181
Homework Problems
1. When youíre done using an electric mixer, you can get most of the
batter off of the beaters by lifting them out of the batter with the motor
running at a high enough speed. Letís imagine, to make things easier to
visualize, that we instead have a piece of tape stuck to one of the beaters.
(a) Explain why static friction has no effect on whether or not the tape
flies off. (b) Suppose you find that the tape doesnít fly off when the motor
is on a low speed, but speeding it up does cause it to fly off. Why would
the greater speed change things.
2. Show that the expression |v|
2
/r has the units of acceleration.
3 . A plane is flown in a loop-the-loop of radius 1.00 km. The plane
starts out flying upside-down, straight and level, then begins curving up
along the circular loop, and is right-side up when it reaches the top . (The
plane may slow down somewhat on the way up.) How fast must the plane
be going at the top if the pilot is to experience no force from the seat or
the seatbelt while at the top of the loop.
4
.
. In this problem, you'll derive the equation |a|=|v|
2
/r using calculus.
Instead of comparing velocities at two points in the particle's motion and
then taking a limit where the points are close together, you'll just take
derivatives. The particle's position vector is r=(r cos
.
)
x
+ (r sin
.
)
y
,
where
x
and
y
are the unit vectors along the x and y axes. By the defini-
tion of radians, the distance traveled since t=0 is r
.
, so if the particle is
traveling at constant speed v=|v|, we have v=r
.
/t. (a) Eliminate
.
to get the
particle's position vector as a function of time. (b) Find the particle's
acceleration vector. (c) Show that the magnitude of the acceleration vector
equals v
2
/r.
5 S. Three cyclists in a race are rounding a semicircular curve. At the
moment depicted, cyclist A is using her brakes to apply a force of 375 N
to her bike. Cyclist B is coasting. Cyclist C is pedaling, resulting in a
force of 375 N on her bike. Each cyclist, with her bike, has a mass of 75
kg. At the instant shown, the instantaneous speed of all three cyclists is 10
m/s. On the diagram, draw each cyclist's acceleration vector with its tail
on top of her present position, indicating the directions and lengths
reasonably accurately. Indicate approximately the consistent scale you are
using for all three acceleration vectors. Extreme precision is not necessary
as long as the directions are approximately right, and lengths of vectors
that should be equal appear roughly equal, etc. Assume all three cyclists
are traveling along the road all the time, not wandering across their lane or
wiping out and going off the road.
SA solution is given in the back of the book.A difficult problem.
A computerized answer check is available.
.
A problem that requires calculus.
Homework Problems
20 m
A
B
C
direction
of travel
Problem 5.
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