PHYS_2014_Lecture_7 - Lecture 7 Circular Motion v r I swing...

Fall 2010
Oklahoma State University
PHYS2014: Benton
Lecture 7, Slide 1
r
v
I swing a ball attached to a string of length r over my head.
The path that the ball takes is a circle of radius with my hand
at the origin. At any given instant, the velocity, , of the ball
is a vector tangent to the circular path, i.e. .
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v
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r
⊥
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v r
Lecture 7: Circular Motion

Fall 2010
Oklahoma State University
PHYS2014: Benton
Lecture 7, Slide 2
Circular or rotational motion , while common in everyday life,
is different from the linear motion we have studied so far.
Circular motion is clearly different from the straight-line
motion of objects sliding or rolling over surfaces (inclined or
otherwise) and is clearly different from projectile motion
wherein objects follow parabolic trajectories.
One difference is that circular motion repeats...the object
traverses the same path, i.e. the same circular trajectory, over
and over again.
r
v

Fall 2010
Oklahoma State University
PHYS2014: Benton
Lecture 7, Slide 3
Because the motion is constrained to a circle, we can specify
all possible positions where the object (ball) might be in
terms of the radius, , and the angle, θ .
Thus, it makes sense to use a polar coordinate system.
2 2
1
ˆ
cos
ˆ
sin
tan
−
=
=
= +
 
=
 
 
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x r i
y r j
r x y
y
x
θ
θ
θ
θ
r
y
x
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r

Fall 2010
Oklahoma State University
PHYS2014: Benton
Lecture 7, Slide 4
r
Δ θ
s
x
The distance traveled by an object undergoing circular
motion can be found by the radius r and the change in the
angle Δ θ .
(radians) s r θ = Δ
(radians)
s
r
θ Δ =
s is not in a straightline . It is the segment of the circumference
of a circle and is referred to as arc length.

Fall 2010
Oklahoma State University
PHYS2014: Benton
Lecture 7, Slide 5
The length of one circular path around my hand is the
circumference of the circle of radius r .
If the ball I’m swinging above my head takes a time T (the
period) to complete one circular path, the the ball’s speed
will be:
2 s r π =
2 s r
v
t T
π Δ
= =
Δ
r
v

Fall 2010
Oklahoma State University
PHYS2014: Benton
Lecture 7, Slide 6
We can apply this to the Earth’s motion around the Sun. The
average distance between the Sun and the Earth is 1.49 × 10
11
m
and the time it takes the Earth to go around (orbit) the Sun one
time is 365.25 days or 3.16 × 10 7 s.
11
4
7
2 1.49 10 m
2.91 10 m s 65,000miles hr
3.16 10 s
v
π ⋅ ×
= = ×
×
≃

Fall 2010
Oklahoma State University
PHYS2014: Benton
Lecture 7, Slide 7
Consider our original definitions for velocity and
acceleration
and
s v
v a
t t
Δ Δ
= =
Δ Δ
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As defined here, position, velocity, and acceleration are all
vectors.
By definition a vector has both magnitude and direction.