ROTATIONAL MOTIONjoetrout/phys182/Phys182_Applied...ROTATIONAL MOTION: ROTATIONAL MOTION: r s=l ≡ s r = l r r r +x +y-y-x 0o 90o 180o 270o 360o 0rad rad 2 rad 2 rad 3 2 rad 2 rad=360o

ROTATIONAL MOTION:

ROTATIONAL MOTION:

r

s=l

≡ sr

= lr

r

r

+x

+y

-y

-x

0o

90o

180o

270o

360o0 rad

2radrad

2rad

32rad

2 rad=360o

+x

+y

-y

-x

0o

90o

180o

270o

360o

0 rad

2 radrad

2rad

32rad

2rad=360o

30o

30o2rad360o =

12rad

+x

+y

-y

-x

0o

90o

180o

270o

360o

0 rad

2 radrad

2rad

32rad

2rad=360o

45rad

45 rad 360o

2 rad =144o

+x

+y

-y

-x

+x

+y

-y

-x

r

Angular Position:

+x

+y

-y

-x

s

r

r

=d d t

Angular Velocity:

+x

+y

-y

-x

s

r

r

=d d t

Angular Velocity:

= dd t

=d 2dt2

Angular Acceleration:

ave= t

= f−it f−t i

ave= t

= f−i

t f−t i= d

dt

= ddt

x

vave= x t

=x f−x it f−t i

aave= v t

=v f−v it f−t i

v= dxdt

a= dvdt

=constant=o t

Special Case: Angular Acceleration = 0 Special Case: Acceleration = 0

v=constantx=xov t

ave= t

= f−it f−t i

ave= t

= f−i

t f−t i

=d dt

=ddt

x

vave= x t

=x f−xit f−t i

aave= v t

=v f−v it f−t i

v=dxdt

a=dvdt

=constant=o t= t

=5 rpm

Disk rotates at a constant rate of 5 revolutions per minute. How much does it rotate in 2 second?

5 revmin 2rad1rev 1min60 s =

6rads

= t=6 rads 2 s =

3rad

=constant=o t

=oo t12 t 2

2=o22

Special Case: Angular Acceleration = constant Special Case: Acceleration = constant

a=constantv=voa t

x=xovo t12a t 2

v2=vo22a x

ave= t

= f−it f−t i

ave= t

= f−i

t f−t i

=d dt

=ddt

x

vave= x t

=x f−xit f−t i

aave= v t

=v f−v it f−t i

v=dxdt

a=dvdt

=20 rads

Disk is rotating at a rate of 20 radians per second and accelerating at a rate of -2 rad/s2 . How long before the disk comes to rest?

=o t

t=−o

=

0 rads

−20 rads

−2 rads2

=10 s

=constant=o t

=oo t12 t 2

2=o22

=−2 rads2

=20 rads

Disk is rotating at a rate of 20 radians per second and accelerating at a rate of -2 rad/s2 . How many revolutions does the disk make before coming to rest?

=oo t12 t2

=20 rads

10 s12 −2 rad

s2 10 s2=100 rad

=constant=o t

=oo t12 t 2

2=o22

=−2 rads2

100 rad 1rev2rad =15.9 rad

t=10s

=20 rads

Disk is rotating at a rate of 20 radians per second and accelerating at a rate of -2 rad/s2 . How many revolutions does the disk make before coming to rest?

2=o22

=2−o

2

2=0 rad

s2

−20 rads

2

2−2 rads2

=100 rad

=constant=o t

=oo t12 t 2

2=o22

=−2 rads2

100 rad 1 rev2 rad =15.9 rad

Another way....

Tangential Velocity:

r

Woman stands still while disk rotates in time ∆t.

r



s

r



s

s=r dsdt=vt=r

d dt

=r

r



s

s=r dsdt

=vt=rd dt

=r

d 2 sdt2 =a t=r

ddt

=r

=0.5 rad /s=constant

r1 =10 m

r2 =5 m

Time t= 0.0 s

=0.5 rad / s=constant

r1 =10 m

lr2 =5 m

Time t= 3.0 s=constant=o t

= t=0.5 rads

3 s =1.5 rad


r1 =10 m

l1r2 =5 m

Time t= 3.0 s=constant=o t

= t=0.5 rads

3 s =1.5 rad

l2

vT 1=r 1

vT 1=5m 0.5rad / s =2.5m / s

l1=2.5m /s 10 s=25m


r1 =10 m

s1r2 =5 m

Time t= 3.0 s=constant=ot

=t=0.5 rads

3 s =1.5rad

s2

vT 1=r1

vT 1=5m 0.5 rad / s =2.5m/ s

s1=2.5m/ s10 s=25m

vT 1=r1

vT 1=5m 0.5 rad / s =2.5m/ s

s1=2.5m/ s10 s=25m

Definitions:

Period ( T ) - Time for one complete rotation or oscillation. Measure in seconds.

Frequency ( f ) - Number of oscillations per second. Measured in Hertz ( 1 Hz = 1 s-1)

f=1T

T=1f

= t

=2T

=2 f

Torque – result of a force causing rotation.Bolt and wrench – Use torque to tighten or loosen the bolt.

The magnitude of the torque will depend on three things:

1. The angle at which the force is applied.2. The distance from the axis of rotation to the force. (the MOMENT ARM or LEVER ARM)3. The magnitude of the force applied.


F

Look at is force. No matter how large the force, the bolt will not turn.




F

Same here. No matter how large the force, the bolt will not turn.




F

In order to get the bolt to turn, the force should be oriented in the proper direction. Here the force turns the bolt in the positive direction.




F

In order to get the bolt to turn, the force should be oriented in the proper direction. Here the force turns the bolt in the negative direction.




Fr

=F X r




Fr

=F X r∣∣=∣F∣∣r∣




Fr

=F X r∣∣=∣F∣∣r∣




Fr

=F X r∣∣=∣F∣∣r∣


Fr

=F X r∣∣=∣F∣∣r∣

F


Fr

=F X r∣∣=∣F∣∣r∣

∣∣=∣F∣∣r∣sin

F

Torque – Another Look.Bolt and wrench – Use torque to tighten or loosen the bolt.

FC

r= F c X r∣∣=∣F c∣∣rc∣

∣∣=∣F c∣∣rc∣sin

Line of Action

rC

Torque – Another Look.Bolt and wrench – Use torque to tighten or loosen the bolt.

F= 10 N

=∣F∣∣r∣sin=10N 2msin 30o=10Nm

r = 2m

Torque:

r

F F=maF=mr

Torque:

r

F

F=maF=mr

=Fr=mr2

Torque:

r

F

F=maF=mr

=Fr=mr2but I=mr 2

=I

NET=12... N=∑i=1

N

NET=∑i=1

N

=I

F xnet=∑i=1

N

F x=max

F ynet=∑i=1

N

F y=ma y

For static equilibrium:

NET=∑i=1

N

=0

F xnet=∑i=1

N

F x=0

F ynet=∑i=1

N

F y=0

NET=12... N=∑i=1

N

m

M

TFH

L

L / 2

Wm

WM

m

M

TF

H

L

L / 2

Wm

WM

F H x

F H yT

y

Tx

Force x comp y comp τT

FH

Wm

WM

F=ma=Fr sin

m

M

TF

H

L

L / 2

Wm

WM

F H x

F H yT

y

Tx


FH

Wm

WM

−T x=−T cos T y=T sin T L sin

F=ma=Fr sin

m

M

TF

H

L

L / 2

Wm

WM

F H yT

y

Tx


FH

Wm

WM

−T x=−T cos

F H x

T y=T sin

F H y

T L sin

0

F H x

m

M

TF

H

L

L / 2

Wm

WM

F H x

F H yT

y

Tx


FH

Wm

WM

−T x=−T cos

F H x

0

T y=T sin

F H y

−mg

T L sin

0

−mg L/2

F=ma=Fr sin

m

M

TF

H

L

L / 2

Wm

WM

F H x

F H yT

y

Tx


FH

Wm

WM

−T x=−T cos

F H x

0

0

T y=T sin

F H y

−mg

−Mg

T L sin

0

−mg L/2

−MgL

F=ma=Fr sin

m

M

TF

H

L

L / 2

Wm

WM

F H x

F H yT

y

Tx

Force x y τT

FH

Wm

WM

−T x=−T cos

F H x

0

0

T y=T sin

F H x

−mg

−Mg

T L sin

0

−mg L/2

−MgL

∑ =0∑ F x=0∑ F y=0

m

M

TF

H

L

L / 2

Wm

WM

F H x

F H yT

y

Tx

Force x y τT

FH

Wm

WM

−T x=−T cos

F H x

0

0

T y=T sin

F H x

−mg

−Mg

T L sin

0

−mg L/2

−MgL

∑=0=TLsin−mg L /2−MgL∑ F x=0=−T cosF H x

∑ F y=0=T sinF H y−mg−Mg

Torque – result of a force causing rotation.

Door and door knob.

Fa

ra

Torque – result of a force causing rotation.

Door and door knob.

Fa

ra

Fb

rb

Force Fb at r

b must be much

larger than Force Fa at r

a .

Rotational Kinetic Energy

m2

m1

v1

v2

r1r2

KE=12m v2 KE total=KE1KE2

KE total=12m1 v1

212m2 v2

2

KE total=12 m1v1

2m2v22


m2

m1

v1

v2

r1r2

KE=12m v2

v=r

KE total=KE 1KE 2

KE total=12m1v1

212m2 v2

2

KE total=12 m1v1

2m2 v22

KE total=12 m1r11

2m2r222


m2

m1

v1

v2

r1r2

KE=12m v2

v=r1=2=

KE total=KE1KE2

KE total=12m1 v1

212m2 v2

2

KE total=12 m1v1

2m2 v22

KE total=12 m1r11

2m2r222

KE total=122 m1 r1

2m2 r22

KE total=12 ∑i=1

2

mi r i22=1

2I2

FT=20 N

5 kg Disk, with a radius of 0.25 m, free to rotate about center, starts from rest. What is the angular velocity after 10 seconds?

net=∑i=1

N

i=∑i=1

N

r iF i sini

=r FT sin =90o

I =r FT

=r FT

I

r

I disk=12M R2

FT=20 N


net=∑i=1

N

i=∑i=1

N

r i F i sini

=r FT sin =90o

I =r FT

=r FT

I

=r FT

12mr2

=2F T

mr

r

I disk=12M R2

FT=20 N


net=∑i=1

N

i=∑i=1

N

r i F i sini

=r F T sin =90o

I =r F T

=r F T

I

=r F T

12mr2

=2F T

mr= 220N

5kg 0.25=32 rad

s2

r

I disk=12M R2

FT=20 N


net=∑i=1

N

i=∑i=1

N

r i F i sini

=r FT sin =90o

I =r F T

=r F T

I

=r F T

12mr2

=2F T

mr=

220N 5kg 0.25

=32 rads2

r

I disk=12M R2

=o t=032 rads2 10 s=320 rad

s

w=mg

10 kg Disk, with a radius of 0.2 m, free to rotate about center, starts from rest ( no friction). A 2 kg mass is attached. What is the speed of the mass after is drops 1 meter?

r

I disk=12M R2

m

1 m

M

w=mg

∑i=1

N

=∑i−1

N

F i r ir

I disk=12M R2

m

1 m

∑i=1

N

F y i=ma y

M

F T

FT


w=mg

∑i=1

N

=∑i−1

N

F i r i

F T r=I r

I disk=12M R2

m

1 m

∑i=1

N

F yi=ma y

mg−F T=ma y

M

FT

FT

+y


w=mg

∑i=1

N

=∑i−1

N

F i r i

F T r=I

F T r=I a y

r r

I disk=12M R2

m

1 m

∑i=1

N

F yi=ma y

mg−F T=ma y

M

FT

FT

+y


w=mg

∑i=1

N

=∑i−1

N

F i r i

F T r=I

FT r=I a y

r FT r

2

I=a y

FT r2

12M r2

=a y

F T=M a y

2

r

I disk=12M R2

m

1 m

∑i=1

N

F y i=ma y

mg−FT=ma y

M

FT

FT

+y


w=mg

∑i=1

N

=∑i−1

N

F i r i

F T r= I

FT r=I a y

r F T r

2

I =a y

F T r2

12M r2

=a y

F T=M a y

2

r

I disk=12M R2

m

1 m

∑i=1

N

F y i=ma y

mg−F T=ma y

mg−M a y

2 =ma y

g− M2m a y=a y

M

FT

FT

+y


w=mg

∑i=1

N

=∑i−1

N

F i r i

F T r= I

FT r=I a y

r F T r

2

I =a y

F T r2

12M r2

=a y

F T=M a y

2

r

I disk=12M R2

m

1 m

∑i=1

N

F y i=ma y

mg−FT=ma y

mg−M a y2 =ma y

g− M2m a y=a yg= M2m a ya yg

1 M2m

=a y=2.8 ms2

M

FT

FT

+y


w=mg

∑i=1

N

=∑i−1

N

F i r i

FT r=I

FT r=I a y

r FT r

2

I =a y

FT r2

12 M r 2

=a y

F T=M a y

2

r

I disk=12M R2

m

1 m

∑i=1

N

F y i=ma y

mg−FT=ma y

mg−M a y

2 =ma y

g− M2m a y=a y

g= M2m a ya y

g

1 M2m

=a y=2.8 ms2

M

FT

FT

+y


v2=vo22a y y

v=2a y y=22.8 ms21m=2.4m / s

w=mg

r

I disk=12M R2

m

1 m

M

FT

FT

+y

10 kg Disk, with a radius of 0.2 m, free to rotate about center, starts from rest ( no friction). A 2 kg mass is attached. What is the speed of the mass after is drops 1 meter? This can also be done with Energy!

E i=E f

KE iPE iKE rot i=KE fPE fKE rot f

mgh=12mv21

2I 2

w=mg

r

I disk=12M R2

m

1 m

M

FT

FT

+y


E i=E f


mgh=12mv21

2I 2

mgh=12m v21

2 12 M r2vr 2

mgh=12mv2 1

4M v2

w=mg

r

I disk=12M R2

m

1 m

M

FT

FT

+y


E i=E f


mgh=12mv21

2I 2

mgh=12mv21

2 12 M r2vr 2

mgh=12mv21

4M v2

v= mh12m1

4M g=2.4 m

s

w=mg

∑i=1

N

=∑i−1

N

F i r i

FT r− friction= I r

I disk=12M R2

m

1 m

∑i=1

N

F yi=ma y

mg−FT=ma yM

FT

FT

+y

10 kg Disk, with a radius of 0.2 m, free to rotate about center, starts from rest ( with friction). A 2 kg mass is attached. What is the speed of the mass after is drops 1 meter? F f

w=mg

∑i=1

N

=∑i−1

N

F i r i

F T r− friction=I

friction=FT r−Ia y

r

r

I disk=12M R2

m

1 m

∑i=1

N

F yi=ma y

mg−F T=ma y

F T=mg−ma y

M

FT

FT

+y


w=mg

∑i=1

N

=∑i−1

N

F i r i

F T r− friction=I

friction=F T r−Ia y

r

friction=mg−ma yr−Ia y

r

friction=mgr2−mr2I a y

r

r

I disk=12M R2

m

1 m

∑i=1

N

F y i=ma y

mg−FT=ma y

FT=mg−ma y

M

FT

FT

+y


ave= t

= f−it f−t i

ave=t

= f−i

t f−t i

x

vave= x t

=x f−x it f−t i

aave=v t

=v f−v it f−t i

=constant=o t

=oo t12 t 2

2=o22

a=constantv=voa t

x=xovo t12 at

2

v2=vo22a x

=0=constant=oo t

a=0v=constantx=xovo t

I=∑i=1

N

mi ri2

=rF sin=I

KE rot=12I2

mF=ma

KE=12mv2

Documents

ROTATIONAL MOTIONjoetrout/phys182/Phys182_Applied...ROTATIONAL MOTION: ROTATIONAL MOTION: r s=l ≡ s r = l r r r +x +y-y-x 0o 90o 180o 270o 360o 0rad rad 2 rad 2 rad 3 2 rad 2 rad=360o