Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Problem-solving examples and concepts related to rotational motion, including conservation of energy and momentum in collisions, angular velocity and acceleration, and rotational kinetic energy. Students will learn how to calculate angular velocity, angular acceleration, and moment of inertia, as well as understand the relationship between linear and rotational motion.
Typology: Study notes
1 / 15
This page cannot be seen from the preview
Don't miss anything!
°
°
=
°
°
= =
60
cos
30 cos
or
60
cos
30
cos
or
2
1
2
xf^1
xi
v
v
v
mv
mv
mv
p
p
°
− °
=
°
− °
=
=
60 sin
30 sin
or 0
60 sin
30 sin
or 0
2
1
2
yf 1
yi
v
v
mv
mv
p
p
X-direction
Y-direction
2 equations, 2 unknowns – solve for one, substitute into the other:
2
1
x
y
i^
f
1i
1f
2f
1i
1f
2f
1i
2
v1f
2
2f
2
1i
2
1f
2
2f
2
1f
2f
o
By Pythagorean theorem, this must be a right triangle
Let’s look at a particle movingon a rotating disk
z
We first need to define theconcept of an
angle
,^
θ
and
angular displacement
Δθ
z
Rotational motion describeshow this angle changes (notdistance) as a function of time
z
Convention:
θ
> 0
counterclockwise,
θ
< 0,
clockwise
z
Units: Radians. 1 revolution =360 degrees = 2
π
radians
z
In SI units, we usually just dropthe “radians” units (so forexample an angle of 180degrees can be expressed as π
radians or just
π
).
θ
Positive direction
If I rotate from point A to point B, my angular displacement
would be positive
(θ
B^
Axis of rotation
radius
(Δθ
θ
final
θ
initial
The
angular velocity
is the
rate of change of the angle
z
Units: rad/s, degrees/s,revolutions per minute (rpm),etc.
z
The time rate of change ofangular velocity is the
angular
acceleration
z
Units: rad/s
2
d^ θ^ dt
ω =
2 2 d^ dt
Be careful about sign conventions!
Demonstration of a rotating object with aconstant angular acceleration
z
What should the angle, angular velocity,and angular acceleration curves look likeas a function of time?
Every point on a rotating object alsohas a tangential velocity andacceleration.
z
How are these related to angularvelocity and angular acceleration?
z
The “tangential velocity” is v
t^
= dx/dt
z
We can define x in terms of the arclength: x = r
θ
, or dx = r d
θ
(constant
r)
z
Therefore, v
t^
= rd
θ
/dt = r
ω
z
There is also a tangentialacceleration a
t^
= dv
/dt = rdt^
ω
/dt = r
α
z
Similarly, each particle has acentripetal acceleration towards thecenter which is a
C
= v
2 /r
z
By substitution, we then have that thecentripetal acceleration is
z
a
C
= (r
ω
/r = r
ω
2
θ
Axis of rotation
Tangential velocity at
Point A
Centripetal acceleration
at point A
a
C
v
t
a
t^
is changing with timet^
a
t
(A)
z
(B)
t Δ
=
α
ω
ω
t Δ
=
α
ω
Starts from rest
ω
0
At t = 5 s :
(
)
(^
)
(^
)
rad/s
s 5
rad/s 8
s 5
2
=
=
ω
α r
a
= t
(
)
(
)
(^
)
2
2
t
m/s
(^960). 0
rad/s 8
m
s 5
= =
a
Tangential acceleration: Centripetal acceleration:
2
c
ω
(^
)^
(^
)(
)
2
2
c
th
2
i i
i^
∑
∑
∑
i^
i^
i
ii
ii
i i^
r m r m v m K
2
2
2 2
2
ω
ω
i
i ri m
I^
2
This depends on the axis ofrotation, and the mass distribution
