rotational motion rotation about a fixed axis. rotational motion translation rotation rolling
TRANSCRIPT
Rotational Motion
Rotation about a fixed axis
Rotational Motion
Translation
Rotation
Rolling
Angular Kinematics
Describing Angular Motion
Angular Position -
Displacement is now defined as angle of rotation.
Polar coordinates (r, ) will make it easier to notate this motion.
Angular Position -
For a circle: =circumference/radius So: = s/r or s = r
s => arc length r => radius => angle, measured in radians
Remember: 360º = 2radians
Displacement:
–
Angular Velocity - Average angular
velocity
Instantaneous angular velocity
t
dt
d
Angular Direction
Right-hand coordinate system is positive when is increasing, which
happens in the counter-clockwise direction
is negative when is decreasing, which happens in the clockwise direction
Angular Acceleration - Average Angular
Acceleration
Instantaneous Angular Acceleration
t
dt
d
example 1
= (t3 - 27t + 4)rad At what time(s) does = 0
rad/s?
example 2
= (5t3 – 4t)rad/s2
At t = 0s: = 5 rad/s and = 2 rad.
Linear to Angular Conversions
Position: s = rVelocity: v = rAcceleration: a = r
Remember, also, centripetal accleration: ac = v2/r =2r
Angular Kinematics*constant
2
2
1
20
21
20
01
tt
t
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 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
Note: r is the perpendicular distance from the particle to the axis of rotation.
2i iI m r
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 forContinuous Mass Distribution
2
2
0
2
limi
i i
i im
I m r
I m r
I r dm
V
mdm dV
dm dA
dm dr
Calculate I for a hoop
A uniform hoop: mass (M); radius (R)
2
2
2
2
1
I r dm
I R dm
I R dm
I MR
Calculate I for a thin rod
A uniform thin rod: mass (M); length (L); rotating about its center of mass.
2
2
22
2
23
2
23
2
2
1
3
1
3
1
12
L
L
L
L
L
L
I r dm
I r dr
I r dr
I r
MI r
L
I ML
Common Moments of Inertia
Parallel Axis Theorem
To calculate the moment of inertia around any axis….
Where d is the distance between the center of mass and the axis of rotation
2cmI I Md
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.
sin
r F
rF
Where r is the distance between the pivot point and the force and 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.
0ˆˆ
0ˆˆ
0ˆˆ
kk
jj
ii
The Right Hand Rule
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
Rotational Direction is defined by the axis that it rotates around.
Cross Product
kijik
jkikj
ijkji
ˆˆˆˆˆ
ˆˆˆˆˆ
ˆˆˆˆˆ
Calculating Cross Product
kBkAjBkAiBkA
kBjAjBjAiBjA
kBiAjBiAiBiABA
kBjBiBkAjAiABA
zzyzxz
zyyyxy
zxyxxx
zyxzyx
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
Finding the Determinant
kFrFrjFrFriFrFrFr
kFF
rrj
FF
rri
FF
rrFr
FFF
rrr
kji
Fr
xyyxxzzxyzzy
yx
yx
zx
zx
zy
zy
zyx
zyx
ˆˆˆ
ˆˆˆ
ˆˆˆ
example 6
Find the cross product of 7i + 5j – 3k and 2i-8j + 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…
r = 0.8m
F = 900 N19º
Nm
rF
680
)71sin()900)(8.0(
sin
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
2. Net torque about any point equals zero
0F
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
Newton’s Second Law
When the sum of the torques does not equal zero
Newton’s Second Law
2
( )
( )
( )
F ma
F r
ma r
m r r
mr
I
example 13
Fpivot
Fnormal
Fgravity
2
2
1
2 3
3/
2
g
I
LF mL
grad s
L
L
example 14
Pulley – M1, R
Block – M2
Find a and FT
FT
Fg
2
2 2
2 1 2
2 2 1
2 2 1
2
2 1
1
2
1
21
2
12
y y
g T y
T y
y y
y y
y
y
F ma
F F M a
M g F M a
M g M a M a
M g M a M a
M g M M a
M ga
M M
2
1
1
21
2 1
1 2
2 1
1
2
1
2
1122
2
T
T
T
T
I
aF R M R
R
F M a
M gF M
M M
M M gF
M M
Angular Momentum
For a particle
Remember linear momentum:
So angular momentum:
mvp
sinrmvL
prL
Angular momentum and Torque
dt
dLt
Lt
mvrFr
t
mv
t
pF
For a system…
IL
rmL
rmrrmL
rv
rvmL
ii
iiiii
iii
2
2)(
To check…
I
Idt
d
dt
dL
IL
Conservation
If net torque is zero, then the angular momentum is constant.
0dt
dL
Conservation
For rotation about a fixed axis, we can say:
1100
10
II
LL
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?