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Rotation of a Rigid Body Rotation of a Rigid Body Readings: Chapter 13 1 Readings: Chapter 13

Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

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Page 1: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Rotation of a Rigid BodyRotation of a Rigid Body

Readings: Chapter 131

Readings: Chapter 13

Page 2: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

How can we characterize the acceleration during rotation?

t l ti l l ti dF

Newton’s second law:

- translational acceleration and

- angular acceleration

F ma=

2

Page 3: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Angular acceleration2v

1v1r

ϕ

1r

Center of rotation

Both points have the same angular velocityϕω Δ

=Both points have the same angular velocity t

ωΔ

1 1v rω= 2 2v rω=Linear acceleration:

11 1

va r

t tωΔ Δ

= =Δ Δ

22 2

va r

t tωΔ Δ

= =Δ Δ

3

t tΔ Δ t tΔ Δ

Both points have the same angular acceleration tωα Δ

Page 4: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Rotation of Rigid Body:

Every point undergoes circular motion with the same angular velocity and the same angular acceleration

tωα Δ

v rω=tΔ

4

Page 5: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

The relation between angular velocity and angular acceleration is the same as the

l i b li l i d lirelation between linear velocity and linear acceleration

Δtωα Δ

vat

Δ=Δ

5

Page 6: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

The Center of Mass

For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the rotation about the special point– the center of mass of the body.

Definition: The coordinate of the center of mass:

1 1 2 2

1 2cm

m x m xx

m m+

=+

Definition: The coordinate of the center of mass:

Rigid body consisting of two particles: 1 2

1 2m m=If then

x x+1 2

2cmx x

x+

=

2 0 0.5 0.5 0.12.0 0.5cmx m⋅ + ⋅

= =+

6

Page 7: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

The Center of Mass

Definition: The coordinate of the center of mass:

x mΔ∑ ∫i ii

cm

x m xdmx

M M

Δ= =∑ ∫

i ii

cm

y m ydmy

M M

Δ= =∑ ∫

cmyM M

7

Page 8: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

The Center of Mass: Example

21 1

L L

L

Mx dx M xdxxdm L L L∫ ∫∫ 20 00

1 12 2 2

L

cm

xdm L L Lx xM M LM L L

= = = = = =∫ ∫∫

The center of mass of a disk is the center O of the disk

8O

Page 9: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Torque: Rotational Equivalent of Force

9

Page 10: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Torque

Th t ti f th b d i d t i d b th tThe rotation of the body is determined by the torque

sinFrτ φ=

10

Page 11: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Torque

φsinFrτ φ=

090φ =Torque is maximum if

Torque is 0 if 0 00 180orφ φ= =

0τ =

1 3τ τ>

1 4τ τ>

11

1 4τ τ>

Page 12: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Torque sinFrτ φ=

Torque is positive if the force is trying to rotate the body counterclockwise

Torque is negative if the force is trying to rotate the body clockwise

1 0τ >3 0τ < 1F

3F

i 2 0τ <4 0τ > 2F4Faxis 2 0τ <4

The net torque is the sum of the torques due to all applied forces:

24

1 2 3 4netτ τ τ τ τ= + + +

12

Page 13: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Torque: Example sinFrτ φ=Fi d th t tFind the net torque

1F3F 5F

3 030

FF

1m

2m

3m

4m 060

045

045

axis 2F4F

1 2 3 4 5 5F F F F F N= = = = =

1 5 3sin(30) 7.5N mτ = ⋅ = ⋅1 2 3 4 5

2 5 4sin(60) 17.3N mτ = − ⋅ = − ⋅3 5 1sin(45) 3.5N mτ = − ⋅ = − ⋅

4 5 2sin(45) 7.1N mτ = ⋅ = ⋅2 ( ) 4 ( )

5 5 0 0N mτ = ⋅ = ⋅

131 2 3 4 4 7.5 17.3 3.5 7.1 6.2net N mτ τ τ τ τ τ= + + + + = − − + = − ⋅

Page 14: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Torque: Relation between the torque and angular acceleration:

v Δ Δr

va r rt t

ω αΔ Δ= = =Δ Δ

1F

ϕ

F ma mrα= =

2Fr mrτ α= =

2m r Iτ α α= =∑net i ii

m r Iτ α α= =∑2I m r= ∑ moment of inertiai i

iI m r= ∑ - moment of inertia

14

Page 15: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Moment of Inertia

21I MLThin rod about center 2

12I ML=Thin rod, about center

Thin rod, about end 213

I ML=

1Cylinder (or disk), about center 21

2I MR=

Cylindrical loop about center2I MR=

15

Cylindrical loop, about center I MR=

Page 16: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Moment of Inertia: Parallel-axis Theorem

If you know the moment of Inertia about the center of mass (point O)

then the Moment of Inertia about point (axis) P will be

2d 2P OI I Md= +

OP Od

2 2( ) ( )I Δ Δ∑ ∑2 2

2 2 2

( ) ( )

( ) ( ) 2 ( )

P i P i i O O P ii i

i O i i O i O i

I x x m x x x x m

x x d m r r d r r d m

= − Δ = − + − Δ =

⎡ ⎤= − + Δ = − + − + Δ =⎣ ⎦

∑ ∑

∑ ∑2 2 2

( ) ( ) 2 ( )

( ) 2 ( )

i O i i O i O ii i

i O i i O i Oi i

x x d m r r d r r d m

r r m d r r m Md I Md

⎡ ⎤+ Δ + + Δ⎣ ⎦

+ − Δ + − Δ + = +

∑ ∑

∑ ∑16

i i

Page 17: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Parallel-axis Theorem: Example 2P OI I Md= +

1Thin rod, about center of mass

2112

I ML=

22 2 2 21 1 1 1

12 2 12 4 3LI ML M ML ML ML⎛ ⎞= + = + =⎜ ⎟

⎝ ⎠12 2 12 4 3⎝ ⎠

17

Page 18: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

net Iτ α=Equilibrium: 0α =

0netτ =

1 2m kg=2 10m kg=

2d m= ?x =Massless rod1n

2n

Two forces (which can results in rotation) acting on the rod

1 1 1n w m g= = 1 1 1n d m gdτ = =1 1 1g

2 2 2n w m g= =1 1 1g

2 2 2n x m gxτ = − = −

Equilibrium: 0τ τ τ= + =18

Equilibrium:1 2 0netτ τ τ= + =

1 2 0m gd m gx− =1

2

0.4m

x d mm

= =

Page 19: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Rotational Energy:

2 2 21 1 2 2 3 3

1 1 1 ...2 2 2rotK m v m v m v= + + + =

2 2 2 2 2 21 1 2 2 3 3

1 1 1 ...2 2 2

m r m r m rω ω ω= + + + =

2 2 2 2 21 1 2 2 3 3

1 1( ...)2 2

m r m r m r Iω ω= + + + =

Conservation of energy (no friction):

2 21 1Mgy Mv I constω+ + =

19

2 2Mgy Mv I constω+ +

Page 20: Rotation of a Rigid BodyRotation of a Rigid Body · For Rigid Body sometimes it is convenient to describe the rotation aboutFor Rigid Body sometimes it is convenient to describe the

Kinetic energy of rolling motion 212rotK Iω=

212rot PK I ω= 2

P cmI I MR= +

1 1 1 1cmv Rω=

2 2 2 2 22

1 1 1 12 2 2 2rot cm cm cm cmK I MR I v Mv

Rω ω= + = +

Cylinder: 212cmI MR= 2 2 21 1 1 3

2 2 2 4rot cm cm cmK Mv Mv Mv= + =

Cylindrical loop: 2cmI MR=

2 2 21 12 2rot cm cm cmK Mv Mv Mv= + =

20Solid sphere: 22

5cmI MR=2 2 21 2 1 7

2 5 2 10rot cm cm cmK Mv Mv Mv= + =