Download File - Mr. Dorsey: Physics

DE Physics
Chapter 8 Torque and Rotation
 8.2 Torque and Stability
 6.5 Center of Mass
 8.3 Rotational Inertia
Dorsey, Adapted from CPO Science
Chapter 8 Objectives


Calculate the torque created by a force.
Solve problems by balancing two torques in
rotational equilibrium.


Using center of mass to find the torque
Calculate the moment of inertia for rotation mass.

Know and use the Moment of Inertia of common
objects
Describe the relationship between torque, angular
acceleration, and rotational inertia.

Chapter 8 Vocabulary Terms
 torque
 center of rotation
 center of mass
 rotational equilibrium
 angular
acceleration
 lever arm
 rotational inertia
 rotation
 translation
 center of gravity
 moment of inertia
 line of action
8.2 Torque
Key Question:
How does force create rotation?
8.2 Torque
 A torque is an action that causes objects
to rotate.
 Torque is not the same thing as force.
Torque
 Motion in which an
entire object moves is
called translation.
 Motion in which an
object spins is called
rotation.
Torque
 The point or line about which an object turns
is its center of rotation.
 An object can rotate and translate.
Torque
 The line of action:
Goes through the point
of application in the
direction of the applied
force
 Torque is created when
the line of action….???
Let’s see.
 Put a book on the table.
 Push it in the middle
 Then push it on the edge
 How does it move both
times?
Torque
 Torque is created when
the line of action, does
not pass through the
center of rotation.
Another Example
 When you came in the
classroom you used
torque to get in.
 What did you have to
do to get in the room?
Opening a Door
 Where is the handle with
respect to the hinges?
 Go to the door as a class:
 try opening it really close
to the hinges
 then as far as you can go.
Opening a Door
 Which was easier
 Opening from the edge
or near the hinges?
Torque
 Maximum torque happens when
pulling perpendicular to the shaft
Torque
Lever arm length (m)
Torque (N.m)
t=rxF
Force (N)
The right hand rule:
Curl your right hand following the torque
arrow with your thumb in the middle.
Thumb pointing out of page is +
Thumb into page is -
Calculate a torque
 A force of 50. newtons is
applied to a wrench that
is 30. centimeters long.
 Calculate the torque if the force is applied
perpendicular to the wrench, making 30 cm lever arm.
Calculate a torque
 A force of 50. newtons is
applied down, counter
clockwise, to a wrench in
a way with a 30. cm long
lever arm.
 τ = (-50. N)(0.30 m) = -15 N.m
The torque is negative
following the right hand rule.
When the force and lever arm are NOT
perpendicular
9.1 Calculate a torque
 Perpendicular Force 50. N
turns this bolt. What force is
needed at 30. degrees:
 Knowns:
r = .20 m
Angle2 = 30 deg
 F1 = 50. N F2 = ?
 Angle1= 0 deg
 Eq: τ = rF.
Calculate a torque
 Perpendicular Force 50 N turns
this bolt. What force is needed
at 30 degrees:
 Knowns: Eq: τ = rF.
 r = .20 m
Angle = 30 deg
 F = 50. N F(perpen) = ?
 Torque required to break τ = (50 N)(0.2 m) = 10 N.m
 To get the same torque with a F at 30 degrees:
 10 N.m = F × (0.2 m)cos30o
 10 N.m = 0.173 F
F = 58 N : More force is required
Rotational Equilibrium
 When an object is in rotational equilibrium, the net
torque applied to it is zero.
 Rotational equilibrium is often used to determine
unknown forces.
Calculate using equilibrium
 A boy and his cat sit on a seesaw.
 The cat has a mass of 4.0 kg and sits 2.0 m from the
center of rotation.
 If the boy has a mass of 50 kg, where should he sit so
that the see-saw will balance?
Calculate using equilibrium
The cat, mass of 4 kg sits 2 m
from the center of rotation.
Where should a boy of mass
50 kg sit to balance the system
 Solve: τcat = (2 m)(4 kg)(9.8 N/kg) = + 78.4 N-m
 τboy = (d)(50 kg)(9.8 N/kg) = - 490 d
 For rotational equilibrium, the net torque = zero.
 78.4 - 490 d = 0
 d = 0.16 m The boy must sit 16 cm from the center.
6.5 Center of Mass
Key Question:
How do objects balance?
Center of Mass
 There are three different axes about which an object
will naturally spin.
 The point at which the three axes intersect is called
the center of mass.
Finding the center of mass
 Center of mass for all objects can be found by
spinning the object and finding the intersection of the
three spin axes.
 There is not always material at an object’s center of
mass.
Finding the center of
gravity
 Center of gravity is different
from center of mass.
 For very tall objects, such as
skyscrapers, the acceleration
due to gravity may be
slightly different at points
throughout the object.
Balance and center of mass
 For an object to remain upright, its center of
gravity must be above its area of support.
 The area of support includes all the area within
the supports.
 An object will topple over if its center of mass is
not above its area of support.
Center of mass and people:
 Start at 1:00
Rotational Inertia
Key Question:
Does mass resist rotation
the same way it resists
linear acceleration?
Rotational Inertia
 Inertia is resistance to a change in
its motion
 Rotational inertia describes an
object’s resistance to a change in
its rotational motion.
 An object’s rotational inertia
depends on the total mass and
the way mass is distributed.
 Look at the pictures on the right.
Mass further means more inertia
Watch this!
Watch this!
Linear and Angular Acceleration
Angular acceleration
(kg)
Linear
acceleration
(m/sec2)
a=ar
Radius of motion
(m)
Rotational Inertia
 Rotational motion’s equation has force replaced by the
torque about the center of rotation.
 The linear acceleration is replaced by the angular
acceleration.
Rotational Inertia
 A rotating mass on a rod
can be described with
variables from linear or
rotational motion.
Rotational Inertia
 The product of mass × radius squared (mr2) is
the rotational inertia for a point mass where r is
measured from the axis of rotation.
Moment of Inertia
 Is analogous to the rotational mass
 Greater the moment of inertia:
the harder it is to change its motion
Moment of Inertia
 For just one particle it is I
= mr2
 Each solid object has mass distributed at
different distances from the center of rotation.
 Mass distribution makes a big difference for
rotational inertia because of the r2 relationship
Moment of Inertia
The moment of inertia
of some simple shapes
rotated around their
center axes
You need to write
these six down and
know them!!!
Watch This:
Why is one faster than the other.
Rotation and Newton's 2nd Law
 If you apply a torque to a wheel, it will spin in the
direction of the torque.
 The greater the torque, the greater the angular
acceleration.
Solving Inertia Example
 Two point mass of 1 kg are on opposite
ends of a 2 m long massless rod.
 If spun around the center, what is the
rotational inertia, I.
 I=m1r2 + m2r2 = 1*12+1*12 = 2 kg*m2
Solving Inertia Example
 Two point mass of 1 kg are on opposite
ends of a 2 m long massless rod.
 If spun at one end, what is I.
 I=m1r2 = 1*22= 4 kg*m2
12.3 Angular Momentum
 Momentum resulting from an
object moving in linear
motion is linear momentum.
 Momentum resulting from
the rotation (or spin) is
called angular momentum.
12.3 Conservation of Angular Momentum
 Angular momentum is
conserved!
 The total angular
momentum of a closed
system stays the same.
12.3 Calculating angular momentum
Moment of
inertia
(kg m2)
Angular
momentum
(kg m/sec2)
L=Iw
Angular
velocity
(rad/sec)
Calculating angular momentum
A 1.0 m, 1.0 kg straight bar and a hoop with a
radius of 0.16 m are spun around the center
with an angular velocity of 1 rad/sec.
Calculate the angular momentum of each and
decide which would be harder to stop.
1. You are asked for angular momentum.
2. You are given mass, shape, and angular velocity.
— Hint: both rotate about y axis.
3. Use L= Iw,
Ihoop = mr2,
Ibar = 1/12 ml2
Calculating angular momentum
3. Solve hoop: Ihoop= (1 kg) (0.16 m)2 = 0.026 kg m2
— Lhoop= (1 rad/s) (0.026 kg m2) = 0.026 kg m2/s
4. Solve bar: Ibar = (1/12)(1 kg) (1 m)2 = 0.083 kg m2
— Lbar = (1 rad/s) (0.083 kg m2) = 0.083 kg m2/s
5.
The bar has more than 3x the angular momentum of the hoop, so it is
harder to stop.
Stability
Slides after this are extra untested
concepts
12.3 Gyroscopes angular momentum
 A gyroscope is a device that contains a spinning object
with a lot of angular momentum.
 Gyroscopes can do amazing tricks because they
conserve angular momentum.
 For example, a spinning gyroscope can easily balance
on a pencil point.
12.3 Gyroscopes angular momentum
 A gyroscope on the space shuttle is mounted at the
center of mass, allowing a computer to measure
rotation of the spacecraft in three dimensions.
 An on-board computer is able to accurately measure
the rotation of the shuttle and maintain its orientation
in space.
Jet Engines
 Nearly all modern airplanes use jet propulsion to fly. Jet
engines and rockets work because of conservation of
linear momentum.
 A rocket engine uses the same principles as a jet, except
that in space, there is no oxygen.
 Most rockets have to carry so much oxygen and fuel that
the payload of people or satellites is usually less than 5
percent of the total mass of the rocket at launch.
Force on bridge supports
 http://www.physics.ucla.edu/demoweb/n
ewdyn/html/static_forces_bridge.htm
 Play with this for no longer than 3 min.
 Notice how the forces on the pillars change
as the truck moves across the bridge.
Force on bridge supports
 Find the fraction of where the center of
mass is to the total length of the bridge
 That is the fraction of the weight that is
on the further pillar
 The closer pillar takes the rest of the
weight
Force on bridge supports
 You can also sum the torques of what is
on the bridge and the torque the CM of
the bridge applies.
Rotational Equilibrium
 When an object is in rotational equilibrium, the net
torque applied to it is zero.
 Rotational equilibrium is often used to determine
unknown forces.
 What are the forces (FA, FB) holding the bridge up at
either end?
Rotational Equilibrium