Rotational Equilibrium

1
Chapter 8

• Rotational Equilibrium
• and
• Rotational Dynamics

2
Force vs. Torque

• Forces cause accelerations
• Torques cause angular accelerations
• Force and torque are related

3
Torque

• The door is free to rotate about an axis through
  O
• There are three factors that determine the
  effectiveness of the force in opening the door
• The magnitude of the force
• The position of the application of the force
• The angle at which the force is applied

4
Torque, cont

• Torque, t, is the tendency of a force to rotate
  an object about some axis
• t r F
• t is the torque
• symbol is the Greek tau
• F is the force
• r is the length of the position vector
• SI unit is N.m

5
Direction of Torque

• Torque is a vector quantity
• The direction is perpendicular to the plane
  determined by the position vector and the force
• 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

6
Multiple Torques

• When two or more torques are acting on an object,
  the torques are added
• As vectors
• If the net torque is zero, the objects rate of
  rotation doesnt change

7
General Definition of Torque

• The applied force is not always perpendicular to
  the position vector
• The component of the force perpendicular to the
  object will cause it to rotate

8
General Definition of Torque, cont

• When the force is parallel to the position
  vector, no rotation occurs
• When the force is at some angle, the
  perpendicular component causes the rotation

9
General Definition of Torque, final

• Taking the angle into account leads to a more
  general definition of torque
• t r F sin q
• F is the force
• r is the position vector
• q is the angle between the force and the position
  vector

10
Lever Arm

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

11
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

12
Example 1
If the torque required to loosen a nut that is
holding a flat tire in place on a car has a
magnitude of 40.0 N m, what minimum force must
be exerted by the mechanic at the end of a
30.0-cm lug wrench to accomplish the task?
13
Torque and Equilibrium

• First Condition of Equilibrium
• 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

14
Torque and Equilibrium, cont

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

15
Equilibrium Example

• The woman, mass m, sits on the left end of the
  see-saw
• The man, mass M, sits where the see-saw will be
  balanced
• Apply the Second Condition of Equilibrium and
  solve for the unknown distance, x

16
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
• Often the nature of the problem will suggest a
  convenient location for the axis
• 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

17
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 a single point

18
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)

19
Calculating the Center of Gravity, cont.

• We assume the object is free to rotate about its
  center
• The torque produced by each particle about the
  axis of rotation is equal to its weight times its
  lever arm
• For example, t1 m1 g x1

20
Calculating the Center of Gravity, cont.

• 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

21
Coordinates of the Center of Gravity

• The coordinates of the center of gravity can be
  found from the sum of the torques acting on the
  individual particles being set equal to the
  torque produced by the weight of the object

22
Center of Gravity of a Uniform Object

• The center of gravity of a homogenous, symmetric
  body must lie on the axis of symmetry.
• Often, the center of gravity of such an object is
  the geometric center of the object.

23
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

24
Notes About Equilibrium

• A zero net torque does not mean the absence of
  rotational motion
• An object that rotates at uniform angular
  velocity can be under the influence of a zero net
  torque
• This is analogous to the translational situation
  where a zero net force does not mean the object
  is not in motion

25
Solving Equilibrium Problems

• Draw a diagram of the system
• Include coordinates and choose a rotation axis
• Isolate the object being analyzed and draw a free
  body diagram showing all the external forces
  acting on the object
• For systems containing more than one object, draw
  a separate free body diagram for each object

26
Problem Solving, cont.

• Apply the Second Condition of Equilibrium
• This will yield a single equation, often with one
  unknown which can be solved immediately
• Apply the First Condition of Equilibrium
• This will give you two more equations
• Solve the resulting simultaneous equations for
  all of the unknowns
• Solving by substitution is generally easiest

27
Example of a Free Body Diagram (Forearm)

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

28
Example of a Free Body Diagram (Beam)

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

29
Example 2
Write the necessary equations of equilibrium of
the object shown in Figure P8.4. Take the origin
of the torque equation about an axis
perpendicular to the page through the point O.
30
Example 3
A meterstick is found to balance at the 49.7-cm
mark when placed on a fulcrum. When a 50.0-gram
mass is attached at the 10.0-cm mark, the fulcrum
must be moved to the 39.2-cm mark for balance.
What is the mass of the meter stick?
31
Example 4
A window washer is standing on a scaffold
supported by a vertical rope at each end. The
scaffold weighs 200 N and is 3.00 m long. What is
the tension in each rope when the 700-N worker
stands 1.00 m from one end?
32
Example 5
A 500-N uniform rectangular sign 4.00 m wide and
3.00 m high is suspended from a horizontal,
6.00-m-long, uniform, 100-N rod as indicated in
Figure P8.17. The left end of the rod is
supported by a hinge, and the right end is
supported by a thin cable making a 30.0 angle
with the vertical. (a) Find the tension T in the
cable. (b) Find the horizontal and vertical
components of force exerted on the left end of
the rod by the hinge.
33
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
• Newtons Second Law

34
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

35
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

36
More About Moment of Inertia

• There is a major difference between moment of
  inertia and mass the moment of inertia depends
  on the quantity of matter and its distribution in
  the rigid object.
• The moment of inertia also depends upon the
  location of the axis of rotation

37
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

38
Other Moments of Inertia
39
Example, Newtons Second Law for Rotation

• Draw free body diagrams of each object
• Only the cylinder is rotating, so apply St I a
• The bucket is falling, but not rotating, so apply
  SF m a
• Remember that a a r and solve the resulting
  equations

40
Example 6
A potters wheel having a radius of 0.50 m and a
moment of inertia of 12 kg m2 is rotating
freely at 50 rev/min. The potter can stop the
wheel in 6.0 s by pressing a wet rag against the
rim and exerting a radially inward force of 70 N.
Find the effective coefficient of kinetic
friction between the wheel and the wet rag.
41
Example 7
A 150-kg merry-go-round in the shape of a
uniform, solid, horizontal disk of radius 1.50 m
is set in motion by wrapping a rope about the rim
of the disk and pulling on the rope. What
constant force must be exerted on the rope to
bring the merry-go-round from rest to an angular
speed of 0.500 rev/s in 2.00 s?
42
Rotational Kinetic Energy

• An object rotating about some axis with an
  angular speed, ?, has rotational kinetic energy
  ½I?2
• Energy concepts can be useful for simplifying the
  analysis of rotational motion

43
Total Energy of a System

• Conservation of Mechanical Energy
• Remember, this is for conservative forces, no
  dissipative forces such as friction can be
  present
• Potential energies of any other conservative
  forces could be added

44
Work-Energy in a Rotating System

• In the case where there are dissipative forces
  such as friction, use the generalized Work-Energy
  Theorem instead of Conservation of Energy
• Wnc DKEt DKER DPE

45
General Problem Solving Hints

• The same basic techniques that were used in
  linear motion can be applied to rotational
  motion.
• Analogies F becomes , m becomes I and a
  becomes , v becomes ? and x becomes ?

46
Problem Solving Hints for Energy Methods

• Choose two points of interest
• One where all the necessary information is given
• The other where information is desired
• Identify the conservative and nonconservative
  forces

47
Problem Solving Hints for Energy Methods, cont

• Write the general equation for the Work-Energy
  theorem if there are nonconservative forces
• Use Conservation of Energy if there are no
  nonconservative forces
• Use v w to combine terms
• Solve for the unknown

48
Example 8
A 10.0-kg cylinder rolls without slipping on a
rough surface. At an instant when its center of
gravity has a speed of 10.0 m/s, determine (a)
the translational kinetic energy of its center of
gravity, (b) the rotational kinetic energy about
its center of gravity, and (c) its total kinetic
energy.
49
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

50
Angular Momentum, cont

• If the net torque is zero, the angular momentum
  remains constant
• 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

51
Conservation Rules, Summary

• In an isolated system, the following quantities
  are conserved
• Mechanical energy
• Linear momentum
• Angular momentum

52
Conservation of Angular Momentum, Example

• With hands and feet drawn closer to the body, the
  skaters angular speed increases
• L is conserved, I decreases, w increases

53
Example 9
A 60.0-kg woman stands at the rim of a horizontal
turntable having a moment of inertia of 500 kg
m2 and a radius of 2.00 m. The turntable is
initially at rest and is free to rotate about a
frictionless, vertical axle through its center.
The woman then starts walking around the rim
clockwise (as viewed from above the system) at a
constant speed of 1.50 m/s relative to the Earth.
(a) In what direction and with what angular speed
does the turntable rotate? (b) How much work does
the woman do to set herself and the turntable
into motion?
54
Example 10
A light rigid rod 1.00 m in length rotates about
an axis perpendicular to its length and through
its center, as shown in Figure P8.45. Two
particles of masses 4.00 kg and 3.00 kg are
connected to the ends of the rod. What is the
angular momentum of the system if the speed of
each particle is 5.00 m/s? (Neglect the rods
mass.)