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Physics 1710—Warm-up Quiz
A jet ski expels water at a rate of 1440. liters per
minute at a velocity of 45.00 m/s. What thrust
does it produce?
1.
2.
3.
4.
45.00 N
1440. N
1080. N
14112. N
86%
8%
5%
0%
1
2
3
4
Physics 1710—Chapter 10 Rotating Bodies
Solution:
F jet ski = - F water jet
F water jet = dp/dt = d(mv)/dt
F water jet = v dm/dt = (45 m/s)(1440/60 kg/s)
= (45 m/s)(80 kg/s) = 1080 N
How does a canoe paddle work?
Physics 1710—Chapter 10 Rotating Bodies
1′ Lecture
• The motion of a system of point particles is a
combination of motion of the center of mass (CM) and the
motion about the CM.
• Angular displacement is angle through which a body has
rotated.
• Instantaneous angular speed is the time rate of angular
displacement.
• Instantaneous angular acceleration is the time rate of
change in angular speed.
Physics 1710—Chapter 10 Rotating Bodies
Center of Mass (CM)
m1
RCM ≡ ∑mi ri / ∑mi
M = ∑mi
r1
RCM
r3
m2
CM
r2
m3
RCM is the mass-weighted mean position.
Physics 1710—Chapter 10 Rotating Bodies
Center of Mass (CM)
dm
RCM ≡ {∫ rdm }/ M
M≡ ∫ dm
CM
r
RCM
Physics 1710—Chapter 10 Rotating Bodies
Total Linear Momentum
RCM ≡ ∑mi ri / M
Thus:
Thus:
vCM = d RCM /dt = (1/M) ∑i mi d ri /dt
vCM = (1/M) ∑i mi vi
PCM = ∑i pi = total p
aCM = d vCM /dt = (1/M) ∑i mi d vi /dt
aCM = (1/M) ∑i mi d vi /dt
FCM= M aCM = ∑i mi ai
 Force acts as if mass all mass were at CM.
Physics 1710—Chapter 10 Rotating Bodies
The center of mass (CM) of a system of particles
of combined mass M
moves like an equivalent particle of mass M
would move under the influence of the resultant
external force on the system.
Physics 1710—Chapter 10 Rotating Bodies
CM
CM
CM
CM
CM
CMCM
CM CM
CM
Physics 1710—Chapter 10 Rotating Bodies
Angular Rotation
s=r⍬
⍬ = s/r
One radian is the angle subtended by an arc length equal
to the radius of the arc.
1 radian = 180⁰/π ≈ 57.32⁰
⍬
s =r ⍬
Physics 1710—Chapter 10 Rotating Bodies
Hold your hand at arm’s length. What angle (in
radians and degrees) does it subtend?
A.
B.
C.
D.
42%
~ 0.01 rad, 1o.
~ 0.1 rad, 5o.
~ 0.2 rad, 10o
~ 1. rad, 60o.
44%
13%
2%
A
B
C
D
Physics 1710—Chapter 10 Rotating Bodies
Solution:
Θ = s/R = 4″/24″ = .17 rad = 0.2 radian ~ 10 o
Physics 1710—Chapter 10 Rotating Bodies
Average ⍵ave and Instantaneous Velocity ⍵ :
Angular Displacement: ∆⍬ = ⍬f -
⍬I
⍵ave = ∆⍬ / ∆t
⍵ = lim∆t →0 ∆⍬ / ∆t = d⍬/dt
⍵ is pronounced “omega.”
Physics 1710—Chapter 10 Rotating Bodies
Average and
Instantaneous (Angular) Acceleration:
⍺ave = ∆⍵ / ∆t
⍺ = lim∆t →0 ∆⍵ / ∆t = d⍵ / dt
⍺ is pronounced “alpha.”
Physics 1710—Chapter 10 Rotating Bodies
Rotational Kinematics:
(constant ⍺)
⍵f = ⍵i + ⍺t
⍬f = ⍬i + ⍵i t + ½ ⍺t 2
⍵f 2 = ⍵I 2 + 2⍺(⍬f -⍬i )
v=r⍵
a=r⍺
a = v 2 /r = r ⍵ 2
Physics 1710—Chapter 10 Rotating Bodies
Moment of Inertia
I = lim∆mi→0 ∑i ri 2 ∆ mi
I = ∫r 2 dm
I = ∭⍴r 2 d V
N.B. : The moment of inertia is the second moment of
the mass distribution, while the center of mass is the
first moment.
Physics 1710—Chapter 10 Rotating Bodies
Moment of Inertia
I = ∑i xi 2 ∆ mi
mo
mo
xi
mo
mo
xi′
A
B
Which has greater moment of inertia about
the vertical axis? A or B?
Physics 1710—Chapter 10 Rotating Bodies
Which has greater moment of inertia about the
vertical axis? A or B?
68%
A.
B.
C. A and B arr equal.
16%
A
16%
B
C
Physics 1710—Chapter 10 Rotating Bodies
Volume Integrals:
Cartesian Coordinates:
V = ∭dx dy dz = x y z
Cylindrical Coordinates:
V = ∭ r dr dz d
= (½ r 2) h (2) =  r 2 h
Spherical Coordinates:
V = ∭ r 2 sin  dr d d
= (⅓ r 3)(2) [cos (0) – cos ()]= 4/3  r 3
Physics 1710—Chapter 10 Rotating Bodies
Parallel Axis Theorem:
The moment of inertia through a point, not the
center of mass, is equal to the moment of
inertia of the Center of Mass about the axis of
rotation (MD 2) and the moment of inertia of
the body about a parallel axis that passes
through the center of mass:
I = ICM + MD 2
Physics 1710—Chapter 10 Rotating Bodies
Torque:
T=r╳F
Ti = rj Fk – rk Fj
Tx = ry Fz – rz Fy
Ty = rz Fx – rx Fz
Tz = rx Fy – ry Fx
Physics 1710—Chapter 10 Rotating Bodies
Torque:
T = r F sin 
In the direction ⊥ to r & ⊥ to F.
Use right hand rule.
Physics 1710—Chapter 10 Rotating Bodies
Torque Ladder Puzzle:
The torques are balanced. If one
moves the mass higher on the ladder,
should one move the weigh closer,
farther or the same distance from the
spar?
Physics 1710—Chapter 10 Rotating Bodies
Newton’s Second Law for Rotation:
T = I d⍵/dt = I ⍺
Physics 1710—Chapter 10 Rotating Bodies
The Ring Race
What will be the outcome in a race
between a disk and a ring of equal
mass and diameter? Will the ring win,
lose or draw? Why?
Physics 1710—Chapter 10 Rotating Bodies
Work and Energy
in Rotational Motion:
T = I ⍺ = I (d⍵/dt) = I (d⍵/d) (d/dt)
∫T d = ∫I ⍵ (d⍵/d) d
Thus:
W = ∫I ⍵ d⍵
= ½ I ⍵f 2 - ½ I ⍵i 2
Physics 1710—Chapter 10 Rotating Bodies
Work – Rotational Energy Relation:
The net work done by external torques
in rotating a symmetric rigid object
about a fixed axis equals the change in
the object’s rotational energy.
∆W = ½ I ⍵f 2 - ½ I ⍵i 2
Physics 1710—Chapter 10 Rotating Bodies
Summary:
•Angular displacement is the angle
through which a body has rotated.
•Instantaneous angular speed is the
time rate of angular displacement.
•Instantaneous angular acceleration is
the time rate of change in angular
speed.
Physics 1710—Chapter 10 Rotating Bodies
Summary (cont’d):
•The moment of inertia is the measure of
the (inertial) resistance to angular
acceleration and equal to the second
moment of the mass distribution.
•Torque (“twist”) is the vector product of a
force and the “moment” arm.
Physics 1710—Chapter 10 Rotating Bodies
Which will have the greater initial velocity?
Scenario A or B?
Think!
Peer Instruction Time
No Talking!
Confer!