Transcript Circular Motion

Circular Motion
Physics
Mr. Padilla
Rotation and Revolution

Both rotation and revolution occur by an
object turning about an axis.
 Rotation - The axis is located within the
rotating body.
– Ex: Merry-go-round

Revolution - Object turns around an
external axis.
– Ex: Earth around the sun
Angles can be measured in
radians

In science, angles are often measured in
radians (rad) rather than degrees.
θ=s
r
 s = arc lentgh
 r = length of radius
Converting from Degrees to
Radians
θ (rad) = __ π__ θ (deg)
180o
Speed

Speed of a rotating object can be measured
in a few different ways.
 Linear speed - distance per unit time
– A.K.A. - Tangential Speed (m/s)

Rotational speed - (angular speed) rate at
which an object rotates
– Rotations per minute (RPM)
– Radians/second
More Speed

Two people sit on a merry-go-round, one on
the outside, one on the inside.
 Which one has a greater tangential speed?
– The one on the outside

Which has a greater rotational speed?
– They both have the same angular speed

θ is analogous to x
Angular Displacement

The angle through which a point line, or
body is rotated in a specified direction and
about a specified axis.
 Can be found by the change in arc length
divided by the distance from the axis of
rotation.
∆Ө = ∆s
r
Angular Displacement

∆Ө is positive when the object rotates
counterclockwise

∆Ө is negative when the object rotates
clockwise.
Sample Problem 7A

While riding on a carousel that is rotating
clockwise, a child travels through an arc
length of 11.5 m. If the child’s angular
displacement is 165o, what is the radius of
the carousel?
Angular Speed

Angular speed is a measure of how fast an
object moves in a circle
 Average angular speed = angular
displacement / time
 Ө is angular displacement
 ω = ∆Ө/∆t
– Ө is measured in radians
– Number of rotations * 2π = Ө (in radians)
– ω is analogous to v
Sample Problem 7B

A child at an ice cream parlor spins on a
stool. The child turns counterclockwise
with an average angular speed of 4.0 rad/s.
In what time interval will the child’s feet
have an angular displacement of 8.0π rad?
Angular Acceleration

The time rate of change of angular speed,
expressed in radians per second squared.
αavg = ∆ = 2 - 1
∆t
t2 – t1
Angular Acceleration Example
What α is needed to increase the speed of a
fan blade from 8.5rad/s to 15.4rad/s in 5.2s?
 αavg = Δω/t
 αavg = (15.4rad/s-8.5rad/s)/5.2s
 αavg = 1.3 rad/s2

Sample Problem 7C

A car’s tire rotates at an initial angular
speed of 21.5 rad/s. The driver accelerates,
and after 3.5 s the tire’s angular speed is
28.0 rad/s. What is the tire’s average
angular acceleration during the 3.5 s time
interval?
Practice Assignment

Practice Q’s:
7A-C odd
Rotational Kinematics
Linear
Rotational
 vf
 ωf
= vi + at
 Δx = ½ at2 + vit
 vf2 = vi2 + 2aΔx
 Δx = ½ (vi + vf)t
= ωi + αt
 Δθ = ½ αt2 + ωit
 ωf2 = ωi2 + 2αΔθ
 Δθ = ½ (ωi + ωf)t
Sample Problem 7D

The wheel on an upside-down bicycle
moves through 11.0 rad in 2.0 s. What is the
wheel’s angular acceleration if its initial
speed is 2.0 rad/s?
Tangential Speed

Tangential speed ~
radial distance x
rotational speed.
 The farther out an
object is, the faster its
tangential speed.
 vt = r
  is in rad/s
 rot/s * 2π = rad/s

If object A is twice as
far from the axis of
rotation as object B,
how much faster is
object A going?
 Object A will be going
twice as fast.
Problems

A wheel rotates at
90 rpm. What is the
tangential speed of an
object .3m from the
center?
– First: 90 rpm = 1.5rot/s
– 1.5rot/s * 2π =
9.42 rad/s

vt = r
 vt = (.3m)(9.42 rad/s)
 vt = 2.83m/s

A stone in a sling
moves with a linear
velocity of 6m/s. If the
sling is .5m metes in
length, how fast is it
being twirled?
 v = r
 6 m/s = (.5m) 
  = 12 rad/s
Sample Problem 7E

The radius of a CD in a computer is 0.0600
m. If a microbe riding on the disc’s rim has
a tangential speed of 1.88 m/s, what is the
disc’s angular speed?
Tangential Acceleration

The instantaneous linear acceleration of an
object directed along the tangent to the
object’s circular path.
 α must be in rad/s2
at = rα
Sample Problem 7F

A spinning ride at a carnival has an angular
acceleration of 0.50 rad/s2. How far from
the center is a rider who has a tangential
acceleration of 3.3 m/s2?
Centripetal Force

Def: Any force that causes an object to
follow a circular path.
– When you swing a ball around on a string, the
string provides the centripetal force.
– When the Earth revolves around the sun,
gravity provides the centripetal force.

Centripetal force is always pulling towards
the center of the circle
Centripetal Acceleration

Acceleration directed toward the center of a
circular path

Centripetal Acceleration
a c = vt 2
r
or
ac = rω2
Sample Problem 7G

A test car moves at a constant speed around
a circular track. If the car is 48.2 m from the
track’s center and has a centripetal
acceleration of 8.05 m/s2, what is its
tangential speed?
Finding Total Acceleration
at
ac
atotal
atotal2 = at2 + ac2
Causes of Circular Motion

A ball of mass m is tied to a string of length
r that is whirled in a horizontal circular
path. Assume the ball moves with a
constant speed.
 Does the ball experience centripetal
acceleration?

The ball does experience centripetal
acceleration because the velocity vector
changes direction continuously during the
motion.
 The ball also experience a change in force.
Centripetal Force

Newton’s 2nd Law can also be applied along
the radial direction
Fc = mac or
Fc = mvt2 or
r
Fc = mr2
Example

Calculate the force
needed to keep a
1000kg car going in a
circle with a radius of
10m, if the car’s
velocity is 40m/s
 What is providing the
force?

ac = v2/r
– ac = (40m/s)2/10m
– ac =160m/s2

Fc = mac
– F = (1000kg)(160m/s2)
– F = 160,000 N

Friction
Sample Problem 7H

A pilot is flying a small plane at 30.0 m/s in
a circular path with a radius of 100.0 m. If a
force of 635 N is needed to maintain the
pilot’s circular motion, what is the pilot’s
mass?
Centrifugal
Means “center fleeing”
 A common misconception is that when you
swing the ball around on the string, the
centrifugal force will cause it to fly out from
its path.

– What really causes the ball to go outward?

Centrifugal force is considered to be a
fictitious force.
More Centrifugal

What is really causing an object to be
“pulled” outward? …its inertia
 From an outside perspective, this is not a
real force, but from a rotating frame of
reference, this force appears real.
 This appearance can be felt as simulated
gravity.
Simulated Gravity



The amount of simulated
gravity depends on how
fast things rotate and the
radius of rotation.
If the object spinning is to
small, it would have to
rotate very quickly to get
an a = g
Too fast of an acceleration
could make a person dizzy
or sick



About 1 rpm is acceptable
for most people.
For a space station to
experience simulated
gravity at 1 rpm, it would
have to be very large.
People at the outer edge
would experience a = g,
those closer would have a
less than g.
– Half way = .5g
– Center = 0g