Transcript Lecture 8: Forces & The Laws of Motion

Lecture 16:
Rotational Motion
Questions of Yesterday
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
Rotational Motion: Angular Quantities
Angular
=
q
Position:
s
r
Average Angular Acceleration:
aav =
Angular Displacement:
Dq = qf - qi
Instantaneous Angular Acceleration:
Average Angular Velocity:
wav =
qf - qi
tf - ti
wf - wi
= Dw
Dt
tf - ti
= Dq
Dt
Instantaneous Angular Velocity:
Dq
w = Dtlim
-> 0
Dt
Dw
a = Dtlim
-> 0
Dt
Motion with Constant a:
w = w0 + at
Dq = w0t + 1/2at2
w2 = w02 + 2aDq
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
Centripetal Acceleration always points towards the CENTER of
the circle
aav =
vf - vi
tf - ti
vf
vi
Dq
vf
-vi
r
Dq
Dv
Centripetal Force
If an object is accelerating what do know about
(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 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
Centripetal Force
In what direction is the net force if an object is 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 in the perpendicular components!!
F = ma
mv2
Fc = mac = r
FT = maT
Practice Problem
An air puck of mass 0.5 kg is tied to a string and allowed to
revolve in a circular radius of 1.0 m on a frictionless horizontal
table. The other end of the string passes through a hole in the
center of the table and a mass of 1.0 kg is tied to it. The
suspended mass remains in equilibrium while the puck on the
tabletop revolves.
What is the tension in the string?
What is the horizontal force
acting on the puck?
What is the speed of the puck?
Practice Problem
Tarzan (m = 100 kg) tries to cross a river by swinging from a 10m-long vine. His speed at the bottom of the swing (as he just
clears the water) is 8.0 m/s. Tarzan doesn’t know that the vine
has a breaking strength of 1500 N.
Does he make it safely across the river?
If not, what is the maximum speed he can have to make it?
If Tarzan continues swinging on the vine what is the highest point
he reaches?
What is the tension in the vine at this highest point?
What is the net force on Tarzan at this point?
Planetary Motion
Why do the planets revolve around the sun, and the moon
revolve around the Earth?
Is there a net force acting on the planets and moons?
How do you know?
What is the direction of the force?
Gravitational Force
Force of attraction between any two objects in the Universe.
Gravitational force causes….
Objects in free fall near the Earth’s surface
to accelerate towards the Earth
the moon to orbit the earth &
the planets to orbit the sun
An astronaut to be able to
jump higher on the Moon
than on Earth
Gravitational Force
If gravity is an attractive force why doesn’t the moon crash into
the Earth?
The moon is constantly falling towards Earth
The planets are constantly falling towards the sun
Gravitational Force
Newton’s Law of Gravitational Force
m1m2
Fg = G 2
r
m1,m2= mass of objects attracting each other
r = distance between the objects
Universal gravitational constant = G = 6.67*10-11 N*m2/kg2
Gravitational Force between two objects is felt equally by both
objects
Fg of m1 exerted by m2 = Fg of m2 exerted by m1
Gravitational Force
What if you have many objects near each other?
E
M
S
∑FgE = FgSE + FgME
The net gravitational force felt on an object is equal to the sum
of the gravitational forces exerted by all the surrounding objects
Practice Problem
Objects with masses of 200 kg and 500 kg are separated by
0.500 m.
Find the net gravitational force exerted by these objects on a
50.0 kg object placed midway between them.
At what position can the 50.0 kg object be placed so as to
experience a net force of zero?
Questions of the Day
You are riding on a Ferris wheel moving at constant speed.
1a) At what point is the net force acting on you the greatest?
a) the top
b) the bottom
c) halfway between top and bottom
d) the force is the same over the whole motion
1b) Is the net force doing work on you?
a) YES
b) NO
2) If the mass of the moon were doubled, what would happen to
its centripetal acceleration?
a) it would increase
b) it would decrease
c) it would stay the same
Practice Problem
A 0.500-kg pendulum bob passes through the lowest part of its
path at a speed of 5.00 m/s.
What is the tension in the pendulum cable at this point if the
pendulum is 100.0 cm long?
When the pendulum reaches its highest point, what angle does
the cable make with the vertical?
What is the tension in the pendulum cable when the pendulum
reaches its highest point?
What is the net force acting on the pendulum at this point?
What is the direction of the acceleration?