By Derek F. Lawden
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Additional resources for A Course in Applied Mathematics, Vol. 1 and 2
Rt w. ) A ship leaves port and steers a straight course at 12 knots for a destina tion that is unknown. Six hours later a ship that can do 20 knots is sent from port in pursuit. It sails at top speed due northwards for 9 hours, and, on failing to find the first ship on this course, it proceeds to steer on such a curve that it would find it whatever the course taken by the first ship. Show that it describes the equiangular spiral r = 1 7. O. ) A particle moves in a plane so that the velocity components along and perpendicular to the radius vector from a fixed origin are c tan �6 and c respectively, where c is a constant.
The velocity of Q is of magnitude aw and is in the direction of the tangent at Q. This tangent makes an angle (wt + if>) with BB'. The resolute of Q's velocity in the direction of BB' is accordingly aw cos (wt + ¢>). 28) with respect to t. 34) = aw. v v = aw cos (wt + if>), Vmax. , when P passes through 0 in the positive direction. , see Example 3 below) . Practically, the simplest way of causing a body to execute SHM is by connecting it to a fixed point by an elastic support. Thus, suppose that a particle P of mass m hangs freely from a fixed point A by an 2] NE WTON ' S LA W S .
Prove that it returns to the point A after a time + �) �i (1 and that its greatest depth below A is (3 + 2y2)l. The particle first falls from A with a constant acceleration g to a point B, where AB = l. The string then becomes taut and the particle's motion is Example 3. simple harmonic until it returns to B, when the string goes slack again. On the two occasions that the particle is at B, its distances from the centre of oscillation 0 are the same. It follows from equation that the particle's speeds at these instants are also identical, though the senses of its velocities are opposite.