College Physics

1
General Physics (PHY 2130)
Lecture VII

• Rotational dynamics
• torque
• moment of inertia
• angular momentum
• conservation of angular momentum
• rotational kinetic energy
• Exam II review

http//www.physics.wayne.edu/apetrov/PHY2130/
2
Lightning Review

• Last lecture
• Rotations
• angular displacement, velocity (rate of change of
  angular displacement), acceleration (rate of
  change of angular velocity)
• motion with constant angular acceleration
• Gravity laws

Review Problem A rider in a barrel of fun
finds herself stuck with her back to the wall.
Which diagram correctly shows the forces acting
on her?
3
Sample Review Problem
An engineer wishes to design a curved exit ramp
for a toll road in such a way that a car will not
have to rely on friction to round the curve
without skidding. She does so by banking the road
in such a way that the force causing the
centripetal acceleration will be supplied by the
component of the normal force toward the center
of the circular path. Find the angle at which the
curve should be banked if a typical car rounds it
at a 50.0-m radius and a speed of 13.4 m/s.
4

• Rotational Equilibrium
• and
• Rotational Dynamics

5
Torque

• Consider force required to open door. Is it
  easier to open the door by pushing/pulling away
  from hinge or close to hinge?

close to hinge
away from hinge
Farther from from hinge, larger rotational effect!
Physics concept torque
6
Torque

• Torque, , is the tendency of a force to rotate
  an object about some axis
• is the torque
• d is the lever arm (or moment arm)
• F is the force

Door example
7
Lever Arm

• The lever arm, d, is the shortest (perpendicular)
  distance from the axis of rotation to a line
  drawn along the the direction of the force
• d L sin F
• It is not necessarily the distance between the
  axis of rotation and point where the force is
  applied

8
Direction of Torque

• Torque is a vector quantity
• The direction is perpendicular to the plane
  determined by the lever arm and the force
• Direction and sign
• 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

Direction of torque out of page
9
An Alternative Look at Torque

• The force could also be resolved into its x- and
  y-components
• The x-component, F cos F, produces 0 torque
• The y-component, F sin F, produces a non-zero
  torque

L
F is the force
L is the distance along the object
F is the angle between force and object
10
Lets watch the movie!
11
ConcepTest 1
You are trying to open a door that is stuck by
pulling on the doorknob in a direction
perpendicular to the door. If you instead tie a
rope to the doorknob and then pull with the same
force, is the torque you exert increased? Will it
be easier to open the door? 1. No 2. Yes
Please fill your answer as question 29 of
General Purpose Answer Sheet
12
ConcepTest 1
You are trying to open a door that is stuck by
pulling on the doorknob in a direction
perpendicular to the door. If you instead tie a
rope to the doorknob and then pull with the same
force, is the torque you exert increased? Will it
be easier to open the door? 1. No 2. Yes
Convince your neighbor!
Please fill your answer as question 30 of
General Purpose Answer Sheet
13
ConcepTest 1
You are trying to open a door that is stuck by
pulling on the doorknob in a direction
perpendicular to the door. If you instead tie a
rope to the doorknob and then pull with the same
force, is the torque you exert increased? Will it
be easier to open the door? 1. No 2. Yes
14
ConcepTest 2
You are using a wrench and trying to loosen a
rusty nut. Which of the arrangements shown is
most effective in loosening the nut? List in
order of descending efficiency the following
arrangements
Please fill your answer as question 31 of
General Purpose Answer Sheet
15
ConcepTest 2
You are using a wrench and trying to loosen a
rusty nut. Which of the arrangements shown is
most effective in loosening the nut? List in
order of descending efficiency the following
arrangements
Convince your neighbor!
Please fill your answer as question 32 of
General Purpose Answer Sheet
16
ConcepTest 2
You are using a wrench and trying to loosen a
rusty nut. Which of the arrangements shown is
most effective in loosening the nut? List in
order of descending efficiency the following
arrangements
2, 1, 4, 3 or 2, 4, 1, 3
17
What if two or more different forces act on lever
arm?
18
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

19
Example 1
N
Determine the net torque
4 m
2 m
Given weights w1 500 N w2 800 N lever
arms d14 m d22 m Find
St ?
500 N
800 N
1. Draw all applicable forces
2. Consider CCW rotation to be positive

Rotation would be CCW
20
Where would the 500 N person have to be relative
to fulcrum for zero torque?
21
Example 2
N
y
2 m
d2 m
Given weights w1 500 N w2 800 N lever
arms d14 m St 0 Find d2 ?
500 N
800 N
1. Draw all applicable forces and moment arms

According to our understanding of torque there
would be no rotation and no motion!
What does it say about acceleration and force?
Thus, according to 2nd Newtons law SF0 and a0!
22
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
• Second Condition of Equilibrium
• The net external torque must be zero
• This is a statement of rotational equilibrium

23
Axis of Rotation

• So far we have chosen obvious 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

24
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

25
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
• 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

26
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
• 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.

27
Example
Find center of gravity of the following system
Given masses m1 5.00 kg m2 2.00 kg m3
4.00 kg lever arms d10.500 m d21.00
m Find Center of gravity

28
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

29
Equilibrium, once again

• 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

30
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

31
Example
Suppose that you placed a 10 m ladder (which
weights 100 N) against the wall at the angle of
30. What are the forces acting on it and when
would it be in equilibrium?
32
Example
Given weights w1 100 N length l10 m angle
a30 St 0 Find f ? n? P?
mg
a
1. Draw all applicable forces
2. Choose axis of rotation at bottom corner (t of
f and n are 0!)
Torques Forces

Note f ms n, so
33
So far net torque was zero. What if it is not?
34
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

35
Torque and Angular Acceleration
torque t
dependent upon object and axis of rotation.
Called moment of inertia I. Units kg m2
The angular acceleration is inversely
proportional to the analogy of the mass in a
rotating system
36
Example 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

37
Other Moments of Inertia
38
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
• 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

39
Lets watch the movie!
40
Example
Consider a flywheel (cylinder pulley) of mass M5
kg and radius R0.2 m and weight of 9.8 N hanging
from rope wrapped around flywheel. What are
forces acting on flywheel and weight? Find
acceleration of the weight.
mg
41
Example
N
T
T
Mg
Given masses M 5 kg weight w 9.8
N radius R0.2 m Find Forces?
mg
1. Draw all applicable forces
Forces Torques
Tangential acceleration at the edge of flywheel
(aat)

42
ConcepTest 3
A force F is applied to a dumbbell for a time
interval Dt, first as in (a) and then as in (b).
In which case does the dumbbell acquire the
greater center-of-mass speed?
1. (a) 2. (b) 3. no difference 4. The answer
depends on the rotational inertia of the
dumbbell.
Please fill your answer as question 33 of
General Purpose Answer Sheet
43
ConcepTest 3
A force F is applied to a dumbbell for a time
interval Dt, first as in (a) and then as in (b).
In which case does the dumbbell acquire the
greater center-of-mass speed?
Convince your neighbor!
1. (a) 2. (b) 3. no difference 4. The answer
depends on the rotational inertia of the
dumbbell.
Please fill your answer as question 34 of
General Purpose Answer Sheet
44
ConcepTest 3
A force F is applied to a dumbbell for a time
interval Dt, first as in (a) and then as in (b).
In which case does the dumbbell acquire the
greater center-of-mass speed?
1. (a) 2. (b) 3. no difference 4. The answer
depends on the rotational inertia of the
dumbbell.
Force acts the same time change of momentum is
the same. Thus CM speed is the same as well.
45
Return to our example
Consider a flywheel (cylinder pulley) of mass M5
kg and radius R0.2 m with weight of 9.8 N
hanging from rope wrapped around flywheel. What
are forces acting on flywheel and weight? Find
acceleration of the weight.
mg
If flywheel initially at rest and then begins to
rotate, a torque must be present
Define physical quantity
46
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 ?
• If the net torque is zero, the angular momentum
  remains constant
• Conservation of Linear Momentum states The
  angular momentum of a system is conserved when
  the net external torque acting on the systems is
  zero.
• That is, when

(compare to )
47
Return to our example once again
Consider a flywheel (cylinder pulley) of mass M5
kg and radius R0.2 m with weight of 9.8 N
hanging from rope wrapped around flywheel. What
are forces acting on flywheel and weight? Find
acceleration of the weight.
mg
Each small part of the flywheel is moving with
some velocity. Therefore, each part and the
flywheel as a whole have kinetic energy!
Thus, total KE of the system
48
Total Energy of Rotating System

• 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
• Conservation of Mechanical Energy
• Remember, this is for conservative forces, no
  dissipative forces such as friction can be present

49
ConcepTest 4
A force F is applied to a dumbbell for a time
interval Dt, first as in (a) and then as in (b).
In which case does the dumbbell acquire the
greater energy?
1. (a) 2. (b) 3. no difference 4. The answer
depends on the rotational inertia of the
dumbbell.
Please fill your answer as question 35 of
General Purpose Answer Sheet
50
ConcepTest 4
A force F is applied to a dumbbell for a time
interval Dt, first as in (a) and then as in (b).
In which case does the dumbbell acquire the
greater energy?
Convince your neighbor!
1. (a) 2. (b) 3. no difference 4. The answer
depends on the rotational inertia of the
dumbbell.
Please fill your answer as question 36 of
General Purpose Answer Sheet
51
ConcepTest 4
A force F is applied to a dumbbell for a time
interval Dt, first as in (a) and then as in (b).
In which case does the dumbbell acquire the
greater energy?
1. (a) 2. (b) 3. no difference 4. The answer
depends on the rotational inertia of the
dumbbell.
Since CM speeds are the same, translational
kinetic energies are the same. But (b) also
rotates, so it also has rotational energy.
52
Note on problem solving

• 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 ?
• Techniques for conservation of energy are the
  same as for linear systems, as long as you
  include the rotational kinetic energy
• Problems involving angular momentum are
  essentially the same technique as those with
  linear momentum
• The moment of inertia may change, leading to a
  change in angular momentum

53
Review before Exam 2

• Useful tips
• Do and understand all the homework problems.
• Review and understand all the problems done in
  class.
• Review and understand all the problems done in
  the textbook (chapters 5-8).
• Come to office hours if you have questions!!!

54
Exam Review

• Chapter 5 (Work and energy)
• Work, kinetic and potential energy. Reference
  levels. Elastic energy.
• Conservation of energy. Power.
• Chapter 6 (Momentum and collisions)
• Impulse-momentum theorem. Conservation of
  momentum.
• Chapter 7 (Circular motion and gravity)
• Angular velocity and acceleration. Centripetal
  acceleration.
• Newtons law of universal gravitation.
• Chapter 8 (Rotational dynamics)

55
Review problem
The launching mechanism of a toy gun consists of
a spring of unknown spring constant, as shown in
Figure 1. If the spring is compressed a distance
of 0.120 m and the gun fired vertically as shown,
the gun can launch a 20.0-g projectile from rest
to a maximum height of 20.0 m above the starting
point of the projectile. Neglecting all resistive
forces, determine (a) the spring constant and (b)
the speed of the projectile as it moves through
the equilibrium position of the spring (where x
0), as shown in Figure 1.
56
(No Transcript)