Transcript T = mv 2 / r

Circular Motion
Rotating
Turning about an
internal axis
Revolving
Turning about an
external axis
Linear speed, v
How far you go in a certain amount of time
Miles per hour, meters per second
Rotational speed, w
How many times you go around in a certain
amount of time
Revolutions per minute, rotations per hour,
radians per second
w = v/r (r = radius)
Which horse has a
larger linear speed
on a merry go round,
one on the outside or
one on the inside?
Outside.
Which horse has a
greater rotational
speed?
Neither, all the horses
complete the circle in
the same amount of
time.
How much faster will a horse at TWICE the
distance from the center of the circle be moving
Since both horses complete a circle in the same
time, they have the same rotational speed, w.
w = v/r
v = wr
TWICE the distance means TWICE the speed
How do you find the velocity if it is not
directly provided?
Velocity = distance / time
In circular motion, the distance traveled is all
around the circle… the circumference.
The circumference = 2pr
So…
v = 2pr / t
The number of revolutions per second is
called the frequency, f.
Frequency is measured in Hertz, Hz.
The time it takes to go all the way around
once is called the period, T.
Frequency is related to period by
f=1/T
Uniform Circular Motion, UCM: moving in a circle
with a constant speed.
Question: Is there a constant velocity when an
object moves in a circle with a constant speed?
No, the direction changes, therefore the velocity
changes.
If the velocity changed, the object is actually
ACCELERATING even while moving at the
same speed.
Suppose an object was moving in a straight line with some
velocity, v.
According to Newton’s 1st Law of Motion, “An object in motion
continues that motion unless a net external force acts on
it”.
If you want the object to move in a circle, some force must
push or pull it towards the center of the circle.
A force that pushes or
pulls an object towards
the center of a circle is
called a centripetal force
Centripetal means “center
seeking”
According to Newton’s 2nd Law, SF = ma, If
there is a centripetal force, there must be
a centripetal acceleration.
ac = v2 / r
Where r is the radius of the circle and v is
the velocity of the object.
Centripetal force
Since SF= ma, the net centripetal force is
given by
2
v
SF  m
r
Lots of forces can help in pushing or pulling
an object towards the center of a circle.
Sometimes it takes more than one force to
get an object to move in uniform circular
motion.
Centripetal force is NOT a new kind of force.
If any force is making an object move in a
circle, it becomes a centripetal force.
When can these forces be
centripetal forces?
Gravity?
Moon revolving around the
Earth
Tension?
Twirling a pail at the end of
a string
Friction?
Cars rounding a curve.
Air Resistance?
Birds flying in a circle.
Normal?
Riders in a carnival ride
Example
A boy twirls a ½ kg rock in a horizontal circle on the end
of a 1.6 meter long string. If the velocity of the rock was
4 m/s, what is the Tension in the string?
m = ½ kg
r = 1.6 m
2
v = 4 m/s
The only centripetal force is Tension.
T = m v2 / r
T = ½ 42 / 1.6
T=5N
v
SF  m
r
Example
How fast was the ½ kg rock moving if the Tension
was 10 N and the string was 1.6 m long?
m = ½ kg
r = 1.6 m
T = 10 N
T = mv2 / r
Tr/m = v2
10 x 1.6 / .5 = v2
v = 5.7 m/s
Only a component of the tension acts as a
centripetal force!
SF = mv2/r
Tsinq = mv2/r
and…
Tcosq = mg
Friction
A 1500 kg race car goes
around a curve at 45 m/s. If
the radius of the curve is
100 m, how much friction is
require to keep the car on
the track? What is m, the
coefficient of friction?
m = 1500 kg
v = 45 m/s
r = 100 m
The centripetal force is friction.
v2
SF  m
r
2
f = mv /r
f = 1500 x 452 / 100
f = 30375 N
f = mN
m= f / N
N = mg = 15000 N
m = 30375 N / 15000 N
m = 2.02
The Normal force
In some cases the normal force can contribute to the net centripetal
force.
For example, on the carnival ride where the riders stand against the
walls of the circular room and the floor drops out! And yet, the rider
does not slide down!
Draw the free-body diagram!
What keeps them from sliding down?
The wall pushes against the rider toward the center of the circle.
SFcentripetal
v2
m
r
f
N
N = m v2 / r
mg
Banked Race Tracks- no friction!
q
q
N
mg
The acceleration is NOT up or down the plane, but pointing toward the center
of the track. We need to look for HORIZONTAL forces.
Without friction, the ONLY force that points to the center of the track is the
horizontal component of the Normal force.
S F = Nsin q = mv2/r
You will have to look at the vertical forces to determine the magnitude of the
Normal force.
Vertical loops
Twirling a rock at the end of a string in a
vertical loop.
At the top of the loop, both the Tension
and the weight point towards the
center of the circle!
SF = T + mg = mv2/r
At the bottom of the loop, the Tension
points toward the center, the weight
away from the center:
SF = T – mg = mv2/r
What about an object on a vertical
track?
At the top of the track, both the
Normal force and the weight
point toward the center of the
circle:
SF = N + mg = mv2/r
At the bottom of the track, the
Normal force points toward the
center and the weight points
away from the center:
SF = N – mg = mv2/r
Air resistance (lift force)
At the bottom of the loop: SF = Flift – mg = mv2/r
Of course, the air lift force is not acting on
the pilot!
For the pilot, the two forces are his weight
and the Normal force from the seat.
N
SF = mv2/r
N – mg = mv2/r
What about at the top of the loop?
N
Loop
the
Loop
What is the minimum speed that a rider must be moving at in order to
complete a loop the loop of radius 12 m?
At the top of the loop, both the Normal force and weight point towards
the center of the circle, so
SFcentripetal = N + mg = mv2 / r
However, at the minimum required speed, contact is lost for a moment at
the top of the loop, so that…
The Normal force goes all the way to ZERO.
The weight is the only centripetal force when the rider is moving at the
minimum required speed.
v2
SF  m
r
2
mg = mv /r
g = v2/r
v2 = rg
v2 = 12 x 10
v = 10.95 m/s
Rounding the top
of a hill
Which force is pointing TOWARD the center? Which force
is pointing AWAY from the center?
SF = mv2/r
mg – N = mv2/r
If you are going fast enough to come up out of your seat,
the Normal force pushing on you is…
ZERO
In that case… mg = mv2/r
“Artificial Gravity”
Occupants of a space station feel weightless because they lack a
support (Normal) force. By spinning the station as just the right
speed, they will experience a “simulated gravity” when the Normal
force of the floor pushing on them becomes a centripetal force. The
closer their centripetal acceleration, v2/r is to g, the acceleration due
to Earth’s gravity, the more they feel the sensation of normal weight.
Only a component of the lift force provides any
kind of centripetal force!
Tarzan plans to cross a gorge by
swinging in an arc from a
hanging vine . If his arms are
capable of exerting a force of
1400 N on the rope, what is the
maximum speed he can tolerate
at the lowest point of his
weight? His mass is 80 kg and
the vine is 4.8 m long.
SF = m v2/r
T – mg = m v2/r
V2 = (T – mg) r/m