KD4 Projectile and

Circular Mechanics

Chapter 3; Chapter 9

Questions for Thought

Which moves faster on a merry-go-round, a Which moves faster on a merry-go-round, a horse on the outside or inside?horse on the outside or inside?

What happens if you are the passenger in a What happens if you are the passenger in a car and the driver takes a right-hand turn?car and the driver takes a right-hand turn?

Left-hand turn?Left-hand turn?

What happens during the spin cycle in a What happens during the spin cycle in a washing machine?washing machine?

Rotation vs. RevolutionRotationRotation (spin) – Motion of an object spinning (spin) – Motion of an object spinning around an internal axisaround an internal axis

Ex) Merry-go-round spins Ex) Merry-go-round spins around its own axis around its own axis

RevolutionRevolution- Motion of an object - Motion of an object turning around an external axisturning around an external axis

Person on merry-go-round Person on merry-go-round revolves around the same axisrevolves around the same axis

Hmmm?

Which moves faster a person near the outside Which moves faster a person near the outside of the merry-go-round or one near the inside of the merry-go-round or one near the inside of the same merry-go-round?of the same merry-go-round?

Rotational vs. Tangential Speed

Rotational speed Rotational speed (angular)-# of rotations/time(angular)-# of rotations/time

Both have the same rotational speedBoth have the same rotational speed

Tangential speed Tangential speed (linear)-Distance moved/time(linear)-Distance moved/time

The person on the outside has a greater The person on the outside has a greater tangential speed because he/she is moving a tangential speed because he/she is moving a greater distancegreater distance

Part 1: Circular Motion

Uniform Circular MotionUniform Circular Motion-The motion of an -The motion of an object traveling in a object traveling in a circularcircular path at a path at a constant constant speedspeed

Ex) A tetherball moving in a circular pathEx) A tetherball moving in a circular path

Is the tetherball accelerating?Is the tetherball accelerating?

Circular Motion

Centripetal accelerationCentripetal acceleration- Acceleration due - Acceleration due to changing directionto changing direction

““Center seeking”Center seeking”

All objects movingAll objects movingin a circle are in a circle are

acceleratingaccelerating(towards the(towards the

center)center)

Centripetal Acceleration

a = va = v²²/R/R

a = 2a = 2ллv/Tv/T

a = 4a = 4лл²R/T²²R/T²

R=RadiusR=Radius

T=period (time for 1 revolution)T=period (time for 1 revolution)

Centripetal Acceleration

An object which experiences an acceleration An object which experiences an acceleration must also experience a must also experience a forceforce

F=maF=ma

Centripetal Force

Force acting towards the Force acting towards the center which causes the center which causes the object to object to seek the centerseek the center

Works Works againstagainst inertia inertia

What is the Centripetal Force?A car moving around a track?A car moving around a track?

Frictional force between tires and Earth forcing Frictional force between tires and Earth forcing car inward keeping it in its pathcar inward keeping it in its path

Planets orbiting the sun?Planets orbiting the sun?

Gravitational force pulling on the Gravitational force pulling on the object keeps it in its pathobject keeps it in its path

Centripetal Force

Formulas:Formulas:

FFcc =(m) (a =(m) (acc))

FFcc = (m) (V = (m) (V22/R)/R)

FFcc = (m) (2πv/T) = (m) (2πv/T)

Fc = (m) (4πFc = (m) (4π22R/TR/T22))

Centrifugal Force

““Center fleeing”Center fleeing”

This is a FAKE forceThis is a FAKE force

Outward force is a Outward force is a misconception duemisconception due to to inertiainertia

If swing a can on the end of a string over your If swing a can on the end of a string over your head and the string breaks, what happens to head and the string breaks, what happens to the can?the can?

The can moves in a straight line tangent to its The can moves in a straight line tangent to its circular pathcircular path

Example:

Calculate the maximum speed a 1200 kg car Calculate the maximum speed a 1200 kg car can travel around a curve of 35 m radius if the can travel around a curve of 35 m radius if the frictional force between the tires and the road frictional force between the tires and the road surface is 2.4 X 10surface is 2.4 X 1033 N. N.

Fc = m vFc = m v22 / r / r

2.4 x 102.4 x 1033 = 1200 v = 1200 v22 / 35 / 35

V = 8.4 m/sV = 8.4 m/s

Centripetal acceleration is always toward the centerCentripetal acceleration is always toward the center

Centrifugal acceleration does NOT exist since centrifugal Centrifugal acceleration does NOT exist since centrifugal force does NOT existforce does NOT exist

What happens to a person in the backseat if driver makes a right turn? Left?

The person goes STRAIGHT (due to inertia), but

you get the misconception you are going outward

Part 2: Projectile Motion

Motion resulting from the sum of 2 Motion resulting from the sum of 2 independentindependent velocities velocities

Horizontal Horizontal constantconstant velocity (neglecting air velocity (neglecting air resistance)resistance)

Vertical Vertical increasingincreasing velocity (due to gravity) velocity (due to gravity)

Projectile Motion

Makes a parabolic pathMakes a parabolic path

Projectile Motion: Equations• Range (horizontal distance)Range (horizontal distance)

– ConstantConstant

– DDxx = v = vxx t t

• Altitude (height)Altitude (height)

– Changes due to gravityChanges due to gravity

– DDyy = ½ g t = ½ g t22

vx

t

Example

• If a bullet is dropped at the same moment a If a bullet is dropped at the same moment a bullet is shot out of a gun (and they both start bullet is shot out of a gun (and they both start at the same height), which bullet hits the at the same height), which bullet hits the ground 1st?ground 1st?

– They both hit at the same time!They both hit at the same time!

Example:

• If a bullet is fired 20 m above the ground, how If a bullet is fired 20 m above the ground, how long does it take to hit the ground?long does it take to hit the ground?

– DDyy = ½ g t = ½ g t22

– 20 = ½ (9.8) t20 = ½ (9.8) t22

– t = 2 st = 2 s

What if the object is fired up at an angle?

Horizontal vs. Vertical

• VerticalVertical

– velocity changes due to gravityvelocity changes due to gravity

– Time calculated would be the same up OR downTime calculated would be the same up OR down

• HorizontalHorizontal

– velocity is constant (neglecting air)velocity is constant (neglecting air)

– Time would be the same as down vertically is the object is Time would be the same as down vertically is the object is projected horizontally (only going down)projected horizontally (only going down)

– Time would be doubled if the object is projected at an angle Time would be doubled if the object is projected at an angle (due to the object traveling horizontally for the time up AND (due to the object traveling horizontally for the time up AND the time downthe time down