Transcript w is rad/s - cloudfront.net

CH 6
Work Work is calculated by
multiplying the force applied by
the distance the object moves
while the force is being applied.
W = Fs
Ex. 4 - A flatbed truck
2
accelerating at a = +1.5 m/s
is carrying a 120 kg crate. The
crate does not slip as the truck
moves s = 65 m. What is the
total work done on the crate
by all the forces acting on it?
The Work-Energy
Theorem.
W = KEf - KE0 =
1/2
2
mvf
- 1/2
2
mv0
Ex. 6 - A 54 kg skier is coasting
down a 25° slope. A kinetic
frictional force of fk = 70 N opposes
her motion. Her initial speed is
v0 = 3.6 m/s. Ignoring air
resistance, determine the speed vf
at a displacement 57 m downhill.
Gravitational Potential
Energy is energy due to
the distance an object is
able to fall.
PE = mgh
PE is also measured in joules.
The work done by the
gravitational force on an
object does not depend on
the path taken by the object.
This makes gravitational
force a conservative force.
Conservation of
Mechanical Energy
The total mechanical energy
(E = KE + PE) of an object remains
constant as the object moves,
provided that the net work done
by external nonconservative forces
is zero.
Ex. 9 - A motorcyclist drives
horizontally off a cliff to leap across
a canyon. When he drives off, he
has a speed of 38.0 m/s. Find the
speed with which the cycle strikes
the ground on the other side if he is
35 m below his starting point when
he strikes the ground.
Ex. 10 - A 6.00-m rope is tied to
a tree limb and used as a swing.
A person starts from rest with
the rope held in a horizontal
orientation. Determine how fast
the person is moving at the
lowest point on the circular arc
of the swing.
Power is the rate at which work is
done.
P = W/t
The unit is the joule/second, which
is called the watt.
1 horsepower = 746 watts
W/t = Fs/t
W/t is power, and s/t is
average speed v, so P = Fv
3
10
Ex. 15 - A 1.10 x
kg car,
starting from rest, accelerates
for 5.00 s. The magnitude of the
acceleration is a = 4.60 m/s2.
Determine the average power
generated by the net force that
accelerates the vehicle.
Energy of all types can be
converted from one form to
another.
The Principle of Conservation of
Energy:
Energy can be neither created nor
destroyed, but can only be
converted from one form to
another.
CH 7
The impulse of a force is the
product of the average force
and the time interval during
which the force acts.
Impulse = Fave Δt
The unit is the newton•second (N•s)
The linear momentum p
of an object is the product
of the object’s mass m and
the velocity v.
p = mv
The unit is the kilogram•meter/second (kg•m/s)
The impulse-momentum
theorem, the impulse is
equal to the change in
momentum.
F Δt = mvf - mv0
impulse
final
momentum
initial
momentum
Ex. 1 - A baseball (m = 0.14 kg) has
an initial velocity of v0 = -38 m/s as it
approaches a bat. The ball leaves the
bat with a velocity of
vf = +58 m/s. (a) Determine the
impulse applied to the ball by the
bat.
(b) If the time of contact is Δt =
1.6 x 10-3 s, find the average force
exerted on the ball by the bat.
This is the principle of conservation
of linear momentum.
The total linear momentum of an
isolated system remains constant.
(mvf1+ mvf2) = (mv01+ mv02)
or:
P f = P0
Ex. 5 - A freight train is being
assembled in a switching yard.
Car 1 has a mass of m1 = 65 x103 kg
and moves with a velocity of
v01 = +0.80 m/s. Car 2, with a mass of
m2 = 92 x 103 kg and a velocity of
v02 = +1.2 m/s, overtakes car 1 and
couples to it. Find the common velocity
vf of the two cars after they become
coupled.
Ex. 7 - When a gun fires
a blank, is the recoil
greater than, the same
as, or less than when
the gun fires a
standard bullet?
An elastic collision is one in
which the total kinetic energy of
the system after the collision is
equal to the total kinetic energy
before the collision.
K.E. is conserved in the collision.
An inelastic collision is one in which
the total kinetic energy of the system
is not the same before and after the
collision; if the objects stick together
after colliding, the collision is said to
be completely inelastic.
Kinetic energy is not conserved.
The coupling boxcars is an example
of an inelastic collision.
Ex. 9 - A ballistic pendulum consists of
a block of wood (mass m2 = 2.50 kg)
suspended by a wire.
A bullet (mass m1 = 0.0100 kg) is fired
with a speed v01. Just after the bullet
collides with it, the block (now
containing the bullet) has a speed vf
and then swings to a maximum height
of 0.650 m above the initial position.
Find the speed of the bullet.
In an isolated system
momentum is conserved,
Pf = P0. Remember that
momentum is a vector quantity;
when a collision in two
dimensions occurs the x and y
components are conserved
separately.
CH 8
The angle through
which a rigid body
rotates about a fixed
axis is called the
angular displacement.
Angular velocity is the
angular displacement
divided by elapsed time.
w = Dq / Dt
The unit is radians
per second.
rad/s
Example 3. A gymnast on a
high bar swings through two
revolutions in time of 1.90 s.
Find the average angular
velocity (in rad/s) of the
gymnast.
Angular acceleration a is
the rate of change of
angular velocity.
a = Dw / Dt
Example 4. A jet engine’s turbine
fan blades are rotating with an
angular velocity of -110 rad/s. As
the plane takes off, the angular
velocity of the blades reaches -330
rad/s in a time of 14 s. Find the
angular acceleration.
The equations for
rotational dynamics are
similar to those for linear
motion.
w = w0 + at
q = w0t + ½
2
w
2
= w0 +
2
at
2aq
The tangential velocity vT
is the speed in m/s
around the arc. The
magnitude is called the
tangential speed.
vT = rw
The centripetal
acceleration formula is
2
ac = vT /r. This can be
expressed in terms of
angular speed since
vT = rw.
2
vT /r
ac =
becomes
2
ac = (rw) /r
ac =
2
rw (w is rad/s)
When objects roll there is
a relationship between
the angular speed of the
object
and the linear speed
at which the object
moves forward.
Linear speed is equal to
tangential speed.
v = rw
It follows that linear
acceleration is equal to
tangential acceleration.
a = ra
Right-Hand Rule. When the
fingers of your right hand
encircle the axis of rotation, and
your fingers point in the
direction of the rotation, your
extended thumb points in the
direction of the angular velocity
vector.
The direction of the
angular acceleration
vector is found the same
way. The direction is
determined by the
change in angular
velocity.
If the angular velocity is
increasing, the angular
acceleration vector
points in the same
direction as the angular
velocity.
If the angular velocity is
decreasing, the angular
acceleration vector
points in the opposite
direction as the angular
velocity.
CH 9
Torque Ƭ is the
magnitude of the
force multiplied by
the lever arm.
Ƭ = Fl
∑Fx = 0 and ∑Fy = 0
∑Ƭ = 0
The above must be true for
all equilibrium conditions.
2
mr a
Ƭ=
The net external torque Ƭ is
directly proportional to the
angular acceleration a. The
constant of proportionality
2
is I = mr , the moment of
2
inertia. (Si unit kgm .)
The moment of inertia
depends on the location
and orientation of the
axis relative to the
particles that make up
the object.
There are different formulas
for moment of inertia. The
moment of inertia depends
on the shape of the object,
the distribution of the mass
in the object, and the
location of the pivot point.
Newton’s second law for a rigid
body rotating about a fixed axis.
Torque = moment of inertia X
angular acceleration
Ƭ=Ia
(a must be in rad/s2)
Work is equal to force times displacement,
W = Fx. Angular displacement q is equal to
linear displacement/radius, x/r. So x = rq.
Thus W = Fx becomes W = Frq. Since Fr is
equal to torque Ƭ, rotational work is equal
to torque multiplied by angular
displacement.
WR = Ƭq
(q must be in radians and the work unit is the
joule J.)
Rotational KE = ½ I
2
w
(w = must be in rad/s and the
unit is the joule J.)
Total kinetic
energy is not just
2
½ mv , but
2
2
½ mv + ½ I w .
The rotational analog to
displacement is angular
displacement q, the rotational
analog to velocity is angular velocity
w. For acceleration it is angular
acceleration a. For work it is
rotational work Ƭq. The rotational
analog to kinetic energy is rotational
kinetic energy ½ I w2
and
the rotational analog to
momentum is angular
momentum. The formula for
momentum is p = mv. In angular
momentum m is replaced with
moment of inertia I, and velocity is
replaced with angular velocity w.
Angular momentum L
= I w.
L=Iw
w must be in rad/s and
2
I must be in kgm .
2
L is in kgm /s
If the net force on an
object is zero, the
momentum remains
constant.
Law of Conservation
of Momentum.
Similarly,
If the net torque on an
object is zero, the
angular momentum
remains constant.
Law of Conservation of
Angular Momentum.
CH 10
A force must be applied
to a spring to stretch or
compress it. By Newton’s
third law, the spring must
apply an equal force to
whatever is applying the
force to the spring.
This reaction force is
often called the
“restoring force” and is
represented by the
equation F = -kx.
When the restoring force
has the mathematical
form given by F = -kx, the
type of motion resulting
is called “simple
harmonic motion”.
2
kx
PEelastic = 1/2
where k is the spring
constant, and x is the
distance the spring is
compressed or stretched
beyond its unstrained length.
The unit is the joule (J).
A simple pendulum is a mass m
suspended by a pivot P. When
the object is pulled to one side
and released, it will swing back
and forth in a motion
approximating simple harmonic
motion.
A series of substitutions
finds that, for small angles,
2πf = √g/L
f is frequency, g is 9.80, and
L is length.
Formulas for frequency
and
period of an oscillating
spring:
2•π•f = √ k/m
2•π/ T = √ k/m
Formulas for
period of an
oscillating spring and
pendulum:
TP = 2π √ L/g
TS = 2π √ m/k