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Analogies between linear and rotational motion, two types of circular motion, centripetal, and centrifugal forces, weightlessness, common situations involving centripetal acceleration.

1.
r

Fc, ac

v

Fc, ac

v

2.
Circular motion is very

similar to linear motion in

many ways.

Linear Angular

Quantity unit Quantity unit

Displacement (x) M Angular Rad

Velocity (v) m/s displacement (ϴ)

Acceleration (a) m/s/s Angular velocity (ω) Rad/sec

Mass (m) Kg Angular Rad/sec/sec

acceleration (α)

Force (F) Kg * m/s or N

Moment of inertia Kg * m*m

(ɪ )

Torque (t) N*m

similar to linear motion in

many ways.

Linear Angular

Quantity unit Quantity unit

Displacement (x) M Angular Rad

Velocity (v) m/s displacement (ϴ)

Acceleration (a) m/s/s Angular velocity (ω) Rad/sec

Mass (m) Kg Angular Rad/sec/sec

acceleration (α)

Force (F) Kg * m/s or N

Moment of inertia Kg * m*m

(ɪ )

Torque (t) N*m

3.
Analogies Between

Linear and Rotational

Motion

Linear and Rotational

Motion

4.
There are two types of circular

motion

An axis is the straight line around

which rotation takes place.

• When an object turns about an internal

axis—that is, an axis located within the

body of the object—the motion is called

rotation, or spin.

• When an object turns about an external

axis, the motion is called revolution.

motion

An axis is the straight line around

which rotation takes place.

• When an object turns about an internal

axis—that is, an axis located within the

body of the object—the motion is called

rotation, or spin.

• When an object turns about an external

axis, the motion is called revolution.

5.
Centripetal acceleration –

acceleration of an object in

circular motion. It is directed

toward the center of the circular

path.

2 ac = centripetal acceleration,

v m/s2

ac =

r v = tangential speed, m/s

r = radius, m

acceleration of an object in

circular motion. It is directed

toward the center of the circular

path.

2 ac = centripetal acceleration,

v m/s2

ac =

r v = tangential speed, m/s

r = radius, m

6.
Centripetal Force – the net inward

force that maintains the circular

motion of an object. It is directed

toward the center.

Fc = centripetal force, N

Fc = m a c m = mass, kg

2

mv ac = centripetal

Fc = acceleration,

r m/s2

v = tangential speed, m/s

r = radius, m

force that maintains the circular

motion of an object. It is directed

toward the center.

Fc = centripetal force, N

Fc = m a c m = mass, kg

2

mv ac = centripetal

Fc = acceleration,

r m/s2

v = tangential speed, m/s

r = radius, m

7.
Two types of speed

Linear speed is the distance traveled per unit of

• A point on the outer edge of the turntable travels

a greater distance in one rotation than a point

near the center.

• The linear speed is greater on the outer edge of a

rotating object than it is closer to the axis.

• The speed of something moving along a circular

path can be called tangential speed because

the direction of motion is always tangent to the

circle.

Linear speed is the distance traveled per unit of

• A point on the outer edge of the turntable travels

a greater distance in one rotation than a point

near the center.

• The linear speed is greater on the outer edge of a

rotating object than it is closer to the axis.

• The speed of something moving along a circular

path can be called tangential speed because

the direction of motion is always tangent to the

circle.

8.
Tangential Speed (linear)

Tangential speed depends on two things

1. rotational speed

2. the distance from the axis of rotation.

Tangential speed depends on two things

1. rotational speed

2. the distance from the axis of rotation.

9.
Which part of the turntable moves faster—the outer part where the

ladybug sits or a part near the orange center?

***It depends on whether you are talking about linear speed or

rotational speed.***

ladybug sits or a part near the orange center?

***It depends on whether you are talking about linear speed or

rotational speed.***

10.
Rotational speed (sometimes called angular

speed) is the number of rotations per unit of

• All parts of the rigid turntable rotate about

the axis in the same amount of time.

• All parts have the same rate of rotation, or

the same number of rotations per unit of

time.

• It is common to express rotational speed

in revolutions per minute (RPM).

speed) is the number of rotations per unit of

• All parts of the rigid turntable rotate about

the axis in the same amount of time.

• All parts have the same rate of rotation, or

the same number of rotations per unit of

time.

• It is common to express rotational speed

in revolutions per minute (RPM).

11.
Rotational Speed

All parts of the turntable rotate at the

same rotational speed.

• A point farther away from the center travels a longer path

in the same time and therefore has a greater tangential

speed. (Linear Speed)

All parts of the turntable rotate at the

same rotational speed.

• A point farther away from the center travels a longer path

in the same time and therefore has a greater tangential

speed. (Linear Speed)

12.
Rotational Speed

Remember All parts of the turntable rotate at the same rotational

speed.

A point farther away from the center travels a longer path in the

same time and therefore has a greater tangential speed.

(linear speed)

Therefore, a ladybug sitting twice as far from the center

moves twice as fast.

Remember All parts of the turntable rotate at the same rotational

speed.

A point farther away from the center travels a longer path in the

same time and therefore has a greater tangential speed.

(linear speed)

Therefore, a ladybug sitting twice as far from the center

moves twice as fast.

13.
Question # 1

At an amusement park, you and a friend sit on a large

rotating disk. You sit at the edge and have a rotational

speed of 4 RPM and a linear speed of 6 m/s. Your

friend sits halfway to the center. What is her rotational

speed? What is her linear speed?

Her rotational speed is also 4 RPM, and her linear

speed is 3 m/s.

At an amusement park, you and a friend sit on a large

rotating disk. You sit at the edge and have a rotational

speed of 4 RPM and a linear speed of 6 m/s. Your

friend sits halfway to the center. What is her rotational

speed? What is her linear speed?

Her rotational speed is also 4 RPM, and her linear

speed is 3 m/s.

14.
Calculating Average Speed

An object moving in uniform circular motion would cover

the same linear distance in each second of time.

When moving in a circle, an object travels a distance

around the perimeter of the circle.

The distance of one complete cycle around the

perimeter of a circle is known as the circumference.

The circumference of any circle is

Circumference = 2*pi*Radius

An object moving in uniform circular motion would cover

the same linear distance in each second of time.

When moving in a circle, an object travels a distance

around the perimeter of the circle.

The distance of one complete cycle around the

perimeter of a circle is known as the circumference.

The circumference of any circle is

Circumference = 2*pi*Radius

15.
For a constant tangential speed:

v = tangential speed, m/s

d 2πr d = distance, m

v= =

t T t = time, s

r = radius, m

T = period, s (time for 1

rev.)

If rpm (revolutions per minute) is given, convert to m/s using

these conversion factors:

1rev 2 r and 1 min. = 60 sec.

Or you can find the period by taking the inverse of the

1 T = period, s – time for one revolution

T= F = frequency, rev/s – number of revolutions per time

f Note: Period and frequency are inverses.

v = tangential speed, m/s

d 2πr d = distance, m

v= =

t T t = time, s

r = radius, m

T = period, s (time for 1

rev.)

If rpm (revolutions per minute) is given, convert to m/s using

these conversion factors:

1rev 2 r and 1 min. = 60 sec.

Or you can find the period by taking the inverse of the

1 T = period, s – time for one revolution

T= F = frequency, rev/s – number of revolutions per time

f Note: Period and frequency are inverses.

16.
Constant Speed, but is there

constant Velocity?

Remember speed is a scalar quantity

and velocity is a vector quantity.

The direction of the velocity vector is

directed in the same direction that the

object moves. Since an object is

moving in a circle, its direction is

continuously changing.

The best word that can be used to describe

the direction of the velocity vector is the

word tangential.

constant Velocity?

Remember speed is a scalar quantity

and velocity is a vector quantity.

The direction of the velocity vector is

directed in the same direction that the

object moves. Since an object is

moving in a circle, its direction is

continuously changing.

The best word that can be used to describe

the direction of the velocity vector is the

word tangential.

17.
BIG IDEA….

Centripetal force keeps an

object in circular motion.

Centripetal force keeps an

object in circular motion.

18.
Centripetal Force

The force exerted on a whirling can is toward the center. NO

outward force acts on the can.

The force exerted on a whirling can is toward the center. NO

outward force acts on the can.

19.
Since centripetal force is a net force,

there must be a force causing it. Some

examples are

A car going around a curve on a flat road:

Fc = Ff (friction force)

there must be a force causing it. Some

examples are

A car going around a curve on a flat road:

Fc = Ff (friction force)

20.
Creates a curved path

Centripetal force holds a car in a curved path.

a. For the car to go around a curve, there must be sufficient friction to

provide the required centripetal force.

b. If the force of friction is not great enough, skidding occurs.

Centripetal force holds a car in a curved path.

a. For the car to go around a curve, there must be sufficient friction to

provide the required centripetal force.

b. If the force of friction is not great enough, skidding occurs.

21.
Since centripetal force is a net force,

there must be a force causing it. Some

examples are

A car going around a curve on a flat road:

Fc = Ff (friction force)

Orbital motion, such as a satellite: Fc = Fg

(weight or force of gravity)

there must be a force causing it. Some

examples are

A car going around a curve on a flat road:

Fc = Ff (friction force)

Orbital motion, such as a satellite: Fc = Fg

(weight or force of gravity)

22.
Since centripetal force is a net force,

there must be a force causing it. Some

examples are

A car going around a curve on a flat road:

Fc = Ff (friction force)

Orbital motion, such as a satellite: Fc = Fg

(weight or force of gravity)

A person going around in a spinning

circular room: Fc = FN (normal force)

there must be a force causing it. Some

examples are

A car going around a curve on a flat road:

Fc = Ff (friction force)

Orbital motion, such as a satellite: Fc = Fg

(weight or force of gravity)

A person going around in a spinning

circular room: Fc = FN (normal force)

23.
Since centripetal force is a net force,

there must be a force causing it. Some

examples are

A car going around a curve on a flat road: Fc

= Ff (friction force)

Orbital motion, such as a satellite: Fc = Fg

(weight or force of gravity)

A person going around in a spinning circular

room: Fc = FN (normal force)

A mass on a string (horizontal circle, i.e..

parallel to the ground): Fc = T (tension in the

string)

there must be a force causing it. Some

examples are

A car going around a curve on a flat road: Fc

= Ff (friction force)

Orbital motion, such as a satellite: Fc = Fg

(weight or force of gravity)

A person going around in a spinning circular

room: Fc = FN (normal force)

A mass on a string (horizontal circle, i.e..

parallel to the ground): Fc = T (tension in the

string)

24.
For a mass on a string moving in a vertical

circle, the centripetal force is due to

different forces in different locations.

At the top of the circle, Fc = T + Fg (tension

plus weight or gravity)

At the bottom of the circle, Fc = T - Fg

(tension minus weight or gravity)

On the outermost side, Fc = T

Anywhere other than above, you would need

to find the component of gravity parallel to

the tension and either add or subtract from

tension depending on the location on the

circular path

circle, the centripetal force is due to

different forces in different locations.

At the top of the circle, Fc = T + Fg (tension

plus weight or gravity)

At the bottom of the circle, Fc = T - Fg

(tension minus weight or gravity)

On the outermost side, Fc = T

Anywhere other than above, you would need

to find the component of gravity parallel to

the tension and either add or subtract from

tension depending on the location on the

circular path

25.
Example :Motion in a

Vertical

v

Circle

v

v Consider the forces on a ball attached to

v +

Bottom

Tmg a string as it moves in a vertical loop.

+

T T Note changes as you click the mouse

T

+ mg to show new positions.

+

mg + mg

T The velocity of the object is constantly

changes depending on which direction

v gravity is pointing compared to

Top

Left of Path

Side velocity.

Top

Top mg

RightRight

Tension is minimum +

The tension required to keep this

as weight

Weight

Maximum has helps FcT, W

no effect

tension object moving in a circle changes

Weight causes small

force

on T

Weight

opposes has

Fcin no effect while it is in it motion as well.

decrease tension T

on T Bottom

Vertical

v

Circle

v

v Consider the forces on a ball attached to

v +

Bottom

Tmg a string as it moves in a vertical loop.

+

T T Note changes as you click the mouse

T

+ mg to show new positions.

+

mg + mg

T The velocity of the object is constantly

changes depending on which direction

v gravity is pointing compared to

Top

Left of Path

Side velocity.

Top

Top mg

RightRight

Tension is minimum +

The tension required to keep this

as weight

Weight

Maximum has helps FcT, W

no effect

tension object moving in a circle changes

Weight causes small

force

on T

Weight

opposes has

Fcin no effect while it is in it motion as well.

decrease tension T

on T Bottom

26.

27.
Calculating Centripetal Forces

Greater speed and greater mass require greater centripetal force.

Traveling in a circular path with a smaller radius of curvature requires a

greater centripetal force.

Centripetal force, Fc, is measured in newtons when m is expressed in

kilograms, v in meters/second, and r in meters.

Greater speed and greater mass require greater centripetal force.

Traveling in a circular path with a smaller radius of curvature requires a

greater centripetal force.

Centripetal force, Fc, is measured in newtons when m is expressed in

kilograms, v in meters/second, and r in meters.

28.
Adding Force Vectors

•A conical pendulum is a bob held in a circular path by a string attached

•This arrangement is called a conical pendulum because the string sweeps

out a cone.

•Only two forces act on the bob: mg, the force due to gravity, and T,

tension in the string.

• Both are vectors.

•A conical pendulum is a bob held in a circular path by a string attached

•This arrangement is called a conical pendulum because the string sweeps

out a cone.

•Only two forces act on the bob: mg, the force due to gravity, and T,

tension in the string.

• Both are vectors.

29.
•The vector T can be resolved into two perpendicular components,

Tx (horizontal), and Ty (vertical).

•Therefore Ty must be equal and opposite to mg.

•Tx is the net force on the bob–the centripetal force. Its magnitude

is mv2/r, where r is the radius of the circular path.

Tx (horizontal), and Ty (vertical).

•Therefore Ty must be equal and opposite to mg.

•Tx is the net force on the bob–the centripetal force. Its magnitude

is mv2/r, where r is the radius of the circular path.

30.
Centripetal Force

Centripetal force keeps the vehicle in a circular path as it rounds a

banked curve.

Centripetal force keeps the vehicle in a circular path as it rounds a

banked curve.

31.
Centrifugal Forces – MISCONCEPTION!!

•When an object moves in a circular motion there MUST be an

outward force.

•NO!!!

•This apparent outward force on a rotating or revolving body is

called centrifugal force. Centrifugal means “center-fleeing,” or

“away from the center.”

•If there was an outward force, we would see something

completely different than what actually happens.

•When an object moves in a circular motion there MUST be an

outward force.

•NO!!!

•This apparent outward force on a rotating or revolving body is

called centrifugal force. Centrifugal means “center-fleeing,” or

“away from the center.”

•If there was an outward force, we would see something

completely different than what actually happens.

32.
Gravity Near the Earth’s Surface

The acceleration due

to gravity varies

over the Earth’s

surface due to

altitude, local

geology, and the

shape of the Earth,

which is not quite

The acceleration due

to gravity varies

over the Earth’s

surface due to

altitude, local

geology, and the

shape of the Earth,

which is not quite

33.
Satellites and “Weightlessness”

Satellites are routinely put into orbit around the

Earth. The tangential speed must be high enough

so that the satellite does not return to Earth, but

not so high that it escapes Earth’s gravity

Satellites are routinely put into orbit around the

Earth. The tangential speed must be high enough

so that the satellite does not return to Earth, but

not so high that it escapes Earth’s gravity

34.
Satellites and “Weightlessness”

The satellite is kept in orbit by its speed—it is

continually falling, but the Earth curves from

underneath it.

The satellite is kept in orbit by its speed—it is

continually falling, but the Earth curves from

underneath it.

35.
Satellites and “Weightlessness”

Objects in orbit are said to experience weightlessness. They

do have a gravitational force acting on them, though!

The satellite and all its contents are in free fall, so there is no

normal force. This is what leads to the experience of

Objects in orbit are said to experience weightlessness. They

do have a gravitational force acting on them, though!

The satellite and all its contents are in free fall, so there is no

normal force. This is what leads to the experience of

36.
Satellites and “Weightlessness”

More properly, this effect is called apparent

weightlessness, because the gravitational force

still exists. It can be experienced on Earth as

well, but only briefly:

More properly, this effect is called apparent

weightlessness, because the gravitational force

still exists. It can be experienced on Earth as

well, but only briefly:

37.
Common situations involving

Centripetal Acceleration

Many specific situations will use

forces that cause centripetal

acceleration

Level curves

Banked curves

Horizontal circles

Vertical circles

Note that Fc, v or ac may not be constant

Centripetal Acceleration

Many specific situations will use

forces that cause centripetal

acceleration

Level curves

Banked curves

Horizontal circles

Vertical circles

Note that Fc, v or ac may not be constant

38.
Level Curves

Friction is the

force that

produces the

centripetal

acceleration

Can find the

frictional force, µ,

or v

v rg

Friction is the

force that

produces the

centripetal

acceleration

Can find the

frictional force, µ,

or v

v rg

39.
Banked Curves

A component of

the normal force

adds to the

frictional force to

allow higher

speeds

v2

tan

rg

or ac g tan

A component of

the normal force

adds to the

frictional force to

allow higher

speeds

v2

tan

rg

or ac g tan

40.
Vertical Circle

Look at the forces

at the top of the

circle

The minimum

speed at the top of

the circle can be

found

v top gR

Look at the forces

at the top of the

circle

The minimum

speed at the top of

the circle can be

found

v top gR