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Rotational Motion
Chapter 6, 8 and 9
Acceleration in a Circle

Acceleration occurs when velocity
changes


This means either speed OR direction
changes
So objects moving in a circle are
accelerating even if speed remains
constant because they are constantly
changing direction
Centripetal Acceleration

In order to accelerate, there
must be a net force in the
direction of acceleration
according to Newton’s 2nd
Law



This means there must be a
center- directed force
This is called centripetal force
Without centripetal force,
inertia would cause the object
to continue in a straight line at
a constant speed
Centrifugal Force

When moving quickly in a circle,
you feel like you are being pushed
outward


The is no outward force, only a
inward force (centripetal force)


This is called centrifugal force
Centrifugal force is an imaginary
force because it doesn’t have a
reaction force to accompany it
You feel the outward force
because inertia wants you to keep
moving in a straight line, but the
centripetal force forces you to
move in a circle instead
Angular velocity (ω)
 A measure of what angle an
object is able to travel per
unit time
 Unit is rad/s
 All parts of a rigid body rotate
with the same ω, that means
object’s near the edge have to
cover more distance in the
same amount of time (have a
higher tangential velocity)
 Angular measures differ from
centripetal measures
because the object is rotating
around it’s center of mass
instead of orbiting an outside
point
Angular Acceleration (α)
 A measure of how quickly angular
velocity is changing
 Unit is rad/s2
 Again, this differs from centripetal
acceleration because it is rotation of an
object around its center of mass as
opposed to revolving around an external
point
Starting Rotation
 Caused by torque (τ) acting on an
object
 This is rotational force
 Unit is a Nm
 Two parts to torque:
 Lever arm
 To get the most effect, effort force
should be exerted as far from the axis
of rotation as possible (why doorknobs
are at the edge of a door)
 L = r, if the force is exerted
perpendicular to the axis of rotation
 Force
 Often the weight of an object (Fw = mg)
Net Torque
 If clockwise torque = counterclockwise
torque, then net torque is zero and no
rotation occurs
 This is called static equilibrium or translational
equilibrium
 There is no velocity or acceleration
Moment of Inertia (I)
 Not only mass matters for
rotation, its location also
matters
 The further from the axis a
mass is, the harder it is to turn
 This is why you choke up on a
baseball bat to make it easier to
swing
 Can change this by changing
the mass or where the mass
is located in relationship to
the axis of rotation
Newton’s 2nd Law
Modified
 Normally, acceleration is equal to force
divided by mass
 In rotational motion, force is replaced by
torque and mass is replaced by moment
of inertia
 The same equation, with distance from
axis of rotation added to account for
circular motion
Center of Mass (COM)
 Each object has a center of mass (COM)
 This COM follows all motion laws, the rest of the object
rotates around this point
 To find COM, suspend the object at 2 different points. Draw
a vertical line down the object from that point. Where the
two lines cross is the COM
 This is typically higher on a male’s body then a female’s
 You can change your COM by changing your shape
 COM can be located in empty space (ex. donut)
Toppling
 Objects topple when their COM is no
longer over its support base (τ net no
longer = 0)
 Considered stable if an external force is
needed to cause toppling
 The lower the COM, the more stable the
object
Angular Momentum (L)
 Like linear momentum, but with all our
modified angular measures
 Is the product of momentum of inertia
and angular velocity
 The product of torque and time is the
angular impulse which causes a
change in angular momentum
 It’s still conserved, like linear momentum
 So, if your decrease your radius (and
therefore your I), your angular velocity must
increase
 This is why you spin faster when you pull in
your arms when ice skating