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Homework Problems

1. When you’re 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. Let’s 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 doesn’t 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.