjsec2 • If g denotes this acceleration in any convenient units, m denotes the mass of the falling particle in corresponding consistent units of mass, the force F of the Earth's attraction on the particle is given by F = mg, .

The total time required to reach top gear is accordingly 6·91 sec. 4. 3(2) that a particle of mass m, which moves along the x-axis under the action of a force m(i}2x directed towards the origin, executes SHM of period 27tf(J). We will now investigate the effect of introducing, in addition, a resistive force N E WT O N ' S LAWS . 2] R E C T I L I N E AR M O T I O N 43 proportional to the particle's speed. A frictional force which varies as the speed of the body to which it is applied is said to be viscous.

Suppose that the curve AB and the force F lie in the plane of rect­ angular axes Ox, Oy and let (X, Y) be the components of F parallel to the axes. If P is the point (x, y) and P' is the point (x + ax, y + ay), ar has components (ax, ay). Hence F · ar = Xax + Yay and W = LB (Xdx + Ydy) . 8) A particle can move in the plane of rectangular axes Ox, Oy. When its coordinates are (x, y), the components of a certain force applied to it are (ky2, kx2) . Calculate the work done by this force if the particle is constrained to move along the parabola y x 2 from the origin to the point (1, 1 ) .

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