Rotational Dynamics

Chapter 9

Expectations

After Chapter 9, students will: calculate torques produced by forces recognize the condition of complete equilibrium calculate the location of the center of gravity of a

collection of objects use the rotational form of Newton’s second law of

motion to analyze physical situations calculate moments of inertia

Expectations

After Chapter 9, students will: calculate the rotational work done by a torque calculate rotational kinetic energy calculate angular momentum apply the principle of the conservation of angular

momentum in an isolated system

Preliminary Definitions

Torque

Complete Equilibrium

Center of Gravity

Torque

Torque: the rotational analog to force

Force produces changes in linear motion (linear acceleration). A force is a push or a pull.

Torque produces changes in angular motion (angular acceleration). A torque is a twist.

Torque

Mathematical definition:

The lever arm is the line

through the axis of

rotation, perpendicular to

the line of action of the

force.

Fltorque

force

length of lever arm

SI units: N·m

Torque

Torque is a vector quantity. It magnitude is given by

and its direction by the right-hand rule:

Fl

F

l l

Frotate l into F

torque vectorpoints out of page

Torque

For a given force, the torque depends on the location of the force’s application to a rigid object, relative to the location of the axis of rotation.

Fl

more torque less torque

Torque

For a given force, the torque depends on the force’s direction.

Fl

Complete Equilibrium

A rigid object is in complete equilibrium if the sum of the forces exerted on it is zero, and the sum of the torques exerted on it is zero.

An object in complete equilibrium has zero translational (linear) acceleration, and zero angular acceleration.

0 0 0 yx FF

Center of Gravity

In analyzing the equilibrium of an object, we see that where a force is applied to an object influences the torque produced by the force.

In particular, we sometimes need to know the location at which an object’s weight force acts on it.

Think of the object as a collection of smaller pieces.

Center of Gravity

In Chapter 7, we calculated the location of the center of mass of this system of pieces:

Multiply numerator and denominator by g:

i

iiC mmm

xmxmxmx

...

...

21

2211

gmgmgm

gxmgxmgxmx

i

iiC

...

...

21

2211

Center of Gravity

But: Substituting:

It is intuitive that the weight force acts at the effective location of the mass of an object.

gmgmgm

gxmgxmgxmx

i

iiC

...

...

21

2211

mgW

i

iiC WWW

xWxWxWx

...

...

21

2211

Newton’s Second Law: Rotational

Consider an object, mass m, in circular motion with a radius r. We apply a tangential force F:

The result is a

tangential acceleration

according to Newton’s second law.

r

F

TmaF

Newton’s Second Law: Rotational

The torque produced by the force is

But the tangential acceleration

is related to the angular

acceleration:

Substituting:r

F

2mrrrm

rmaFr T

raT

Newton’s Second Law: Rotational

This is an interesting result.

If we define the quantity

as the moment of inertia,

we have

the rotational form of Newton’s second law.

r

F

2mr

2mrI

I

Moment of Inertia

The equation

gives the moment of inertia of a “particle” (meaning an object whose dimensions are negligible compared with the distance r from the axis of rotation).

Scalar quantity; SI units of kg·m2

2mrI

Moment of Inertia

Not many real objects can reasonably be approximated as “particles.” But they can be treated as systems of particles …

iii

ii

rmI

rmrmrmI

2

2222

211 ...

Moment of Inertia

The moment of inertia of an object depends on: the object’s total mass the object’s shape the location of the axis of rotation

Rotational Work and Energy

By analogy with the corresponding translational quantities:

Translational Rotational

2

2

1

cos

mvKE

FsW

2

2

1

IKE

W

R

R

SI units: N·m = J

SI units: (kg·m2) / s2 = N·m = J

Total Mechanical Energy

We now add a term to our idea of the total mechanical energy of an object:

mghImvE 22

2

1

2

1

total energy

translational

kinetic energy

rotational

kinetic energy

gravitational

potential energy

Angular Momentum

By analogy with linear momentum:

Angular momentum is a vector quantity. Its magnitude is given by

and its direction is the same as the direction of .

must be expressed in rad/s.

ILmvp

IL SI units: kg·m2 / s

Angular Momentum: Conservation

If a system is isolated (no external torque acts on it), its angular momentum remains constant.

[If a system is isolated (no external force acts on it), its linear momentum remains constant.]