Download Rotational Motion

Rotational Motion
Rotation about a fixed axis
Rotational Motion

Translation

Rotation

Rolling
Angular Kinematics
Describing Angular Motion
Angular Position - q


Displacement is now defined as angle of
rotation.
Polar coordinates (r, q) will make it easier to
notate this motion.
Dq
Angular Position - q
For a circle: 2p=circumference/radius
So: q = s/r or s = qr





s => arc length
r => radius
q => angle, measured in radians


Remember: 360º = 2p radians
Displacement:
Dq = q1 – q0
Angular Velocity - w

Average angular
velocity

Instantaneous angular
velocity
Dq
w=
Dt
dq
w=
dt
Angular Direction

Right-hand coordinate system


w is positive when q is increasing, which
happens in the counter-clockwise
direction
w is negative when q is decreasing,
which happens in the clockwise direction
Angular Acceleration - a

Average Angular
Acceleration

Instantaneous Angular
Acceleration
Dw
a =
Dt
dw
a=
dt
example 1
q = (t3 - 27t + 4)rad



w=?
a=?
At what time(s) does w = 0
rad/s?
example 2
a=
3
(5t
–
2
4t)rad/s
At t = 0s: w = 5 rad/s and q = 2 rad.
w
=?
q = ?
Linear to Angular Conversions
s = qr
 Velocity: v = wr
 Acceleration: a = ar
 Position:

Remember, also, centripetal
accleration: ac = v2/r =w2r
Angular Kinematics
*constant a
w1 = w0  at
1 2
Dq = w 0 t  a t
2
2
2
w1 = w0  2aDq
example 3
A turntable starts from rest and begins rotating
in a clockwise direction. 10 seconds later, it
is rotating at 33.3 revolutions per minute.
 What is the final angular velocity (rad/s)?
 What is the average angular acceleration?
 How far did a point 10 cm from the center
travel in that 10 seconds, both angularly and
linearly?
example 4
In an astronaut training centrifuge (r = 15m):



What constant w would give 11g’s?
How fast is this in terms of linear speed?
What is the translational acceleration to
get to this speed from rest in 2 minutes?
Moment of Inertia
“Rotational Mass”
Moment of Inertia


Description of the distribution of mass
Measure of an object’s ability to resist a
change in rotation
I =  mi ri

2
Note: r is the perpendicular distance from the particle to
the axis of rotation.
example
Two children (m=40 kg) are on the teacup ride
at King’s Dominion. The childrens’ center of
mass is 0.75 m from the middle of the cup.
What is the moment of inertia for children in
the teacup?
Moment of Inertia for
Continuous Mass Distribution
I =  mi ri
2
I = lim  Dmi ri
Dmi 0
I =  r dm
2
2
V
=
m
dm =  dV
dm =  dA
dm =  dr
Calculate I for a hoop

A uniform hoop: mass (M); radius (R)
I =  r dm
2
I =  R dm
2
I = R  1dm
2
I = MR
2
Calculate I for a thin rod

A uniform thin rod:
mass (M); length (L);
rotating about its
center of mass.
I =  r 2 dm
I =  r 2  dr
L
2
L

2
I =

r 2  dr

L
2
1
I =  r3
3 L
2
M
I =
 L

L
2
1 3
 r L
3 
2
I=
1
ML2
12
Common Moments of Inertia
Parallel Axis Theorem

To calculate the moment of inertia around
any axis….
I = I cm  Md

2
Where d is the distance between the center
of mass and the axis of rotation
Calculate I for a thin rod
rotating around the end

A uniform thin rod:
mass (M); length (L);
rotating about one end.
Rotational Energy
The kinetic energy an object has
due to its rotational velocity.
Rotational Kinetic Energy

example
If you roll a disk and a hoop (of the same mass)
down a ramp, which will win?
Torque
The tendency of a force to cause
angular motion
Torque

Torque is dependent
on the amount and
location of the force
applied to an object.
 = r  F
 = rF sin q

Where r is the distance between the pivot point
and the force and q is the angle between r and F.
example 5



A one piece cylinder has a core section that
protrudes from a larger drum. A rope
wrapped around the large drum of radius, R,
exerts a force, F1, to the right, while a rope
wrapped around the core, radius r, exerts a
force, F2 downward..
Calculate the net torque, in variables.
If F1=5 N, R = 1 m, F2=6 N, and r = 0.5 m,
calculate the net torque.
Cross Product
The “other” vector multiplication
Cross product


Results in a vector quantity
Calculates the perpendicular product of two
vectors

The product of any two parallel vectors will
always be zero.
iˆ  iˆ = 0
ˆj  ˆj = 0
kˆ  kˆ = 0
The Right Hand Rule

Rotational Direction is defined by the axis
that it rotates around.

Point your fingers along
the radius
Curl them in the
direction of the force
Your thumb will be
pointing in the direction
of the rotation.


http://hyperphysics.phy-astr.gsu.edu/hbase/tord.html
Cross Product
ˆi  ˆj = kˆ =  ˆj  iˆ
ˆj  kˆ = iˆ =  kˆ  ˆj
ˆ
ˆ
ˆ
ˆ
ˆ
k  i = j = i  k
Calculating Cross Product


 
A  B = Axiˆ  Ay ˆj  Az kˆ  Bx iˆ  B y ˆj  Bz kˆ

 
 
 
A  B = Axiˆ  Bx iˆ  Axiˆ  B y ˆj  Axiˆ  Bz kˆ
 

 A kˆ  B iˆ  A kˆ  B ˆj  A kˆ  B kˆ 

 
 Ay ˆj  Bxiˆ  Ay ˆj  B y ˆj  Ay ˆj  Bz kˆ
z
x
z
y
z
z


Finding the Determinant
kˆ
rz
Fz
iˆ
 
r  F = rx
Fx
ˆj
ry
Fy
  ry
r F =
Fy
rz
rx
iˆ 
Fz
Fx
rx
rz
ˆj 
Fx
Fz
ry
kˆ
Fy
 
r  F = ry Fz  rz Fy iˆ  rx Fz  rz Fx  ˆj  rx Fy  ry Fx kˆ
example 6

Find the cross product of 7i + 5j – 3k and 2i8j + 7k.

What is the angle between these two
vectors?
example 7

A plumber slips a piece of scrap pipe over his
wrench handle to help loosing a pipe fitting.
He then applies his full weight (900 N) to the
end of the pipe by standing on it. The
distance from the fitting to his foot is 0.8 m,
and the wrench and pipe make a 19º angle
with the ground. Find the magnitude and
direction of the torque being applied.
example 7 continued…
 = rF sin q
 = (0.8)(900) sin( 71)
 = 680 Nm
r = 0.8m
F = 900 N
q = 19º
example 8
One force acting on a machine part is F = (-5i +
4j)N. The vector from the origin to the point
applied is r = (-0.5i + 0.2j)m.
 Sketch r and F with respect to the origin
 Determine the direction of the force with the
right hand rule.
 Calculate the torque produced by this force.
 Verify that your direction agrees with your
calculation.
Equilibrium
Conditions for Equilibrium
1.
Net force in all
directions equals zero
F = 0
2.
Net torque about any
point equals zero
 = 0
example 9
A 2 kg seesaw has one child (30 kg) sitting 2.5 from
the pivot. At what distance from the pivot should a
25 kg child sit to balance the seesaw?
example 10
Find Tcable.
example 11
Find T1 and T2.
example 12
Find m.
Newton’s Second Law
When the sum of the torques
does not equal zero
Newton’s Second Law
 F = ma
 = F r
 = (ma )r
 = (ma r )r
 = (mr )a
 = Ia


2
example 13

a=?
 = I a
 L 1 2
 Fg   =  mL  a
 2 3

3g
a =  rad / s 2
2L
Fpivot
L
Fnormal
Fgravity
example 14
F
y
FT
= ma y
 Fg  FT = M 2 a y
 M 2 g  FT = M 2 a y
 1

 M 2 g    M 1a y  = M 2 a y
 2

Pulley – M1, R
1
 M 2 g = M 2 a y  M 1a y
Block – M2
2
Find a and FT  M 2 g =  M 2  1 M 1  a y
2


M2g

= ay
1


M

M
 2
1
2


Fg
 = Ia
1
 a 
 FT R =  M 1 R 2  
2
 R 
1
FT =  M 1a
2


 M2g 
1
FT = M 1 

1
2
 M 2  M1 

2

M 1M 2 g
FT =
2M 2  M 1
Angular Momentum
For a particle

Remember linear momentum:

So angular momentum:
p = mv
  
L=rp
L = rmv sin q
Angular momentum and
Torque
Dp Dmv
F=
=
Dt
Dt
Dmvr
Fr =
Dt
DL
=
Dt
dL
 =
dt
For a system…
L = mi vi ri
v = wr
L = mi (wri )ri = mi ri w
2
L =  mi ri w
2
L = Iw
To check…
L = Iw
dL d
= Iw
dt dt
 = Ia
Conservation

If net torque is zero,
then the angular
momentum is constant.
dL
 =
=0
dt
Conservation

For rotation about a
fixed axis, we can say:
L0 = L1
I 0w0 = I1w1
example

A star with a radius of 10,000 km rotates
about its axis with a period of 30 days. If it
undergoes a supernova explosion and
collapses into a neutron star with a radius of
3 km, what is its new period?