Lecture 15: Rotational Motion

Questions of Yesterday

1) A piece of clay traveling north with speed v collides perfectly inelastically with an identical piece of clay traveling east with speed v. What direction does the resultant piece of clay travel?a) northb) eastc) 45o N of Ed) 45o S of W

2) Ball 1 of mass m, traveling with speed v, collides with Ball 2 of mass 2m and comes to rest, what is the speed of Ball 2 after the collision?a) 2vb) vc) v/2d) v/(√2)

Linear Motionx = xf - xiDisplacement:

xt

v =Velocity:

Acceleration:vt

a =

Constant a Equations:

v = v0 + atx = v0t + 1/2at2

v2 = v02 + 2ax

F = maForce

(2nd Law):

p = m*vMomentum:

p = FtImpulse:

Equations/Concepts valid for straight line

motion betweenpoints in space (x-y plane)

Circular Motion

Circumference = 2r

How do you define “position” and “displacement” when motion is circular?

r

s

s = r*Arc length:

Angle Unit = Radian

2 Radians = 360o

= 0

= /2

= 2

=

= 3/2

sr =Angular

Position:

Circular Motion

r

i

ti

sr =Angular

Position:

f

tf

=f - i

Angular Displacement:

av= f - i

tf - ti

t

=

Average Angular Velocity:

SI Units: Radians (rad)

SI Units: Radians per

second (rad/s)

limt -> 0

= t

Instantaneous Angular Velocity

Circular Motion

r

i

tif

tf

av= f - i

tf - ti

t

=

Average Angular Acceleration: SI Units:

Radians per second squared (rad/s2)

limt -> 0

= t

Instantaneous Angular Acceleration

Constant Angular Acceleration

Linear Motion with Constant a:

v = v0 + at

x = v0t + 1/2at2

v2 = v02 + 2ax

= 0 + t

Rotational Motion with Constant :

= 0t + 1/2t2

2 = 02 + 2

Rotational Motion

Which position has a greater angular displacement in a given time interval?

What about angular speed? Angular acceleration?

Rotational Motion

Which position has a greater angular displacement in a given time interval?

What about angular speed? Angular acceleration?

Angular and Linear Quantities

rti

tf

sr

=

Displacement:Direction of linear velocity v of an object moving in a circular path is

always TANGENT to the path

svT=r

Tangential Speed:

aT=r

Tangential Acceleration:

Centripetal Acceleration

r

If you’re jogging on a circular track with constant tangential speed is your acceleration ZERO? Why or

Why not?

vf

vf - vi

tf - ti

aav =

vi

During circular motion at constant speed your direction is constantly changing so you

still have an acceleration

CENTRIPETAL ACCELERATIONAcceleration associated with

constant speed circular motion

Centripetal Acceleration

r

vf

Centripetal Acceleration always points towards the CENTER of the circle

vf - vi

tf - ti

aav =

vi

-vi

vf

v

Centripetal AccelerationCentripetal Acceleration always points towards the

CENTER of the circle

r

vf

vi

-vi

vf

v

r

sr

vv = v

taav=

v2

rac

==r2

Similar Triangles

s

Centripetal AccelerationWhat if your tangential speed is NOT constant?

r

vf

vi

-vi

vf

v

rAcceleration has both tangential and centripetal

components!

a = (ac2 + aT

2)1/2

v vc

vT

v2

rac

=

aT=r

Rotational Motion: Practice ProblemA race car starts from rest on a circular track of radius 400 m. The car’s speed increases at the constant rate

of 0.500 m/s2. At the point where the magnitudes of the centripetal

and tangential accelerations are equal, what is…

the tangential speed of the car?

the angular speed of the car?

the distance traveled?

the number of revolutions made?

the elapsed time?

Centripetal Force

F = ma

If an object is accelerating what do know about it (think Newton’s 2nd law)?

Can an object be moving in a circular path if no forces are acting on?

If an object is undergoing constant speed circular motion what direction is the net force acting on the

object?

mv2

rFc = mac

=

Centripetal Force

FT = maT

What if an object undergoing circular motion and changing its tangential speed?

mv2

rFc = mac

=

-vi

vf

va ac

aT

F

FT

FC

F = ma

Just like linear motion (∑Fx = max, ∑Fy = may)…must split vector equation into perpendicular

components!!

Centripetal ForceAs you round the bend at constant

speed in what direction..

is your net acceleration? Why?Is your net force? Why?

do you feel yourself being pulled? Why?

Remember Newton’s 1st law??

What force is acting on you and your car to let you round the

bend?

Centripetal Force

Remember Newton’s 1st law??

What force is acting on you and your car to let you round the

bend?

N

Fg

ff

As you round the bend at constant speed in what direction..

is your net acceleration? Why?Is your net force? Why?

do you feel yourself being pulled? Why?

Practice ProblemSuppose that a 1800-kg car passes over a bump in a

roadway that follows the arc of circle of radius 20.0 m.

What force does the road exert on the car as the car passes the highest point of the bump if the car travels at

9.00 m/s?

What is the maximum speed the car can have without losing contact with the road as it passes this highest

point?

Questions of the Day1) You are going through a vertical loop on roller coaster at a

constant speed. At what point is the force exerted by the tracks on you (and the cart you are in) the greatest? a) at the highest pointb) at the lowest pointc) halfway between the highest and lowest pointd) the force is equal over the whole loop

2) You are on a merry-go-round moving at constant speed. If you move to the outer edge of the merry-go-round, what happens to the net centripetal force keeping you on the merry-go-round? a) it increases

b) it decreasesc) it stays the samed) there is no net centripetal force acting on you