James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Problems Plus 1. A particle P moves with constant angular speed around a circle whose center is at the
v
r
v t
FIGURE for problem 1
y
x
origin and whose radius is R. The particle is said to be in uniform circular motion. Assume
that the motion is counterclockwise and that the particle is at the point sR, 0d when t − 0.
The position vector at time t > 0 is rstd − R cos t i 1 R sin t j.
(a) Find the velocity vector v and show that v ? r − 0. Conclude that v is tangent to the
circle and points in the direction of the motion.
(b) Show that the speed | v | of the particle is the constant R. The period T of the particle
is the time required for one complete revolution. Conclude that
T − 2R
| v |
− 2
(c) Find the acceleration vector a. Show that it is proportional to r and that it points toward
the origin. An acceleration with this property is called a centripetal acceleration. Show
that the magnitude of the acceleration vector is | a | − R2 .
(d) Suppose that the particle has mass m. Show that the magnitude of the force F that is
required to produce this motion, called a centripetal force, is
| F | − m | v | 2
R
¨
F
mg
2. A circular curve of radius R on a highway is banked at an angle so that a car can safely
traverse the curve without skidding when there is no friction between the road and the tires.
The loss of friction could occur, for example, if the road is covered with a film of water
or ice. The rated speed v R of the curve is the maximum speed that a car can attain without
skidding. Suppose a car of mass m is traversing the curve at the rated speed v R. Two forces
are acting on the car: the vertical force, mt, due to the weight of the car, and a force F
exerted by, and normal to, the road (see the figure).
The vertical component of F balances the weight of the car, so that | F |
cos − mt. The
horizontal component of F produces a centripetal force on the car so that, by Newton’s
Second Law and part (d) of Problem 1,
FIGURE for problem 2
| F | sin − mv 2 R
R
(a) Show that v 2 R − Rt tan .
(b) Find the rated speed of a circular curve with radius 400 ft that is banked at an angle
of 128.
(c) Suppose the design engineers want to keep the banking at 128, but wish to increase the
rated speed by 50%. What should the radius of the curve be?
3. A projectile is fired from the origin with angle of elevation and initial speed v 0. Assuming
that air resistance is negligible and that the only force acting on the projectile is gravity,
t, we showed in Example 13.4.5 that the position vector of the projectile is
884
y
y
rstd − sv 0 cos dt i 1 fsv 0 sin dt 2 1 2 tt 2 g j
_R 0 R x
We also showed that the maximum horizontal distance of the projectile is achieved when
− 45° and in this case the h range is R − v0yt.
2
(a) At what angle should the projectile be fired to achieve maximum height and what is the
maximum height?
0(b) Fix the initial speed v 0 and consider the parabola x 2 1 2Ry 2 R 2 − 0, whose graph is
D x
shown in the figure at the left. Show that the projectile can hit any target inside or on
the boundary of the region bounded by the parabola and the x-axis, and that it can’t hit
FIGURE for problem 3
any target outside this region.
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