Transcript File

Uniform Circular Motion
Physics
What is Uniform Circular Motion?
Velocity exists not only in linear
equations, but also in circular paths.
Objects can rotate and revolve.
A good example of this is Earth.
Rotation
• Rotation: When an object spins or rotates
around an internal axis. Example: It takes
Earth 24 hours to rotate on it’s axis.
Revolution
• Revolution: when an object turns or revolves
around an external axis. Example: It takes
Earth 365 days to revolve around the sun.
Period: the time (T) to complete one cycle
of motion. (rotation or revolution)
Frequency: the number of cycles of motion
(revolutions or rotations) in one second.
(measured in Hz) – Also, angular speed (w):
Linear speed and tangential speed
• linear speed: distance moved per unit of
time (speed). Ex: Merry Go Round
• tangential speed (Vt) : the speed of an
object that is moving along a circular path.
The direction of motion is tangent to the
circle.
When dealing with motion in a
circular path, velocity changes are
really direction changes, and
acceleration changes are equal to
force.
Centripetal Force: this causes circular
motion, and is directed towards the
center of the circle.
Example of centripetal force: the moon
being held in orbit by gravity.
Centrifugal Force: an imaginary force
that seems to pull away from the center
of the circle which is caused by inertia.
Example of centrifugal force: carnival
rides.
Key Points for Uniform Circular
Motion
• All points on a rotating rigid object will have the
same angular speed (and thus the same angular
acceleration).
• The tangential speed of an object placed on a
rotating body will increase as it is moved away
from the center of the rigid body.
• On object that is following a circular path has a
net force (Fc) and acceleration (ac) that are acting
towards the center of the object. (Think Geometry)
Key points continued
• Velocity is always tangent to the circle and is
always changing.
• Acceleration is towards the center of the circle
• Look at the following rotating disk. Which point
has the greater tangential velocity? Which point
has the greater angular velocity?
Equations:
V=2πr/T
a= v2/r or 4∏2r/T2
So….F = m x v2 / r
F= (m)(4π2)(r)
T2
Sample Problem I
• What is the frequency and period of a bug
that makes 5 revolutions on a DVD in 2
seconds?
Sample Problem II
• What is the tangential speed of a dude
rotating at a frequency of 5 Hz while sitting
3 m from the center of a carousel? If the
dude’s magical unicorn where to instantly
disappear, what direction would he fly?
Sample Problem III
• Some random guy is twirling his grandfathers 2 kg time piece which is tied to the
end of a .8 m chain. The time piece travels
at a frequency of 3 revolutions every 2
seconds. What is the cetripetal acceleration
of the time piece? What is the Tension
(Centriptal FORCE) in the chain?
Angular speed, force, period
• angular speed (w): the number of rotations
per unit of time. Also called rotational
speed or the objects FREQUENCY!
• centripetal force (Fc): any force that will
cause an object to take a circular path.
Tension, Normal Force &
Circular Motion
• Remember: Velocity is always tangent to the
circle and is always changing. Acceleration goes
towards the center of the circle
• ∑F’s are written
– Center of the Circle is “+”, Always write
positive motion in the equation first.
– g is still g in ΣF Equations
– Always = mv2/r
Sample Problem - Tension
A ball on the end of a string is revolved at a uniform
rate in a vertical circle of radius 75 cm. If its speed is
4.4 m/s and its mass is .35 kg, calculate the tension
in the string when the ball is (a) at the top of its
path, and (b) at the bottom of its path.
Sample Problem – Flat Curve
• A car of mass m is attempting to round an
unbanked curve with a radius of r. If the
coefficient of static friction between the
tires and the road is ms, what is the
maximum speed the driver can have and
successfully negotiate the curve?
Other Ideas
• Normal Force
• Centrifugal Force?
Sample Problem – Banked Curve
A car of mass m is attempting to round a curve
with a banking angle of q, and a radius of r. If
there is no friction on the road, what is the
speed the driver can have to successfully
negotiate the curve?
Rotational Mechanics
• Torque is the tendency of a force to produce
rotation around an axis.
• Torque = Force x distance
• Distance is measured from the pivot point
or fulcrum to the location of the force on the
lever arm.
• The longer the lever arm, the greater the
torque.
Balancing Torques
• The unit for torque is the newton-meter.
• When torques are balanced or in
equilibrium, the sum of the torques = 0
• t = (F1 x d1) + (F2 x d2) + …..
• The Center of Gravity of an object is the
point on an object that acts like the place at
which all the weight is concentrated.
Rotational Inertia
• Rotational Inertia or moment of inertia is
the resistance of an object to changes in its
rotational motion. (rotating objects keep
rotating, non-rotating objects tend to stay
still)
• The further the mass is located from the
axis of rotation, the greater the rotational
inertia.
• Greater rotational inertia means more
laziness per mass.
Rotational Inertia
• All objects of the same shape have the same
laziness per mass.
• You can change your rotational inertia
when spinning by extending your arms or
legs.
• Angular momentum is the measure of how
difficult it is to stop a rotating object.
• Angular momentum = mass x velocity x
radius