Download Holt Physics--Chapter 8 Rotational Equilibrium and Dynasmics

Holt Physics--Chapter 8 Rotational Equilibrium and Dynamics
I. Section 8.1: Torque
A. Def: "The ability of a force to rotate an object around some axis
is measured by a quantity called torque." (p.279)
B. Torque is a rotational force
Insert fig 8-3
NOTICE: The distance d is called the "lever arm" (a.k.a. moment arm)
QUESTION: What is the relationship between the magnitude of the force "F"
and the length of the lever arm "d"?
C. Torque is related to force, length of the lever arm and the angle
between the force and the lever arm.
TORQUE
τ = Fd(sin Θ)
torque = force x lever arm



The lower-case Greek letter "tau" (τ) is the symbol for torque
Θ is the angle between the force and the distance from the axis
The unit of torque is the Newton-meter (N x m)
insert fig 8-5
1
D. Torque is a vector
1. positive torque = counterclockwise rotation
2. negative torque = clockwise rotation
II. Section 8-2: Rotation and Inertia
A. Center of Mass
1. An unattached object will rotate around its center of mass
2. The center of mass is its "balancing point"
3. The center of mass moves as the object's "point mass" (see fig 8-7)
4. For most situations the center of mass and the "center of gravity" are
the same
B. Moment of Inertia
1. An object can rotate around many axes
2. The resistance of an object to changes in its rotation is measured by a
quantity called the "moment of inertia"
3. The farther the mass of an object is, on average, from the axis of
rotation, the greater is the object's moment of inertia and the more
difficult it is to rotate the object.
4. To calculate a moment of inertia use the table below
Insert table 8-1
*the units for moments of inertia are (kg)(m2)
2
C. Rotational Equilibrium
1. Remember from chapter 4 that an object is at equilibrium when the net
force acting on the object is zero.
2. HOWEVER, an object can still rotate while at equilibrium! How is this
possible?
3. To truly be at equilibrium an object must also have a net torque of zero
(known as the second condition for equilibrium)
III. Section 8-3: Rotational Dynamics
A. Newton's second law for rotation
1. There is a relationship between the torque on an object and that
object's angular acceleration.
2. We can apply Newton's second law to rotational dynamics
Translational Dynamics
Rotational Dynamics
F=ma
τ = Iα
net torque = moment of inertia x angular acceleration
B. Momentum
1. Rotating objects have angular momentum because all rotating objects
resist changes to their rotation (rotational inertia).
ANGULAR MOMENTUM
L=Iω
angular momentum = moment of inertia x angular speed
2. The units for angular momentum are (kg)(m2/s)
3. Angular momentum may be conserved (the figure skater)
C. Kinetic Energy
1. Rotating objects have rotational kinetic energy
2. Remember the formula for translational kinetic energy from chapter 5
KE = ½mv2
4. The same principle can be applied to rotational motion as well
3
ROTATIONAL KINETIC ENERGY
KErot = ½Iω2
rotational kinetic energy = ½ x moment of inertia x angular speed2
5. Mechanical energy must now include rotational kinetic energy!
ME = KEtrans + KErot + PEg
IV. Simple Machines
A. A machine is any device that transmits or modifies a force
1. All machines are combinations of or modifications of 6 fundamental
types of machines, called simple machines.
2. The 6 simple machines are: The lever, the pulley, the inclined plane,
the wheel and axle, the wedge and the screw (which is really an
inclined plane wrapped around a cylinder!)
Insert table 8-6
B. Using simple machines
1. If machines "transmit or modify" a force, we compare the input force
to the output force to judge the effectiveness of the machine
3. This ratio is called the machine's "mechanical advantage"
4
ME = Fout/Fin
Mechanical advantage = output force ÷ input force
C. Mechanical energy is conserved (in the absence of friction) for
machines as well. This implies that as force increases, distance
decreases (and vice versa) and work remains constant.
Insert fig 8-14
D. Efficiency is a measure of how well a machine works
1. Real machines lose a lot of energy to friction and sound.
2. Efficiency is the ratio of work input to work output
Mechanical Efficiency
eff = Wout/Win
mechanical efficiency = work output ÷ work input
3. If a machine is frictionless, then mechanical energy is conserved.
4. The mechanical efficiency of an ideal (i.e. frictionless) machine is
1.00 or 100%
5. Since there is no such thing as an ideal machine, all real machines
have an efficiency of less than 1.
5