Class 18 Rotational motion PHY 231 Fall 2004

Class 18
Rotational motion
PHY 231
Fall 2004
Prof. S. Billinge
Concepts overview
1. Angles in Radians
2. Rotational motion
3. Rotational – linear motion
dictionary
PHY 231
Fall 2004
Prof. S. Billinge
Rotational motion is
easy!
•
•
•
•
•
Something moving in a
circle
x->θ
v−>ω
a−>α
Then, just use the same
equations!
PHY 231
Fall 2004
Prof. S. Billinge
Equations of motion
Equations of motion
Linear motion
Angular motion
x(t)=x(0)+v(0)t+½at2
v(t)=v(0)+at
θ(t)= θ(0)+ω(0)t+ +½αt2
ω(t)= ω(0)+αt
GOTCHA’s
• ALWAYS ALWAYS ALWAYS put
your angles into RADIANS
before using the equations
(θ radians) = length of circumference
segment/length of radius
–
= s/r
• Conversion:
– 1 radian = (360 deg)/(2π)
–
= (180 deg)/(π)
Then use your unit conversion
techniques
PHY 231
Fall 2004
Prof. S. Billinge
The units of angle are radians, what are the
dimensions ([L],[M],[T]) of angle?
1. [L]
2. [degrees]
3. [L]2
4. [L]/[M]
5. dimensionless
PHY 231
Fall 2004
Prof. S. Billinge
Angular speed, ω = ∆θ/∆t. What are the
dimensions of ω?
1. Dimensionless
2. [L]/[T]
3. m/s
4. 1/[T]
5. [L]/[T]2
PHY 231
Fall 2004
Prof. S. Billinge
A ladybug sits at the outer edge of a merrygoround, and a gentleman bug sits halfway
between her and the axis of rotation. The
merry-go-round makes a complete revolution
once each second.The gentleman bug’s
angular speed is
1. half the ladybug’s.
2. the same as the ladybug’s.
3. twice the ladybug’s.
4. impossible to determine
PHY 231
Fall 2004
Prof. S. Billinge
Converting between
angular and linear
(cartesian) motion
v is the tangential
component of the velocity
a
α=
r
a is the tangential
component of the
acceleration
PHY 231
Fall 2004
Prof. S. Billinge
GOTCHA 3
• Don’t mix up angular
acceleration/tangential
acceleration with
centripetal acceleration
• Tangential is tangential but
centripetal is radial
• Centripetal acceleration is
always towards the middle
of the circle
PHY 231
Fall 2004
Prof. S. Billinge
Centripetal acceleration occurs when an object
moves in a circle. It is:
1. An angular acceleration
2. A cartesian (linear) acceleration
3. Something in between
PHY 231
Fall 2004
Prof. S. Billinge
A ladybug sits at the outer edge of a merrygoround, that is turning and slowing down.
At the instant shown in the figure, the radial
component of the ladybug’s (Cartesian)
acceleration is
1. in the +x direction.
2. in the –x direction.
3. in the +y direction.
4. in the –y direction.
5. zero.
A ladybug sits at the outer edge of a merrygoround, that is turning at constant speed.
At the instant shown in the figure, the
ladybug’s angular acceleration is acting
1. in the +x direction.
2. in the –x direction.
3. in the +y direction.
4. in the –y direction.
5. zero.
Consider the uniformly rotating object
shown below. If the object’s angular velocity
is a vector (in other words, it points in a
certain direction in space) is there a particular
direction we should associate with the
angular velocity?
1. yes, ±x
2. yes, ±y
3. yes, ±z
4. yes, some other direction
5. no, the choice is really arbitrary
Relationship of centripetal
acceleration to angular
motion
•
ac=v2/r
v is the tangential
component of the velocity
• ac=ω2r
PHY 231
Fall 2004
Prof. S. Billinge
GOTCHA – 3: Angular
velocity
• Velocity is a vector; it has
magnitude and direction
• We need to define an
“angular velocity” which is
like “angular speed” but
with a direction specified.
• Angular speed is the rateof-change of angle. What is
the direction?
PHY 231
Fall 2004
Prof. S. Billinge
A ladybug sits at the outer edge of a merrygoround that is turning and is slowing down.
The vector expressing her angular velocity is
1. in the +y direction.
2. in the –y direction.
3. in the +z direction.
4. in the –z direction.
5. zero.
A ladybug sits at the outer edge of a merrygoround that is turning and slowing down.
The tangential component of the ladybug’s
(Cartesian) acceleration is
1. in the +x direction.
2. in the –x direction.
3. in the +y direction.
4. in the –y direction.
5. zero.