Rotational Equilibrium

1
Chapter 8

• Rotational Equilibrium
• and
• Rotational Dynamics

Conceptual questions 1,2,11,12,16,17 Quick
quizzes 1,2,3,4,6,7,8 Problems 9,24,55
2
Torque

• Torque, , is the equivalence of a force but in
  rotational motion
• is the torque
• F is the force
• d is the lever arm (or moment arm)
• UNITS
• SI
• Newton x meter Nm
• US Customary
• foot x pound ft lb

3
Direction of Torque

• Torque is a vector quantity
• The direction is perpendicular to the plane
  determined by the lever arm and the force
• For two dimensional problems, torque is directed
  into or out of the plane
• If the turning tendency of the force is
  counterclockwise, the torque will be positive
• If the turning tendency is clockwise, the torque
  will be negative

4
Lever Arm

• The lever arm, d, is the perpendicular distance
  from the axis of rotation to a line drawn from
  the axis of rotation to a line along the the
  direction of the force
• d L sin F

From the components of the force or from the
lever arm, F is the force L is the distance
along the object F is the angle between the force
and the object D is the lever arm
5
Torque and Equilibrium
Net Torque
The net torque is the sum of all the torques
produced by all the forces Remember to account
for the direction of the tendency for
rotation Counterclockwise torques are
positive Clockwise torques are negative

• First Condition of Equilibrium states that the
  net external force must be zero
• This is a necessary, but not sufficient,
  condition to ensure that an object is in complete
  mechanical equilibrium
• This is a statement of translational equilibrium

6
Torque and Equilibrium, cont

• To ensure mechanical equilibrium, you need to
  ensure rotational equilibrium as well as
  translational
• The Second Condition of Equilibrium states that
  the net external torque must be zero

7
Mechanical Equilibrium

• The First Condition of Equilibrium is satisfied
• The Second Condition is not satisfied
• Both forces would produce clockwise rotations

8
Axis of Rotation

• If the object is in equilibrium, it does not
  matter where you put the axis of rotation for
  calculating the net torque
• The location of the axis of rotation is
  completely arbitrary
• When solving a problem, you must specify an axis
  of rotation
• Once you have chosen an axis, you must maintain
  that choice consistently throughout the problem

9
Example of aFree Body Diagram

• Isolate the object to be analyzed
• Draw the free body diagram for that object
• Include all the external forces acting on the
  object

10
Center of Gravity

• The force of gravity acting on an object must be
  considered
• In finding the torque produced by the force of
  gravity, all of the weight of the object can be
  considered to be concentrated at one point
• We wish to locate the point of application of the
  single force , whose magnitude is equal to the
  weight of the object, and whose effect on the
  rotation is the same as all the individual
  particles.
• This point is called the center of gravity of the
  object

11
Calculating the Center of Gravity

• The object is divided up into a large number of
  very small particles of weight (mg)
• Each particle will have a set of coordinates
  indicating its location (x,y)
• The torque produced by each particle about the
  axis of rotation is equal to its weight times its
  lever arm

12
Coordinates of the Center of Gravity

• The coordinates of the center of gravity can be
  found from

13
Experimentally Determining the Center of Gravity

• The wrench is hung freely from two different
  pivots
• The intersection of the lines indicates the
  center of gravity
• A rigid object can be balanced by a single force
  equal in magnitude to its weight as long as the
  force is acting upward through the objects
  center of gravity

14
Solving Equilibrium Problems

• Draw a diagram of the system
• Isolate the object being analyzed and draw a free
  body diagram showing all the external forces
  acting on the object
• Establish convenient coordinate axes for each
  object. Apply the First Condition of Equilibrium
• Choose a convenient rotational axis for
  calculating the net torque on the object. Apply
  the Second Condition of Equilibrium
• Solve the resulting simultaneous equations for
  all of the unknowns

15
Example of aFree Body Diagram

• The free body diagram includes the directions of
  the forces
• The weights act through the centers of gravity of
  their objects

16
Problem 8-9
A cook holds a 2.00-kg carton of milk at arms
length (Fig. P8.9). What force FB must be exerted
by the biceps muscle? (Ignore the weight of the
forearm.)
17
Review
18
Problem 8-24
A 15.0-m, 500-N uniform ladder rests against a
frictionless wall, making an angle of 60.0 with
the horizontal. (a) Find the horizontal and
vertical forces exerted on the base of the ladder
by Earth when an 800-N fire fighter is 4.00 m
from the bottom. (b) If the ladder is just on
the verge of slipping when the fire fighter is
9.00 m up, what is the coefficient of static
friction between ladder and ground?
19
F2
15 m
800 N
d3
d
500 N
F1
60o
f
d315.0m sin60.0o d2d cos60.0o d17.50m cos60.0o
d1
d2
20
Part a d4.00 m 1st condition SFx0 f-F20 SFy0
F1-500N-800N0 2nd condition St0 F2d3-800N
d2-500N d10 find F2 Next, calculate F1 and
f Part b For d9.00 m find F2 Then find F1 and f.
fmaxmstatic F1
21
Conceptual questions

• 1. Why cant you put your heels firmly against
  the wall and then bend over without falling?
• 17. A ladder rest against the wall. Would you
  feel safer climbing the ladder if you were told
  that the floor is frictionless but the wall is
  rough, or that the wall is frictionless but the
  floor is rough?

f
22
Torque and Angular Acceleration

• When a rigid object is subject to a net torque
  (?0), it undergoes an angular acceleration
• The angular acceleration is directly proportional
  to the net torque
• The relationship is analogous to ?F ma

23
Moment of Inertia

• The angular acceleration is inversely
  proportional to the analogy of the mass in a
  rotating system
• This mass analog is called the moment of inertia,
  I, of the object
• SI units are kg m2

24
Newtons Second Law for a Rotating Object

• The angular acceleration is directly proportional
  to the net torque
• The angular acceleration is inversely
  proportional to the moment of inertia of the
  object

25
Quick quiz

• 8.1 Using a screwdriver, you try to loosen a
  stubborn screw from a piece of wood and you fail.
  In order to succeed, you should find a
  screwdriver that (a) is longer, (b) is shorter,
  (c) has a narrower handle, or (d) has a fatter
  handle.
• 8.2 A constant net torque is applied to an
  object. One of the following will definitely not
  be a constant. It is the objects (a) angular
  acceleration, (b) angular velocity, (c) moment of
  inertia, or (d) center of gravity.

26
Moment of Inertia of a Uniform Ring

• Image the hoop is divided into a number of small
  segments, m1
• These segments are equidistant from the axis

27
Other Moments of Inertia
28
Quick quiz 8-3

• The two rigid objects shown in Figure 8.16 have
  the same mass, radius, and angular speed. If the
  same braking torque is applied to each, which
  takes longer to stop? (a) A, (b) B, (c) not
  enough information to say. (Hint See Table 8.1
  for the moment of inertia of these objects.)

29
Rotational Kinetic Energy

• An object rotating about some axis with an
  angular speed, ?, has rotational kinetic energy
• KE½I?2
• Conservation of mechanical energy

30
Rolling down an incline

• At the top potential energy
• PEmgh
• At the bottom - kinetic energy
• KEKEtKErmv2/2Iw2/2

31
Example 8-12

• mgh(mv2Iw2)/2
• Isphere(2/5)mr2
• vwr
• mgh(1/2)mv2(1/5)mv2(7/10)mv2
• v(10gh/7)1/2

32
Quick quiz

• 8.4 Two spheres, one hollow and one solid, are
  rotating with the same angular speed about an
  axis through their centers. Both spheres have
  the same mass and radius. Which sphere, if
  either, has the higher rotational kinetic energy?
  
• The hollow sphere.
• The solid sphere.
• They have the same kinetic energy.

33
Other Moments of Inertia
34
Quick quiz 8.5 Which arrives at the bottom
first? (a) A ball rolling without sliding down
incline A, (b) a solid cylinder rolling without
sliding down incline A,(c) a box of the same
mass as the ball sliding down a frictionless
incline A. Assume that each object is released
from rest at the top of the incline.
35
Angular Momentum

• Similarly to the relationship between force and
  momentum in a linear system, we can show the
  relationship between torque and angular momentum
• Angular momentum is defined as
• L I ?
• and

36
Conservation of Angular Momentum

• Conservation of Angular Momentum states The
  angular momentum of a system is conserved when
  the net external torque acting on the systems is
  zero.
• That is, when
• If the net torque is zero, the angular momentum
  remains constant

37
Quick quiz

• 8.6 A horizontal disk with moment of inertia I1
  rotates with angular speed ?0 about a vertical
  frictionless axle. A second horizontal disk,
  with moment of inertia I2 and initially not
  rotating, drops onto the first. Because the
  surfaces are rough, the two eventually reach the
  same angular speed ?. Is the ratio ?/ ?0 equal
  to (a) I1/ I2, (b) I2/ I1, (c) I1/
  ( I1 I2), or (d) I2/ ( I1 I2)?
• 8.7 If global warming continues to occur, it is
  likely that some ice from Earths polar ice caps
  will melt and the water will be distributed
  closer to the Equator. If this occurs, would the
  length of the day (one revolution) (a) increase,
  (b) decrease, or (c) remain the same?

38
Problem 8-55
A cylinder with moment of inertia I1 rotates with
angular velocity ?0 about a frictionless vertical
axle. A second cylinder, with moment of inertia
I2, initially not rotating, drops onto the first
cylinder (Fig. P8.55). Since the surfaces are
rough, the two eventually reach the same angular
speed ?. (a) Calculate ?. (b) Show that kinetic
energy is lost in this situation, and calculate
the ratio of the final to the initial kinetic
energy.
39
Conceptual questions
8.2 Why does a tall athlete have an advantage
over a smaller one when the two are competing in
the high jump? 8.11 Stars originate as large
bodies of slowly rotating gas. Because of
gravity, these clumps of gas slowly decrease in
size. What happens to the angular speed of a
star as it shrinks?
40
Conceptual questions
8.12 If a high jumper positions his body
correctly when going over the bar, the center of
gravity of the athlete may actually pass under
the bar. Explain. 8.16 A cat usually lands on
its feet regardless of the position from which it
is dropped. Why does this type of motion occur?
41
Problem 8-51
The puck in Figure P8.51 has a mass of 0.120 kg.
Its original distance from the center of rotation
is 40.0 cm, and the puck is moving with a speed
of 80.0 cm/s. The string is pulled downward 15.0
cm through the hole in the frictionless table.
Determine the work done on the puck. (Hint
Consider the change of kinetic energy of the
puck.)