Dynamics of Rotational Motion
Torque: the rotational analogue of force
Torque = force x moment arm
t = Fl
moment arm = perpendicular distance through which the force acts
a.k.a. lever arm
l
l
F
F
l
l
F
F
t = Fl = F r sin f = Ftan r
  
τ  r F
Phys211C10 p1
Example: To loosen a difficult nut, a wrench with a piece of pipe for additional leverage is
used. If the distance from the center of the nut to the end of the pipe is 0.80 m, and the
wrench and pipe make an angle of 20º with respect to the horizontal, determine the
magnitude and direction of the torque applied by a 900 N man standing on the edge of the
pipe.
Phys211C10 p2
Torque and Angular Acceleration
single particle contribution
y-axis
axis of rotation
F1,y
r1
F1, tan  m1a1, tan
since t 1  r1 F1, tan
F1,tan
F1,rad
a1, tan  r1
r
t 1  m1r12
for all component particles :
t i  mi ri   I
2
t  I
Phys211C10 p3
Problem solving tactics:
•Draw a diagram
•For each body, draw a free body diagram
•Choose coordinates and directions (including for rotations)
•Relate torques to angular accelerations and forces to linear
accelerations
•for more than one body, repeat and note any geometrical
relation between motions of bodies
Phys211C10 p4
A cord is wrapped around a solid 50 kg cylinder which has a diameter
of 0.120 m, and which rotates (frictionlessly) about an axis through
its center. A 9.00 N force is applied to the end of the cable, causing
the cable to unwind and the drum (initially at rest) to rotate. What is
the angular acceleration of the drum?
9.00 N
A mass m is suspended by a string wrapped around a pulley of radius R and
moment of inertia I. The mass and pulley are initially at rest. Determine the
acceleration of the disk and of the suspended mass in terms of the given
parameters (m, I, R). Examine the special case where the pulley is a uniform disk
of mass M.
Phys211C10 p5
A mass m2 is suspended by a string wrapped around a pulley of radius R and moment of
inertia I. The string is then connected to a mass m1 which slides on a horizontal frictionless
surface. The masses and pulley are initially at rest. Determine the accelerations of the masses
and the pulley in terms of the the given parameters (m1, m2, I, R). Examine the special case
where the pulley can be treated as a thin cylindrical shell of mass M.
m1
M
m2
Phys211C10 p6
y-axis
cm axis
axis of rotation
More on Combining Translation and Rotation
F1,y
v’
1
1   1
 
 
2
r
 mi vi  mi vi  vi  mi (vi 'v cm )  (vi 'v cm )
v
2
2
2
1
 
2
2
v
 mi (vi ' 2vi 'v cm  vcm )
2

1
 
2
m v ' is velocity of cm relative to cm!
 K i  mi (vi '2 2vi 'v cm  vcm )
2
1
1
 
2
 mi vcm  mi vi 'v cm  mi vi '2
2
2
1
1
2
 mi vcm  0  mi ri '2  2
2
2
1
1
2
 mvcm  I cm 2
2
2


 Macm
τ  I
vcm
i
Ki
i
i
cm
K
K

Fext
i i
Phys211C10 p7
example: A primitive yo-yo is made of a solid cylinder of mass M with string wrapped
around its radius R.
What is the speed of the cylinder after it has dropped a distance h?
What is acceleration of the cylinder and the tension in the string?
example: A solid bowling bowl rolls without slipping down a ramp inclined at an angle b
with respect to the horizontal.
What is the ball’s acceleration?
Phys211C10 p8
Rolling Friction: deforming surfaces
Stiff surface; normal force
through cm, no torque
Distorted surface; normal
force offset from cm, net
torque slows rotation
Phys211C10 p9
Work and Power in Rotational Motion
Force applied to a point on a rotating object
dW  Ftan Rdq
Ftan
s
R
q
Ftan
 tdq
W   tdq
W  tq when torqu e is constant
Phys211C10 p10
d
dq
tdq  Idq  I
dq  I d  Id
dt
dt
W   tdq   Id 
1
1 2
2
W  I f  Ii
2
2
dW
dq
P
t
dt
dt
P  t
Work - Energy
Power
Phys211C10 p11
If the power output of an engine is 200 hp at 6000rpm, what is the corresponding torque?
A motor produces a torque of 10 N-m with disk mounted to its shaft. The disk has a moment
of inertia of 2.0 kg m2. If the system starts from rest, determine the work done by the motor
in 8.0 s, and the kinetic energy of the disk at the end of this time. What is the average power
delivered during this 8 seconds?
Phys211C10 p12
Angular momentum
for a single particle, defined as
L  r × p = r × mv
depends upon origin!
mv
f
mv
r
L  mvr sinf = mvr = mvr
r



dL d r
dv 

 
 
  mv  r  m  v  mv  r  ma
dt dt
dt

dL   
 rF  τ
L
dt



Right hand
L  Iω

Generalize to composite object
In the absence of external torques,
angular momentum is conserved
rule for
direction
Phys211C10 p13
Two disks with moments of inertia Ia and Ib share a common axis of rotation, but have
different angular velocities (a and b ). The disks are brought together so that they
eventually reach a common final angular velocity. Derive an expression for that final
angular velocity .
Suppose the two disks are uniform with masses of 2.0 kg and 4.0 kg, radii of .20 m and .10
m and initial angular speeds of 50 rad/s and 200 rad/s. Determine the final angular speed
and the change in the kinetic energy of the system.
Phys211C10 p14
A bullet (10 g and initial speed 400 m/s) is fired into a door from a direction perpendicular
to the door. The door has a mass of 15 kg and is 1.0 m wide. Determine the angular
velocity of the door after the bullet imbeds itself in the center of the door.
Phys211C10 p15
Gyroscopes and Precession
  
τ  r F
L
 

dL  τdt
 geometry
 
df | dL | | L | t wr


 
dt
dt
L I
t
w
Phys211C10 p16