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Lecture 17

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- Chapter 12
- Understand the equilibrium dynamics of an extended object in response to forces
- Analyze rolling motion
- Employ “conservation of angular momentum” concept

Goals:

Assignment:

- HW7 due March 25th
- After Spring Break Tuesday:
Catch up

- A “special” net force is necessary and that action depends on the location from the axis of rotation
- For translational motion and acceleration the position of the force doesn’t matter (doesn’t change the physics we see)
- IMPORTANT: For rotational motion and angular acceleration: we always reference the specific position of the force relative to the axis of rotation.
- Vectors are simply a tool for visualizing Newton’s Laws

A force applied at a distance from the rotation axis gives a torque

a

FTangential

NET = |r| |FTang| ≡|r||F| sin f

F

Fradial

r

- Only the tangential component of the force matters
- With torque the position of the force matters

a

FTangential

F

Fradial

r

A force applied at a distance from the rotation axis

NET = |r| |FTang| ≡|r||F| sin f

- Torque is the rotational equivalent of force
Torque has units of kg m2/s2 = (kg m/s2) m = N m

NET = r FTang = r m aTang

= r m r a

= (m r2) a

For every little part of the wheel

a

FTangential

F

Frandial

r

The further a mass is away from this axis the greater the inertia (resistance) to rotation

- This is the rotational version of FNET = ma
- Moment of inertia, I≡ m r2 , (here I is just a point on the wheel) is the rotational equivalent of mass.
- If I is big, more torque is required to achieve a given angular acceleration.

A force applied at a distance from the rotation axis gives a torque

a

FTangential

NET = |r| |FTang| ≡|r||F| sin f

F

Fradial

r

- A constant torque gives constant angularacceleration iff the mass distribution and the axis of rotation remain constant.

- Recall that linear motion involves x, y and/or z vectors
- Angular motion can be quantified by defining a vector along
the axis of rotation.

- The axis of rotation is the one set of points that is fixed with respect to a rotation.
- Use the right hand rule

F cos(90°-q) = FTang.

line of action

r

a

90°-q

q

F

F

r sin q

F

Fradial

r

r

r

- Magnitude is given by |r| |F| sin q
or, equivalently, by the |Ftangential | |r|

or by |F| |rperpendicular to line of action |

- Direction is parallel to the axis of rotation with respect to the “right hand rule”
- And for a rigid object= I a

- Recall the Work Kinetic-Energy Theorem: K = WNET
- This applies to both rotational as well as linear motion.
- So for an object that rotates about a fixed axis
- For an object which is rotating and translating

- Case 1
- Case 2
- Same

- In which of the cases shown below is the torque provided by the applied force about the rotation axis biggest? In both cases the magnitude and direction of the applied force is the same.
- Remember torque requires F, rand sin q
or the tangential force component times perpendicular distance

L

F

F

L

axis

case 1

case 2

L

F

F

L

axis

case 1

case 2

- In which of the cases shown below is the torque provided by the applied force about the rotation axis biggest? In both cases the magnitude and direction of the applied force is the same.
- Remember torque requires F,rand sin f
or the tangential force component times perpendicular distance

(A)case 1

(B)case 2

(C) same

L

m

- A uniform rod of length L=0.5 m and mass m=1 kg is free to rotate on a frictionless pin passing through one end as in the Figure. The rod is released from rest in the horizontal position. What is
(A) its angular speed when it reaches the lowest point ?

(B) its initial angular acceleration ?

(C) initial linear acceleration of its free end ?

Ball has radius R

M

M

M

M

M

h

M

v ?

q

M

- A cylinder is about to roll down an inclined plane. What is its speed at the bottom of the plane ?

Ball has radius R

M

M

M

M

M

h

M

v ?

q

M

- A cylinder is about to roll down an inclined plane. What is its speed at the bottom of the plane ?
- Use Work-Energy theorem

Mgh = ½ Mv2 + ½ ICMw2

Mgh = ½ Mv2 + ½ (½ M R2)(v/R)2 = ¾ Mv2

v = 2(gh/3)½

- Again consider a cylinder rolling at a constant speed.

Both with

|VTang| = |VCM |

Rotation only

VTang = wR

Sliding only

2VCM

VCM

CM

CM

CM

VCM

- We have shown that for a system of particles,
momentum

is conserved if

- What is the rotational equivalent of this (rotational “mass” times rotational velocity)?
angular momentum

is conserved if

v1

m2

j

m1

r2

r1

i

v2

r3

v3

m3

- Consider a rigid distribution of point particles rotating in the x-y plane around the z axis, as shown below. The total angular momentum around the origin Is the sum of the angular momentum of each particle:
- Even if no connecting rod we can deduce an Lz

( ri and vi, are perpendicular)

Using vi = ri, we get

z

z

F

- A disk of mass M and radius R rotates around the z axis with angular velocity 0. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity F.

0

z

z

0

F

- A disk of mass M and radius R rotates around the z axis with initial angular velocity 0. A second identical disk, at rest, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity F.

No External Torque so Lz is constant

Li = Lf I wii = I wf½ mR2w0 = ½ 2mR2wf

- A student sits on a stool, initially at rest, but which is free to rotate. The moment of inertia of the student plus the stool is I. They throw a heavy ball of mass M with speed v such that its velocity vector moves a distance d from the axis of rotation.
- What is the angular speed F of the student-stool system after they throw the ball ?

M

v

F

d

I

I

Top view: before after

- The concept of angular momentum is also valid on a submicroscopic scale
- Angular momentum has been used in the development of modern theories of atomic, molecular and nuclear physics
- In these systems, the angular momentum has been found to be a fundamental quantity
- Fundamental here means that it is an intrinsic property of these objects

Assignment:

- HW7 due March 25th
- Thursday: Review session