# Relative and Circular Motion

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14 Νοε 2013 (πριν από 4 χρόνια και 8 μήνες)

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Relative and Circular Motion

Mechanics Lecture 3, Slide
1

a
) Relative motion

b) Centripetal acceleration

Mechanics Lecture 3, Slide
2

What is the speed of Mike relative to the station?

A.
-
1 m/s

B.
30 m/s

C.
29 m/s

D.
31 m/s

Mechanics Lecture 3, Slide
3

Relative Motion in 1 dimension

Mechanics Lecture 3, Slide
4

Mechanics Lecture 3, Slide
5

Relative Position and
R
eference
F
rames

Position of Mike in the ground frame is the vector
sum of the position vector of Mike in the train
reference frame and the position vector of the train in
the ground reference frame.

Mechanics Lecture 3, Slide
6

Relative Motion and
R
eference
F
rames

Differentiate the position vectors to obtain the
velocity vectors

Mechanics Lecture 3, Slide
7

Relative Motion and
R
eference
F
rames

Mechanics Lecture 3, Slide
8

Relative Motion and
R
eference
F
rames

Prelecture

3, Questions 1 again

A.

B.

C.

D.

Mechanics Lecture 3, Slide
9

0%
0%
0%
0%
Mechanics Lecture 3, Slide
10

v
belt,ground

= 2 m/s

v
dog,belt

= 8 m/s

A)

6 m/s

B)

8 m/s

C)

10 m/s

CheckPoint

A girl stands on a moving sidewalk that moves to the right at
2 m/s

relative to the ground. A dog runs toward the girl in the opposite direction

along the sidewalk at a speed of
8 m/s

relative to the sidewalk.

What is the speed of the dog relative to the ground?

Mechanics Lecture 3, Slide
11

v
dog, ground

=
v
dog, belt

+
v
belt, ground

= (
-
8 m/s) + (2 m/s) =
-
6 m/s

+
x

v
belt,ground

= 2 m/s

v
dog,belt

= 8 m/s

What is the speed of the dog relative to the ground?

A)

6 m/s

B)

8 m/s

C)

10 m/s

Mechanics Lecture 3, Slide
12

About 55% of you got this right

lets try it again.

CheckPoint

A girl
stands

on a moving sidewalk that moves to the right at 2 m/s

relative to the ground. A dog runs toward the girl in the opposite direction

along the sidewalk at a speed of 8 m/s relative to the sidewalk.

What is the speed of the dog relative to the
girl
?

v
belt,ground

= 2 m/s

v
dog,belt

= 8 m/s

A)

6 m/s

B)

8 m/s

C)

10 m/s

What is the speed of the dog relative to the girl?

A.

B.

C.

Mechanics Lecture 3, Slide
13

0%
0%
0%
v
belt,ground

= 2 m/s

v
dog,belt

= 8 m/s

A)

6 m/s

B)

8 m/s

C)

10 m/s

C)

The dog and girl are running towards each other so when you add the two
velocities together it would be 8+2.

A)

Because the girl is actually moving and the two vectors are opposite, so
together they make 6 m/s

B)

Because the girl is not moving relative to the belt, and the dog is going 8 m/s
relative to the belt, the dog is also moving 8 m/s relative to the girl..

Mechanics Lecture 3, Slide
14

B)

Because the girl is not moving relative to the belt, and the dog is going 8 m/s
relative to the belt, the dog is also moving 8 m/s relative to the girl.

Using the velocity formula:

v
dog, girl

=
v
dog, belt

+
v
belt, girl

=
-
8 m/s + 0 m/s

=
-
8 m/s

What is the speed of the dog relative to the girl?

v
belt,ground

= 2 m/s

v
dog,belt

= 8 m/s

A)

6 m/s

B)

8 m/s

C)

10 m/s

Relative Motion in 2 Dimensions

Mechanics Lecture 3, Slide
15

Speed

relative to shore

Relative Motion in 2 Dimensions

Mechanics Lecture 3, Slide
16

Direction w.r.t shoreline

Relative Motion in 2 Dimensions

Mechanics Lecture 3, Slide
17

Caveat !!!!

as shown only works for speeds
much less than the speed of
light…need special relativity at
v~c
.

Moving Sidewalk Question

A.

B.

C.

D.

Mechanics Lecture 3, Slide
18

0%
0%
0%
0%
A man starts to walk along the dotted line painted on a moving sidewalk

toward a fire hydrant that is directly across from him. The width of the

walkway is
4 m
, and it is moving at
2 m/s

relative to the fire
-
hydrant. If

his walking speed is
1 m/s
, how far away will he be from the hydrant

when he reaches the other side?

A)
2 m

B)
4 m

C)
6 m

D)
8 m

Mechanics Lecture 3, Slide
19

Time to get across:

D
t

= distance / speed

= 4m / 1m/s

= 4 s

If the sidewalk wasn’t moving:

sidewalk
man
v
,

Mechanics Lecture 3, Slide
20

Just the sidewalk:

hydrant
sidewalk
v
,

Mechanics Lecture 3, Slide
21

Combination of motions:

hydrant
sidewalk
sidwalk
man
hydrant
sidewalk
v
v
v
,
,
,

Mechanics Lecture 3, Slide
22

D

= (speed of sidewalk)

(time to get across)

= (2 m/s)

(4 s) = 8 m

D

Swim Race

A.

B.

C.

Mechanics Lecture 3, Slide
23

0%
0%
0%
0%
Three swimmers can swim equally fast relative to the water. They
have a race to see who can swim across a river in the least
time. Relative to the water,
Beth

flow,
Ann

swims upstream at 30 degrees, and
Carly

downstream at 30 degrees.

Who gets across the river first?

A) Ann

B) Beth

C) Carly

x

y

Ann

Beth

Carly

Mechanics Lecture 3, Slide
24

A

B

C

V
y
,
Beth

= V
o

30
o

30
o

V
y
,
Ann

= V
o
cos(30
o
)

V
y
,
Carly

= V
o
cos(30
o
)

Time to get across

=

D
/
V
y

D

Look at just water & swimmers

x

y

Mechanics Lecture 3, Slide
25

x

y

Clicker Question

Three swimmers can swim equally fast relative to the water.
They have a race to see who can swim across a river in the
least time. Relative to the water,
Beth

to the flow,
Ann

swims upstream at 30 degrees, and
Carly

Who gets across the river second?

A) Ann

B) Carly

C) Both same

Ann

Carly

Mechanics Lecture 3, Slide
26

Accelerating (Non
-
Inertial) Frames of Reference

Accelerating Frame of Reference

Confusing due to the fact that the
acceleration can result in what appears to
be a “push or pull”.

Accelerated Frames of Reference

Mechanics Lecture 3, Slide
27

Accelerating Frame of Reference

Accelerometer can detect change in
velocity

Inertial Frames of Reference

Mechanics Lecture 3, Slide
28

Inertial Frames of Reference

Non
-
accelerating

frames

of

reference

in

a

state

of

constant,

rectilinear

motion

with

respect

to

one

another
.

An

accelerometer

moving

with

any

of

them

would

detect

zero

acceleration
.

Mechanics Lecture 3, Slide
29

A girl twirls a rock on the end of a
string around in a horizontal circle
above her head as shown from above
in the diagram.

If the string breaks at the instant
shown, which of the arrows best
represents the resulting path of the
rock?

A

B

C

D

Top view looking down

After the string breaks, the rock will have no
force acting on it, so it cannot accelerate.
Therefore, it will maintain its velocity at the time
of the break in the string, which is directed
tangent to the circle.

CheckPoint

Mechanics Lecture 3, Slide
30

Show
Prelecture

https://www.smartphysics.com/Course/PlaySlideshow?unitItemID=192821

Mechanics Lecture 3, Slide
31

Rotating Reference Frames

Direction Changing

S
peed is constant.

Acceleration

Mechanics Lecture 3, Slide
32

Rotating Reference Frames

Mechanics Lecture 3, Slide
33

Rotating Reference Frames

Mechanics Lecture 3, Slide
34

Rotating Reference Frames

Mechanics Lecture 3, Slide
35

Rotating Reference Frames

Mechanics Lecture 3, Slide
36

Centripetal Acceleration

Constant speed in circular path

Acceleration directed toward center
of circle

What is the magnitude of
acceleration?

Proportional to:

1.
Speed

1.
time rate of change of angle
or angular velocity

Mechanics Lecture 3, Slide
37

Centripetal Acceleration

Mechanics Lecture 3, Slide
38

Centripetal Acceleration

Mechanics Lecture 3, Slide
39

Centripetal Acceleration

Mechanics Lecture 3, Slide
40

v =
w
R

Once around:

w

 D
q
/

D
t

=
2
p
/
T

v

 D
x /

D
t

=
2
p
R / T

w

is the rate at which the angle

q

changes:

q

dt
d
q
w

Mechanics Lecture 3, Slide
41

d
q

v dt

R

=R d
q

Another way to see it:

v =
w
R

v = R
w

dt
d
R
v
q

Mechanics Lecture 3, Slide
42

Centripetal Acceleration
-
Example

Mechanics Lecture 3, Slide
43

Centripetal Acceleration due to Earth’s rotation

Mechanics Lecture 3, Slide
44