Download Introduction to Mechanics Non-uniform Circular Motion Introducing

Introduction to Mechanics
Non-uniform Circular Motion
Introducing Energy
Lana Sheridan
De Anza College
Mar 10, 2016
Last time
• applying the idea of centripetal force
Overview
• non-uniform circular motion and tangential acceleration
• energy and work
mass? Explain. (c) What is the direction and magnitude of your
acceleration when you are at the bottom of the wheel? Assume
that its rotational speed has remained constant.
Banked Turn Related Problems
71. •• A Conical Pendulum A 0.075-kg toy airplane is tied to the
ceiling with a string. When the airplane’s motor is started, it
This situation
iswith
called
a “conical
But notice,
moves
a constant
speed of pendulum”.
1.21 m/s in a horizontal
circle of
radius 0.44 m, as illustrated
in Figurein
6–40.
Find (a) the angle the
at an angle
of
actually
a banked-turn-style
problem
disguise!
string makes with the vertical and (b) the tension in the string.
ox is attached
as shown in
tween the box
unt the spring
it is
!
▲ FIGURE 6–40 Problem 71
•• Awas
tugboat
tows by
a barge
constantforce
speedin
with
The role72.that
played
the atnormal
thea 3500-kg
banked turn
cable, as shown in Figure 6–41. If the angle the cable makes
problem is with
nowthe
played
by
the
tension
in
the
string.
horizontal where it attaches to the barge and the
tugboat is 22°, find the force the cable exerts on the barge in the
forward direction.
1
See prob 85, Ch 6.
m of the test tube a linm the axis of rotation,
Non-uniform Circular Motion
A particle can speed up or slow down while following a circular arc.
It it does this it must have a component of its acceleration along
the direction of its velocity.
560 g
that are many thoudevices referred to as
ion g. Even in the relvolved in a centrifuge
est tube have a mass
bottom of the tube is
y
at
atotal
acp
x
O
or decrease its speed.
ntial to its path that
!
icular to its path, a cp,
▲ FIGURE 6–15 A particle moving in a
rated in Figure 6–15.
circular path with tangential acceleration
!
1
Figure
from Walker,
d a cp. We will
explore
In this“Physics”.
case, the particle’s speed is
Non-uniform Circular Motion
= 9560 g
y
at
ions that are many thouact, devices referred to as
million g. Even in the rels involved in a centrifuge
the test tube have a mass
the bottom of the tube is
ease or decrease its speed.
ngential to its path that
!
pendicular to its path, a cp,
llustrated in Figure 6–15.
!
centripetal
. We will
explore
t and a cpThe
and changes
atotal
acp
O
x
▲ FIGURE 6–15 A particle moving in a
circular path with tangential acceleration
acceleration
acp
is toward
the center
In this case, the
particle’s
speed is
increasing at
ratevelocity.
given by at.
the direction
ofthe
the
of the circle
The tangential acceleration at is tangent to the circle and causes a
change of speed.
Problems
Radial and Tangential Accelerations
105
10 cm
minate
cenbe at
te the
41. A train slows down as it rounds a sharp horizontal
M turn, going from 90.0 km/h to 50.0 km/h in the 15.0 s
it takes to round the bend. The radius of the curve is
150 m. Compute the acceleration at the moment the
train speed reaches 50.0 km/h. Assume the train continues to slow down at this time at the same rate.
ate of
small
edge).
42. A ball swings counterclockwise in a vertical circle at
the end of a rope 1.50 m long. When the ball is 36.9°
past the lowest point on its way up, its total acceleration
is 1 222.5 i^ 1 20.2 j^ 2 m/s2. For that instant, (a) sketch a
vector diagram showing the components of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of
the ball.
earch
ontal,
n Figin a
away.
cond,
elera-
43. (a) Can a particle moving with instantaneous speed
3.00 m/s on a path with radius of curvature 2.00 m
have an acceleration of magnitude 6.00 m/s2? (b) Can
1
Pageit93,
Serway
Jewett.
have
an&acceleration
of magnitude 4.00 m/s2? In
Problems
Radial and Tangential Accelerations
10 cm
minate
cenbe at
te the
105
41. A train slows down as it rounds a sharp horizontal
M turn, going from 90.0 km/h to 50.0 km/h in the 15.0 s
it takes to round the bend. The radius of the curve is
150 m. Compute the acceleration at the moment the
train speed reaches 50.0
km/h.
Chapter
4 Assume
159 the train continues to slow down at this time at the same rate.
ate of Sketch:
42. A ball swings counterclockwise in a vertical circle at
gsmall
both its speed
and
the end
of a rope 1.50 m long. When the ball is 36.9°
n
vector
will
be
the
edge).
past the lowest point on its way up, its total acceleration
is 1 222.5 i^ 1 20.2 j^ 2 m/s2. For that instant, (a) sketch a
tial
and
radial
earch
vector diagram showing the components of its acceler.ontal,
The tangential
ation, (b) determine the magnitude of its radial accelFig- the changing
dnfrom
eration, and (c) determine the speed and velocity of
in a the radial
while
the ball.
away.
d from the radius of
43. (a) Can a particle moving with instantaneous speed
cond,
speed.
3.00 m/s on a path with radius of curvature 2.00 m
elera-
havekm/h
an acceleration of magnitude 6.00 m/s2? (b) Can
eed units1 from
2
ANS. FIG. P4.41
Pageit93,
Serway
Jewett.
have
an&acceleration
of magnitude 4.00 m/s ? In
Radial and Tangential Accelerations
Problems
105
10 cm
minate
cenbe at
te the
41. A train slows down as it rounds a sharp horizontal
M turn, going from 90.0 km/h to 50.0 km/h in the 15.0 s
it takes to round the bend. The radius of the curve is
150 m. Compute the acceleration at the moment the
train speed reaches 50.0 km/h. Assume the train continues to slow down at this time at the same rate.
ate of
small
edge).
42. A ball swings counterclockwise in a vertical circle at
the end of a rope 1.50 m long. When the ball is 36.9°
past the lowest point on its way up, its total acceleration
is 1 222.5 i^ 1 20.2 j^ 2 m/s2. For that instant, (a) sketch a
vector diagram showing the components of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of
the ball.
earch
ontal,
n Figin a
away.
cond,
elera-
43. (a) Can a particle moving with instantaneous speed
3.00 m/s on a path with radius of curvature 2.00 m
1
an acceleration
of magnitude 6.00 m/s2? (b) Can
Pagehave
93, Serway
& Jewett.
Radial and Tangential Accelerations
Problems
105
10 cm
minate
cenbe at
te the
41. A train slows down as it rounds a sharp horizontal
M turn, going from 90.0 km/h to 50.0 km/h in the 15.0 s
it takes to round the bend. The radius of the curve is
150 m. Compute the acceleration at the moment the
train speed reaches 50.0 km/h. Assume the train continues to slow down at this time at the same rate.
ate of
small
edge).
in a vertical circle at
42. A ball swings counterclockwise
90.0
the end
a rope
long.m/s
When
the m/s
ball is 36.9°
vi = of
90.0
km/h1.50 m
=
= 25.0
3.6way up, its total acceleration
past the lowest point on its
2. For
that instant, (a) sketch a
is 1 222.5 i^ 1 20.2 j^ 2 m/s
50.0
vector
diagram
showing
the
components
its accelerm/s = 13.9ofm/s
vf = 50.0 km/h =
ation, (b) determine the3.6
magnitude of its radial acceleration, and (c) determine the speed and velocity of
the ball.
earch
ontal,
n Figin a
away.
cond,
elera-
43. (a) Can a particle moving with instantaneous speed
3.00 m/s on a path with radius of curvature 2.00 m
1
an acceleration
of magnitude 6.00 m/s2? (b) Can
Pagehave
93, Serway
& Jewett.
Radial and Tangential Accelerations
Problems
10 cm
minate
cenbe at
te the
105
41. A train slows down as it rounds a sharp horizontal
M turn, going from 90.0 km/h to 50.0 km/h in the 15.0 s
it takes to round the bend. The radius of the curve is
150 m. Compute the acceleration at the moment the
train speed reaches 50.0 km/h. Assume the train continues to slow down at this time at the same rate.
in a vertical circle at
42. A ball swings counterclockwise
90.0
the end
a rope
long.m/s
When
the m/s
ball is 36.9°
vi = of
90.0
km/h1.50 m
=
= 25.0
3.6way up, its total acceleration
past the lowest point on its
2. For
that instant, (a) sketch a
is 1 222.5 i^ 1 20.2 j^ 2 m/s
earch
50.0
vector
diagram
showing
the
components
its accelerm/s = 13.9ofm/s
vf = 50.0 km/h =
ontal,
3.6
ation,
(b)
determine
the
magnitude
of
its
radial
acceln Figeration, and (c) determine the speed and velocity
of
in a Tangential
accel. corresponds to changing speed: at,avg = ∆v
∆t
the
ball.
away.
ate of
small
edge).
v2
43. (a) Can
particle moving
with instantaneous
cond, Centripetal
accel.a corresponds
to changing
direction: acp =speed
3.00 m/s on a path with radius of curvature 2.00r m
elera1
an acceleration
of magnitude 6.00 m/s2? (b) Can
Pagehave
93, Serway
& Jewett.
Radial and Tangential Accelerations
10 cm
minate
cenbe at
te the
Problems
105
41. A train slows down as it rounds a sharp horizontal
M turn, going from 90.0 km/h to 50.0 km/h in the 15.0 s
it takes to round the bend. The radius of the curve is
150 m. Compute the acceleration at the moment the
train speed reaches 50.0 km/h. Assume the train continues to slow down at this time at the same rate.
ate of
42. A ball swings counterclockwise in a vertical circle at
small
the end of2a rope 1.50 m long.2 When the ball is 36.9°
; apoint
(calling
outward
positive)
r = −1.29
edge). at = −0.741
past them/s
lowest
on its m/s
way up,
its total
acceleration
2
^
^
is 1 222.5 i 1 20.2 j 2 m/s . For that instant, (a) sketch a
earch
vector
components
2
ontal,
a =diagram
1.48 m/sshowing
inward the
at an
angle 29.9◦of its acceleration,
(b)
determine
the
magnitude
of its radial acceln Figbackward
from
the
direction
of travel
eration,
and
(c)
determine
the
speed
and
velocity
of
in a
the ball.
away.
cond,
elera-
43. (a) Can a particle moving with instantaneous speed
Page3.00 93, Serway
& Jewett.
m/s on
a path with radius of curvature 2.00 m
1
Energy
Energy is a difficult concept to define, but it is very important for
physics.
Energy can take many different forms.
Knowing the amount of energy a system has can tell us about
what states or configurations we can find the system in.
Energy
Energy is a difficult concept to define, but it is very important for
physics.
Energy can take many different forms.
Knowing the amount of energy a system has can tell us about
what states or configurations we can find the system in.
One way that energy is often described is that it represents the
ability of a system to do work.
We need to know what work is!
Work
Work is an amount of energy.
The amount of work done on an object depends on the applied
force and the displacement of the object as the force acts.
we do work. The greater the force, the greater the work; the grea
tance, the greater the work. These simple ideas form the basis for ou
Work of work.
!
Toamount
be specific,
suppose we push a box with a! constant force F, a
Work is an
of energy.
Figure 7–1. If we move the box in the direction of F through a displace
work W we have done is Fd:
The amount of work done on an object depends on the applied
Definition
of Work, W, of
When
Constant
Is in the
Direction of Displac
force and
the displacement
theaobject
as Force
the force
acts.
W = Fd
If the force is in the same direction as the displacement,
SI unit: newton-meter 1N # m2 = joule, J
W = Fd
orce in
through
case,
nt are in
e on the
d
F
F
Work
W = Fd
Units of Work?
They have a special name: Joules, symbol J.
1 joule = 1 J = 1 Nm
Work is not a vector. Work is a scalar.
The work done by
gravitational force
the agent on the
Example
rest through a ver
book–Earth system is
What is the work done in mgy
lifting
a 3.0-kg
book 0.50 m?
to our discussion o
f " mgy
i.
Physics
S
S
!r
Fapp
Physics
yf
S
yi
mg
Figure 7.15 An external agent
lifts a book slowly from a height yi
to a height yf .
appear as an incre
the work and is at
the kinetic energy
Because the en
the work-kinetic e
appear as some fo
book, we could rel
(and therefore, the
that was done in l
tem had the potenti
allowed to fall. Th
is released potenti
only be associated
The amount of po
the system. Movin
may change the co
The work done by
gravitational force
the agent on the
Example
rest through a ver
book–Earth system is
What is the work done in mgy
lifting
a 3.0-kg
book 0.50 m?
to our discussion o
f " mgy
i.
appear as an incre
the work and is at
the kinetic energy
S
Because the en
Fapp
S
!r
the work-kinetic e
Physics
appear as some fo
yf
book, we could rel
S
mg
(and therefore, the
yi
that was done in l
tem had the potenti
allowed to fall. Th
is released potenti
Figure 7.15 An external agent
To lift a book withlifts
constant
velocity,
mg be associated
a book slowly
fromrequires
a heightFyiapp = only
The amount of po
to a height yf .
W = Fd = (3.0 kg)(9.81 m/s2 )(.50 m) = 14.7
theJ system. Movin
may change the co
Physics
Summary
• non-uniform circ. motion and tangential acceleration
• energy and work
Homework
Walker Physics:
• Ch 6, onward from page 177. Problems: 29, 105, 109, 110
• Ch 7, onward from page 210. Questions: 1, 3, 5; Problems: 5,
7