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1
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Chapter 8: Torque and Angular Momentum
•Rotational Kinetic Energy
•Rotational Inertia
•Torque
•Work Done by a Torque
•Equilibrium (revisited)
•Rotational Form of Newton’s 2nd Law
•Rolling Objects
•Angular Momentum
2
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§8.1 Rotational KE and Inertia
For a rotating solid body:
n
iiinn vmvmvmvmK
1
22222
211rot 2
1
2
1
2
1
2
1
For a rotating body vi=ri where ri is the distance from the rotation axis to the mass mi.
22
1
2
1
2rot 2
1
2
1
2
1 IrmrmKn
iii
n
iii
3
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The quantity
n
iiirmI
1
2 is called rotational inertia or moment of inertia.
Use the above expression when the number of masses that make up a body is small. Use the moments of inertia in table 8.1 for extended bodies.
4
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example: (a) Find the moment of inertia of the system below. The masses are m1 and m2 and they are separated by a distance r. Assume the rod connecting the masses is massless.
r1 r2m1 m2
r1 and r2 are the distances between mass 1 and the rotation axis and mass 2 and the rotation axis (the dashed, vertical line) respectively.
5
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
2222
211
2
1
2 m kg 67.0
rmrmrmIi
ii
(b) What is the moment of inertia if the axis is moved so that is passes through m1?
Example continued:
2222
2
1
2 m kg 00.1
rmrmIi
ii
Take m1 = 2.00 kg, m2 = 1.00 kg, r1= 0.33 m , and r2 = 0.67 m.
6
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 8.4): What is the rotational inertia of a solid iron disk of mass 49.0 kg with a thickness of 5.00 cm and a radius of 20.0 cm, about an axis through its center and perpendicular to it?
From table 8.1:
222 m kg 98.0m 2.0kg 0.492
1
2
1 MRI
7
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§8.2 Torque
A rotating (spinning) body will continue to rotate unless it is acted upon by a torque.
hinge
Push
Q: Where on a door do you push to open it?
A: Far from the hinge.
8
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Torque method 1:
rF
Hinge end
FF
||F
r = the distance from the rotation axis (hinge) to the point where the force F is applied.
F is the component of the force F that is perpendicular to the door (here it is Fsin).
Top view of door
9
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The units of torque are Nm (not joules!).
By convention:
•When the applied force causes the object to rotate counterclockwise (CCW) then is positive.
•When the applied force causes the object to rotate clockwise (CW) then is negative.
10
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Torque method 2: Fr
r is called the lever arm and F is the magnitude of the applied force.
Lever arm is the perpendicular distance to the line of action of the force.
11
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Line of action of the force
Hinge end
F
r
Top view of door
Lever arm
The torque is:
sin
sin
rrr
r
sinrF
Fr
Same as before
12
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 8.12): The pull cord of a lawnmower engine is wound around a drum of radius 6.00 cm, while the cord is pulled with a force of 75.0 N to start the engine. What magnitude torque does the cord apply to the drum?
F=75 N
R=6.00 cm
Nm 5.4N 0.75m 06.0
rF
Fr
13
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example: Calculate the torque due to the three forces shown about the left end of the bar (the red X). The length of the bar is 4m and F2 acts in the middle of the bar.
4510
30
F1=25 N
F3=20 N
F2=30 N
X
14
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The lever arms are: m 695.010sinm4
m 73.160sinm2
0
3
2
1
r
r
r
Lever arm for F3
Lever arm for F2
4510
30
F1=25 N
F3=20 N
F2=30 N
X
Example continued:
15
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The torques are: Nm 9.13N 20m 0.695
Nm 9.51N 30m 73.1
0
3
2
1
The net torque is + 65.8 Nm and is the sum of the above results.
Example continued:
16
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§8.3 Work done by a Torque
.WThe work done by a torque is
where is the angle (in radians) the object turns through.
17
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 8.25): A flywheel of mass 182 kg has a radius of 0.62 m (assume the flywheel is a hoop).
(a) What is the torque required to bring the flywheel from rest to a speed of 120 rpm in an interval of 30 sec?
rad/sec 6.12sec 60
min 1
rev 1
rad 2
min
rev 120
f
Nm 4.2922
2
tmr
tmr
tmrrrmmarrF
fif
18
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
(b) How much work is done in this 30 sec period?
J 560022
tt
tW
ffi
av
Example continued:
19
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§8.4 Equilibrium
The conditions for equilibrium are: 0
0
τ
F
20
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 8.36): A sign is supported by a uniform horizontal boom of length 3.00 m and weight 80.0 N. A cable, inclined at a 35 angle with the boom, is attached at a distance of 2.38 m from the hinge at the wall. The weight of the sign is 120.0 N. What is the tension in the cable and what are the horizontal and vertical forces exerted on the boom by the hinge?
21
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
FBD for the bar:
Apply the conditions for equilibrium to the bar:
0sin2
)3(
0sin )2(
0cos )1(
bar
bar
xTLFL
w
TFwFF
TFF
sb
sbyy
xx
Example continued:
T
wbar
Fy
Fx
X
Fsb
x
y
22
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Equation (3) can be solved for T:
N 352sin
2
x
LFL
wT
sbbar
N 288cos TFxEquation (1) can be solved for Fx:
Equation (2) can be solved for Fy:N 00.2
sin
TFwF sbbary
Example continued:
23
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§8.5 Equilibrium in the Human Body
Example (text problem 8.43): Find the force exerted by the biceps muscle in holding a one liter milk carton with the forearm parallel to the floor. Assume that the hand is 35.0 cm from the elbow and that the upper arm is 30.0 cm long. The elbow is bent at a right angle and one tendon of the biceps is attached at a position 5.00 cm from the elbow and the other is attached 30.0 cm from the elbow. The weight of the forearm and empty hand is 18.0 N and the center of gravity is at a distance of 16.5 cm from the elbow.
24
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example continued:
Fb
wFca
“hinge” (elbow joint)
N 130
0
1
32
321
x
xFwxF
xFwxxF
cab
cab
25
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§8.6 Rotational Form of Newton’s 2nd Law
I
Compare to aF m
26
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 8.57): A bicycle wheel (a hoop) of radius 0.3 m and mass 2 kg is rotating at 4.00 rev/sec. After 50 sec the wheel comes to a stop because of friction. What is the magnitude of the average torque due to frictional forces?
2MRI 0
rad/sec 1.25rev 1
rad 2
sec
rev00.4
f
i
2rad/s 50.0
tt
if
Nm 09.02 MRav
27
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§8.7 Rolling Objects
An object that is rolling combines translational motion (its center of mass moves) and rotational motion (points in the body rotate around the center of mass).
For a rolling object:
22cm
rotTtot
2
1
2
1 Imv
KKK
If the object rolls without slipping then vcm = R.
28
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example: Two objects (a solid disk and a solid sphere) are rolling down a ramp. Both objects start from rest and from the same height. Which object reaches the bottom of the ramp first?
h
The object with the largest linear velocity (v) at the bottom of the ramp will win the race.
29
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
2
22
2222
2
2
1
2
1
2
1
2
1
2
100
R
Im
mghv
vR
Immgh
R
vImvImvmgh
KUKU
EE
ffii
fi
Apply conservation of mechanical energy:
Example continued:
Solving for v:
30
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
2sphere
2disk
5
22
1
mRI
mRI
For the disk:
For the sphere:
ghv3
4disk
ghv7
10sphere
Since Vsphere> Vdisk the sphere wins the race.
ghv 2box Compare these to a box sliding down the ramp.
Example continued:
The moments of inertia are:
31
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
How do objects in the previous example roll?
FBD:
Both the normal force and the weight act through the center of mass so =0. This means that the object cannot rotate when only these two forces are applied.
N
w
y
x
32
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
FBD:
Add friction:
0cos
sin cm
wNF
maFwF
IrF
y
sx
sy
x
N
w
Fs
Also need acm = R and
The above system of equations can be solved for v at the bottom of the ramp. The result is the same as when using energy methods. (See text example 8.13.)
xavv 220
2
It is static friction that makes an object roll.
33
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§8.8 Angular Momentum
vp
pF
mtt
0
net lim
ωL
L
Itt
0
net lim
Units of p are kg m/s Units of L are kg m2/s
When no net external forces act, the momentum of a system remains constant (pi = pf)
When no net external torques act, the angular momentum of a system remains constant (Li = Lf).
34
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 8.70): A turntable of mass 5.00 kg has a radius of 0.100 m and spins with a frequency of 0.500 rev/sec. What is the angular momentum? Assume a uniform disk.
rad/sec 14.3rev 1
rad 2
sec
rev500.0
/sm kg 079.02
1 22
MRIL
35
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 8.79): A skater is initially spinning at a rate of 10.0 rad/sec with I=2.50 kg m2 when her arms are extended. What is her angular velocity after she pulls her arms in and reduces I to 1.60 kg m2?
He/she is on ice, so we can ignore external torques.
rad/sec 6.15rad/sec 0.10m kg 60.1
m kg 50.22
2
if
if
ffii
fi
I
I
II
LL
36
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§8.9 The Vector Nature of Angular Momentum
Angular momentum is a vector. Its direction is defined with a right-hand rule.
37
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Curl the fingers of your right hand so that they curl in the direction a point on the object moves, and your thumb will point in the direction of the angular momentum.
38
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Consider a person holding a spinning wheel. When viewed from the front, the wheel spins CCW.
Holding the wheel horizontal, they step on to a platform that is free to rotate about a vertical axis.
39
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Initially, nothing happens. They then move the wheel so that it is over their head. As a result, the platform turns CW (when viewed from above).
This is a result of conserving angular momentum.
40
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Initially there is no angular momentum about the vertical axis. When the wheel is moved so that it has angular momentum about this axis, the platform must spin in the opposite direction so that the net angular momentum stays zero.
Is angular momentum conserved about the direction of the wheel’s initial, horizontal axis?
41
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
It is not. The floor exerts a torque on the system (platform + person), thus angular momentum is not conserved here.
42
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Summary
•Rotational Kinetic Energy
•Moment of Inertia
•Torque (two methods)
•Conditions for Equilibrium
•Newton’s 2nd Law in Rotational Form
•Angular Momentum
•Conservation of Angular Momentum