Rotation

1 Rotational Variables

Rotation correspond to the motion of an object about a

given axis.

The proper study of this type of motion requires several

elements:

•Deﬁne a good reference system. In the ﬁgure, zaxis

coincides with the rotation axis.

•Deﬁne angular variables that simplify the math for

describing the position of the object respect the ref-

erence system.

Def. 1 (Rigid Body).A rigid body is a particle system where

the distances between all of them remain ﬁxed at all times. In

other words, the body does not suffer deformations.

A rigid body is sometimes called also solid.

Why we concentrate on the study of rigid bodies? Because

we can study the motion of the complete body paying at-

tention to a single point of the object, and then refer the

rest of the body to that point.

A very useful reference system selection is one where one

of the axis coincides with the rotation axis.

On that case, we can describe the position of each part of

the object using the angle between a given part of the ob-

ject and an arbitrary axis x.

Def. 2 (Angular Position).The angular position is the angle

of the reference line relative to a ﬁxed direction (xaxis), which

we take as the zero angular position.

The angular position is denoted by the letter θand its measured

in the counterclockwise direction.

θ=s

r[rad](1)

where sis the arc length deﬁned by the reference point and ris

the distance from this point to the rotation axis, which coincides

with the origin of the reference system.

Angles are measured in radians ([rad]) instead of degrees

or revolutions. A complete revolution describes an arc of

s= 2πr so the angle is

θ=2πr

r= 2π.

Def. 3 (Angular Displacement).The angular displacement

correspond to the angle difference measured at two different

times. It is represented by ∆θand its value is

∆θ=θ2−θ1.

Obs. 1. Angle displacements can be positive or negative de-

pending on the direction of the rotation.

•Positive: the body rotates counterclockwise.

•Negative: the body rotates clockwise.

Def. 4 (Angular Velocity).Let θ1be the angular position at

t=t1and θ2be the angular position at t=t2.

The average angular velocity on the interval [t1, t2]is

ωavg =θ2−θ1

t2−t1

=∆θ

∆t.

1

Def. 5 (Instantaneous Angular Velocity ω).The instanta-

neous angular velocity, denoted by ω, is the limit of the the av-

erage angular velocity when ∆t→0:

ω= lim

∆t→0

∆θ

∆t≡dθ

dt rad

s.(2)

The magnitude of the |ω|is the angular speed.

The angular acceleration corresponds to the variation of

angular velocity for a rotating body.

Def. 6 (Average Angular Acceleration).Let ω1and ω2be the

angular velocities at t1and t2, respectively. Then, the average

angular velocity αavg is:

αavg =ω2−ω1

t2−t1

=∆ω

∆t.

Def. 7 (Instantaneous Angular Acceleration).The instan-

taneous angular acceleration, denoted by α, is the limit average

angular acceleration when ∆t→0, i.e.,

α= lim

∆t→0

∆ω

∆t=dω

dt =d2θ

dt2rad

s2.(3)

Obs. 2 (Dot Notation for Time Derivatives).In mechanics

it is very common to use an special notation for time derivatives.

This is the “dot notation” for time differentiation. In other

words, we write

˙

θ≡dθ

dt

¨

θ≡d2θ

dt2

2 Rotation with Constant Angular Ac-

celeration

This is the ﬁrst type of rotational motion we are going to

study.

In this case, αis constant, so we can integrate (3) to obtain:

ω(t) = ω0+αt (4a)

θ(t) = θ0+ω0t+1

2αt2(4b)

ω2(t) = ω2+ 2αθ(t)−θ0(4c)

where:

•θ0is the angular displacement at t= 0.

•ω0is the angular velocity at t= 0.

These terms are known as initial conditions.

3 Relating Linear and Angular Vari-

ables

Let’s remember that the description of the motion of a par-

ticle respect a ﬁxed reference system is done using three

variables:

•~r or position

•~v or velocity

•~a or acceleration

But in this chapter we have introduced three additional

variables to describe rotation:

•θor angular position

•ωor angular velocity

•αor angular acceleration

Let’s consider the case of a particle Pmoving along a circle

arc of radius r.

2

We want to express ~r,~v, and ~a in terms of r,θ,ωand α.

One of the issues with making these computations is the

fact that although the distance from Pto the origin is ﬁxed,

the direction of the vector ~r changes at each instant.

Therefore, this direction change implies that the velocity

of Pis not zero.

How do we know this?

3.1 Detour: Unitary Vectors

The deﬁnition of a coordinate system x, y, z such as the

one depicted in the ﬁgure induces the existence of a base

of three unitary vectors ˆ

i,ˆ

j, and ˆ

kthat are parallel to the

axis x,y, and zrespectively.

These vectors are ﬁxed in time and they have the following

properties:

ˆ

i·ˆ

i=ˆ

j·ˆ

j=ˆ

k·ˆ

k= 1

ˆ

i·ˆ

j=ˆ

i·ˆ

k=ˆ

j·ˆ

k= 0.

In other words, the three vectors are perpendicular - hence

the dot product is zero between them - and there magni-

tude is 1.

3.2 Position of P

We can express the position of Pin terms of the unitary

vectors ˆ

iand ˆ

jpreviously deﬁned. Thus,

~r =rcos(θ)ˆ

i+rsin(θ)ˆ

j.

Note that we can deﬁne a unitary vector ˆralong ~r such

that |ˆr|= 1 if and only if

ˆr= cos(θ)ˆ

i+ sin(θ)ˆ

j.

Hence, the position of Pcan be expressed compactly by

~r =rˆr. (5)

3.3 Velocity of P

The velocity vector is

~v =d~r

dt

=drˆr

dt

=dr

dt ˆr+rdˆr

dt

Since ˆr= cos(θ)ˆ

i+ sin(θ)ˆ

jdepends on time indirectly

through θ, we cannot dismiss this term.

In fact:

dˆr

dt =d

dtcos(θ)ˆ

i+ sin(θ)ˆ

j

=−sin(θ)dθ

dtˆ

i+ cos(θ)dθ

dt ˆ

j

=dθ

dt −sin(θ)ˆ

i+ cos(θ)ˆ

j.

We can deﬁne a second unitary vector ˆ

θby

ˆ

θ≡ − sin(θ)ˆ

i+ cos(θ)ˆ

j(6)

This vector is perpendicular to ˆrsince ˆr·ˆ

θ= 0. In addition,

|ˆ

θ|= 1 as well.

Therefore, the velocity of Pis

~v =dr

dt ˆr+rdθ

dt ˆ

θ. (7)

Since Premains at constant distance from the origin, then

dr/dt = 0 and

~v =rωˆ

θ. (8)

Note that the speed of Pis quite simple to compute since

it’s the term that’s next to ˆ

θ:

v=|~v|=rω.

3.4 Acceleration of P

The acceleration of Pcan be computed taking a time

derivative of ~v.

~a =d~v

dt

=d

dtrωˆ

θ

=rdω

dt ˆ

θ+rω dˆ

θ

dt

The second term can be computed by determining dˆ

θ/dt.

dˆ

θ

dt =d

dt−sin(θ)ˆ

i+ cos(θ)ˆ

j

=−cos(θ)dθ

dtˆ

i−sin(θ)dθ

dt ˆ

j

=−dθ

dt ˆr

=−ωˆr

Therefore,

~a =−rω2ˆr+rαˆ

θ(9)

The acceleration of Pit’s due to two effects:

•The change on direction of P; this is the centripetal

acceleration of magnitude rω2.

•The angular acceleration αwhich induces a tangential

acceleration rα.

3

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