Circular Motion TermsThe point or line that is the center of the

circle is the axis of rotation. If the axis of rotation is inside the

object, the object is rotating (spinning).

If the axis of rotation is outside the object, the object is revolving.

Linear/Tangential VelocityObjects moving in a circle still have a

linear velocity = distance/time.This is often called tangential velocity,

since the direction of the linear velocity is tangent to the circle.

v

Rotational/Angular VelocityObjects moving in a circle also have a

rotational or angular velocity, which is the rate angular position changes.

Rotational velocity is measured in degrees/second, rotations/minute (rpm), etc.

Common symbol, (Greek letter omega)

Rotational/Angular Velocity

• Rotational velocity = Change in angletime

Rotational & Linear Velocity If an object is rotating:

All points on the object have the same rotational (angular) velocity.

All points on the object do not have the same linear (tangential) velocity.

Rotational & Linear VelocityLinear velocity of a point depends on:

The rotational velocity of the point. More rotational velocity means more linear

velocity. The distance from the point to the axis of

rotation. More distance from the axis means more linear

velocity.

Rotational & Linear Velocity In symbols:

v = r

v

r

AccelerationAs an object moves around a circle, its

direction of motion is constantly changing.

Therefore its velocity is changing.Therefore an object moving in a circle is

constantly accelerating.

Centripetal AccelerationThe acceleration of an object moving in

a circle points toward the center of the circle.

This is called a centripetal (center pointing) acceleration.

a

Centripetal AccelerationThe centripetal acceleration depends

on: The speed of the object. The radius of the circle.

Acent = v2

r

Centripetal ForceNewton’s Second Law says that if an

object is accelerating, there must be a net force on it.

For an object moving in a circle, this is called the centripetal force.

The centripetal force points toward the center of the circle.

Centripetal Force In order to make an object revolve

about an axis, the net force on the

object must pull it toward the center of the circle.

This force is called a centripetal (center seeking) force.

Fnet

Centripetal ForceCentripetal force on an object depends

on: The object’s mass - more mass means

more force. The object’s speed - more speed means

more force. And…

Centripetal ForceThe centripetal force on an object also

depends on: The object’s distance from the axis

(radius). If linear velocity is held constant, more

distance requires less force. If rotational velocity is held constant, more

distance requires more force.

Centripetal Force In symbols:

Fcent=mv2

r= mr2

Work Done by the Centripetal Force

Since the centripetal force on an object is always perpendicular to the object’s velocity, the centripetal force never does work on the object - no energy is transformed.

v

Fcent

“Centrifugal Force” “Centrifugal force” is a fictitious force -

it is not an interaction between 2

objects, and therefore not a real force.

Nothing pulls an object away from the center of the circle.

“Centrifugal Force”What is erroneously attributed to

“centrifugal force” is actually the action of the object’s inertia - whatever velocity it has (speed + direction) it wants to keep.

Rotational Motion of an Object

Rotational Motion All Spinning Objects Axis of Rotation

The line about which everything rotates. Speed of Rotation

Period of rotation The time of a single complete rotation (T)

Frequency of rotation The number of cycles completed in a given time (f = hertz)

Period = 1/Frequency or Frequency = 1/Period T= 1/f f=1/T

Rotational Motion

Spinning

Angular speed

= s/r ( is measured in radians)For 360 degreed, s = 2r

3600 = 2 radians

Angular speed = 2f

t

Time

angle Rotational

Spinning Angular speed

= s/r ( is measured in radians)

For 360 degreed, s = 2r

3600 = 2 radians

Angular speed = 2f

Torque

Angular MomentumMoment of inertia

In general, the farther away a mass is from the axis the greater its moment of inertia is.

I = kmr2

Angular Momentum (Cont.) Momentum of inertia

times angular speed L = I

Conservation of Angular momentum

Direction of Rotation The right hand rule

Moment of Inertia Momentum of inertia

equals the resistance to motion

I = mr2

Moment of Inertia = mass times the distance from the axis squared

Angular acceleration

Center of GravityCenter of Mass

"All of science is nothing more than the refinement of everyday thinking."-- Albert Einstein

Center of Gravity

Point of an object located at the average position of weight.

Center of Gravity

Point of an object located at the average position of weight.

Center of GravityPoint of an object located at the average position of weight.

Center of Mass

The Average position of matter

Center of Mass

The Average position of matter

Center of Mass

The Average position of matter

Center of Mass

The Average position of matter

Toppling

Toppling occurs when the center of gravity extends beyond the support base.

StabilityUnstable – CG is lowered with

displacementStable – work must be done to raise the

CGNeutral – displacement neither raises or

lowers the CG

Coriolis “force” An apparent force

that seems to deflect a moving object from its path

Only observed in rotating references

Related to Centifrugal “force”

Coriolis “force” An apparent force

that seems to deflect a moving object from its path

Only observed in rotating references

Related to Centifrugal “force”

Coriolis “force” An apparent force

that seems to deflect a moving object from its path

Only observed in rotating references

Related to Centifrugal “force”

Objects rotate around their center of gravity.

Center of Gravity

Center of GravityThrow a ball through the air and it travels a

smooth parabolic path. Throw a bat through the air and it wobbles all over the place (class example: marker). However if you watch the path of the bat, the middle

of it follows the same path that the ball followed. The bat is a sum of two motions.

A spin around the center point A movement through the air as if all

the weight were concentrated at this point.

The Center of Gravity for an object is the point located at the object’s average position of weight.

For a symmetrical object this point would be located at the center

For an irregular shaped object, the center of gravity is toward the heavier end ( i.e. bat)

Center of Mass

Center of Gravity is often called Center of Mass, which is the average positions of all the particles of mass that make up an object.

The Center of mass or center of gravity can lie outside of the object (i.e. Donut, tire, banana, chair

Finding center of mass for a 1-D situation. We can use the equation:

Xcm = (m1x1 + m2x2 + …) / (m1 + m2 + …) 2-D is easy to follow the same trend, but

use Ycm as well.

Locating the Center of GravityUsing a plumb line and bob, you can

suspend the object from some other point and constructing a second vertical line. The Center of Gravity is where the two lines intersect.

Toppling The rule for Toppling:

If the center of gravity of an object is above the area of support, the object will remain upright. If the Center of Gravity extends outside the area of support, the object will topple.

Example: When a male tries to push a penny with his nose on the floor. The center of gravity extends beyond the supports and he will fall over.

The Leaning Tower of Pisa does not topple over, WHY??

Stability We say that an object balanced so that any

displacement lowers its center of gravity is in Unstable Equilibrium. An example would be a cone that was point down, if it

is moved, it’s center of gravity would lower and it would then topple.

We say an object that is balanced so that any displacement raises its center of gravity is in Stable Equilibrium. An example would be a cone that was point up. Any

movement would cause the center of gravity to rise up. So that would need to be overcome before toppling can happen.

Place the cone on its side and its center of gravity is neither raised nor lowered with displacement.

This is called Neutral Equilibrium.

A book that is standing is at stable Equilibrium and so is a book laying flat. Which one is more stable and why?

Why does a tightrope walker use a long poll that bends downward?

Center of Gravity of People What happens when we touch our toes?

Isn’t it true that we push our but back to touch our toes? Why?

When we stand our center of gravity is generally a few cm’s below our navel. Women are typically lower than men.

What happens when we stand against the wall and then try to lean forward and touch our toes?

The End