Transcript Lecture 8: Forces & The Laws of Motion

Lecture 15:
Rotational Motion
Questions of Yesterday
1) A piece of clay traveling north with speed v collides perfectly
inelastically with an identical piece of clay traveling east with speed v.
What direction does the resultant piece of clay travel?
a) north
b) east
c) 45o N of E
d) 45o S of W
2) Ball 1 of mass m, traveling with speed v, collides with Ball 2 of mass
2m and comes to rest, what is the speed of Ball 2 after the collision?
a) 2v
b) v
c) v/2
d) v/(√2)
Linear Motion
Displacement:
Dx = xf - xi
Velocity:
v = Dx
Dt
Acceleration:
a = Dv
Dt
Constant a Equations:
v = v0 + at
Dx = v0t + 1/2at2
v2 = v02 + 2aDx
Force
(2nd Law):
F = ma
Momentum:
p = m*v
Impulse:
Dp = FDt
Equations/Concepts
valid for straight line
motion between
points in space (x-y plane)
Circular Motion
How do you define “position” and “displacement”
when motion is circular?
q = p/2
q=p
Arc length:
r
q
Angle Unit = Radian
s
q=0
q = 2p
Circumference = 2pr
2p Radians = 360o
Angular
Position:
q = 3p/2
s = r*q
s
q= r
Circular Motion
Angular Displacement:
s
q= r
Angular
Position:
Dq = qf - qi
SI Units:
Radians (rad)
tf
r
qf
qi
SI Units:
Radians per
second (rad/s)
ti
Average
Angular Velocity:
qf - qi
= Dq
wav =
Dt
tf - ti
Instantaneous Angular Velocity
Dq
w = Dtlim
-> 0
Dt
Circular Motion
Average Angular Acceleration:
wf - wi
= Dw
aav =
Dt
tf - ti
Instantaneous
Angular Acceleration
tf
r
qf
qi
SI Units:
Radians per
second squared
(rad/s2)
ti
Dw
a = Dtlim
-> 0
Dt
Constant Angular Acceleration
Linear Motion with
Constant a:
Rotational Motion with
Constant a:
v = v0 + at
w = w0 + at
Dx = v0t + 1/2at2
Dq = w0t + 1/2at2
v2 = v02 + 2aDx
w2 = w02 + 2aDq
Rotational Motion
w
Which position has a greater angular displacement
in a given time interval?
What about angular speed? Angular acceleration?
Rotational Motion
w
Which position has a greater angular displacement
in a given time interval?
What about angular speed? Angular acceleration?
Angular and Linear Quantities
Displacement:
Direction of linear velocity v of an
object moving in a circular path is
always TANGENT to the path
Ds
Dq = r
Tangential Speed:
tf
r
Dq
vT = rw
Ds
ti
Tangential Acceleration:
aT = ra
Centripetal Acceleration
If you’re jogging on a circular track with constant tangential
speed is your acceleration ZERO? Why or Why not?
aav =
vf
vi
r
Dq
vf - vi
tf - ti
During circular motion at constant
speed your direction is constantly
changing so you still have an
acceleration
CENTRIPETAL ACCELERATION
Acceleration associated with
constant speed circular motion
Centripetal Acceleration
Centripetal Acceleration always points towards the CENTER of
the circle
aav =
vf - vi
tf - ti
vf
vi
Dq
vf
-vi
r
Dq
Dv
Centripetal Acceleration
Centripetal Acceleration always points towards the CENTER of
the circle
vi
vf
vf
Ds
Dv
Dq
Similar
-vi
Triangles
Dq
r
r
Dv
v
ac
=
=
Ds
r
v2
= rw2
r
aav= Dv
Dt
Centripetal Acceleration
What if your tangential speed is NOT constant?
vi
vf
Dq
r
ac
=
vf
v2
r
aT = ra
Dq
r
Dv
-vi
Acceleration has
both tangential
and centripetal
components!
Dv
DvT
a = (ac2 + aT2)1/2
Dvc
Rotational Motion: Practice Problem
A race car starts from rest on a circular track of radius 400 m.
The car’s speed increases at the constant rate of 0.500 m/s2.
At the point where the magnitudes of the centripetal and
tangential accelerations are equal, what is…
the tangential speed of the car?
the angular speed of the car?
the distance traveled?
the number of revolutions made?
the elapsed time?
Centripetal Force
If an object is accelerating what do know about it
(think Newton’s 2nd law)?
F = ma
Can an object be moving in a circular path if no forces are
acting on?
If an object is undergoing constant speed circular motion what
direction is the net force acting on the object?
mv2
Fc = mac = r
Centripetal Force
What if an object undergoing circular motion and changing its
tangential speed?
vf
Dq
-vi
Dv
a
F
ac
FC
FT
aT
Just like linear motion (∑Fx = max, ∑Fy = may)…
must split vector equation into perpendicular components!!
F = ma
mv2
Fc = mac = r
FT = maT
Centripetal Force
As you round the bend at constant speed
in what direction..
is your net acceleration? Why?
Is your net force? Why?
do you feel yourself being pulled? Why?
Remember Newton’s 1st law??
What force is acting on you and your
car to let you round the bend?
Centripetal Force
As you round the bend at constant speed
in what direction..
is your net acceleration? Why?
Is your net force? Why?
do you feel yourself being pulled? Why?
N
Remember Newton’s 1st law??
What force is acting on you and your
car to let you round the bend?
ff
Fg
Practice Problem
Suppose that a 1800-kg car passes over a bump in a roadway
that follows the arc of circle of radius 20.0 m.
What force does the road exert on the car as the car passes the
highest point of the bump if the car travels at 9.00 m/s?
What is the maximum speed the car can have without losing
contact with the road as it passes this highest point?
Questions of the Day
1) You are going through a vertical loop on roller coaster at a constant
speed. At what point is the force exerted by the tracks on you (and
the cart you are in) the greatest?
a) at the highest point
b) at the lowest point
c) halfway between the highest and lowest point
d) the force is equal over the whole loop
2) You are on a merry-go-round moving at constant speed. If you move
to the outer edge of the merry-go-round, what happens to the net
centripetal force keeping you on the merry-go-round?
a) it increases
b) it decreases
c) it stays the same
d) there is no net centripetal force acting on you