Calculus: Analytic Geometry and Calculus, with Vectors

3.8 Related rates 191

to the spool atthe point T. Find the rate of increase of the distance from the
axis of the spool to the end E of the thread. Hint: It is not necessary to make an
extensive study of the path traversed by the end E; it is sufficient to construct
and use an appropriate right triangle .4ns.: .4s//R2 + s2 centimeters per
second.
7 A particle is moving with constant speed k along the graph of y = sin x
in such a way that its x coordinate is always increasing. Derive the formulas
dy dx dx k dy k cos x
dt =COS xdt' dt - 1 + cos2 dt - 1 + os xco

involving the scalar components of the velocity. Show also that

d2x k2 sin x cos x

dt2 (1 + cost x)2

8 A particle of mass m starts from rest (that is, starts with speed 0) at the

point (xo,yo) of Figure 3.892 and, with the earth's gravitational field pulling it

downward, slides without friction on the graph of the equation y = x2. We

grasp an opportunity to see how basic scientific concepts can be employed to

obtain information about the motion of the particle.

When 0 S y < yo and the particle is at the point (x,y),

the loss in potential energy is mg(yo - y) and the gain

in kinetic energy isIv12, so

Iv12 = 2g(yo - y).

With the aid of this information, derive the formulas

0X2 Yo - Y (dy)2 x2(yo - y)

F) = 2g 1 + 4x2' dt / = 8g 1 + 4x2

Figure 3.892

which determine (except for algebraic sign) the horizontal and vertical scalar

components of the velocity of the particle when it is at the point (x,y).

Remark: There is a reason why the formulas refuse to tell the signs of dx/dt and

dy/dt. As time passes, the particle oscillates to and fro over an arc of the

parabola in such a way that the scalar components of the velocity are sometimes

positive and sometimes negative.

9 Figure 3.893 shows a connecting rod of length b which earns its name by

connecting a piston (which is free to

move to and fro in a cylinder) to a P b

point P on a crankshaft which is free a Piston

to rotate in a circle of radius a having^0 x

its center at 0. We should not be too

busy to observe that b exceeds 2a in Figure 3.893

ordinary engines and pumps. Obtain

a formula relating dx/dt, the scalar velocity of the piston, to dB/dt, the angular

speed of the crankshaft. Hint: Use the law of cosines in the form

b2=a2 + x2- tax cos 8.

dx ax sin 0 dO

flns.:dt x - -ac 0 s 0 dt