Topic 2.1 ExtendedL – Angular acceleration

When we discussed circular motion we looked at two variables:We will also define a new rotational variable called angular acceleration α.

x

yConsider the irregularly-shaped peanut-like solid shown here. Two points are shown on an an arbitrary reference line. We can superimpose the Cartesian coordinate axes for reference: θ

r

Note that every point on the reference line has the same angular position θ. And that every point on the reference line has a different radius r.

r

Angular position θ, and angular speed ω.

Topic 2.1 ExtendedL – Angular acceleration

Note that each point on our reference line traces out an arc of length s:Recall the relationship between r, θ, and s:

x

y

sr

r sθ

s = rθθ in radians

Arc length

Note that each point covers a different distance s in the same amount of time.Thus each point moves at a different velocity v.

Topic 2.1 ExtendedL – Angular acceleration

x

y

θ1

Definition of average angular speed

Consider the position of the red dot at two times:

<ω> =ΔθΔt

θ2

t1

t2

Δθ

We can define a change in angular position Δθ:

We define the average angular speed <ω>:

This is the same definition we had when we talked about circular motion.Note that the units of <ω> are radians per second. We define the instantaneous angular speed ω:

ω =ΔθΔt

limΔt→0

=dθdt

Definition of angular speed

Topic 2.1 ExtendedL – Angular acceleration

The relation between linear velocity v and angular velocity ω comes from s = rθ:

Just as we define average acceleration a for the linear variables,

v =ΔsΔt

Relation between linear and angular speed

= r ΔθΔt

= rω

v = rω

a =ΔvΔt

we define average angular acceleration α for the angular variables.

α =ΔωΔt

Definition of average angular acceleration

FYI: Angular acceleration is measured in radians per second squared.

FYI: In the limit as t0, average values become instantaneous values.

Topic 2.1 ExtendedL – Angular acceleration

The instantaneous angular acceleration is given by

α =ΔωΔt

limΔt→0

=dωdt

Definition of angular acceleration

The relation between linear acceleration a and angular acceleration α comes from v = rω:

a =ΔvΔt

Relation between linear and angular acceleration

= r ΔωΔt

= rα

at = rα

FYI: We call this linear acceleration at the tangential acceleration because it occurs along the arc length, and is thus tangent to the circle the point is following.

Don’t forget the centripetal acceleration for any object moving in a circle:

ac = rω2 = v2

rCentripetalacceleration

FYI: at and ac are mutually perpenducular.

Question: at = 0 in UCM. Why?

FYI: In the limit as t0, average values become instantaneous values.

Topic 2.1 ExtendedL – Angular acceleration

Recall that velocity is a speed in a particular direction.Thus, angular velocity is angular speed in a particular direction.We define angular direction with another right-hand-rule:Consider the record spinning on the axis (a):Curl the fingers of the right hand as (c):Your thumb points in the direction of the angular velocity (b):

FYI: Angular direction always points along the rotational axis of the turning object.

FYI: We can usually get by with clockwise (cw) and counterclockwise (ccw) when working with angular direction.

Topic 2.1 ExtendedL – Angular acceleration

Suppose the angular acceleration α is constant. Thusis also constant so thatat = rα at = constant

and all of the following equations are true:

s = so + vot + att212

v = vo + att

v2 = vo2 + 2ats

s = rθ

v = rω

at = rα

The linear equations Angular to linear relations

Substitution of s, v, and at yields the following:

rθ = rθo + rωot + rαt212

rω = rωo + rαt

r2ω2 = r2ωo2 + 2rα(rθ)

θ = θo + ωot + αt212

ω = ωo + αt

ω2 = ωo2 + 2αθ

Constant angular acceleration

s or x v a

Topic 2.1 ExtendedL – Angular acceleration

Suppose a grinding wheel of 10-cm radius is turned on, and reaches a speed of 6000 rpm in 20 seconds.(a) What is the final angular speed of the wheel?

ω = 6000 revmin

= 628.3 rad/s·1 min60 s

·2 rad1 rev

(b) What is constant angular acceleration of the wheel during the first 20 seconds?

ω = ωo + αt628.3 = 0 + α(20)

α = 31.42 rad/s2

(c) Through what angle does the wheel rotate during the first 20 seconds?

ω2 = ωo2 + 2αθ

628.3 2 = 02 + 2(31.42)θ

θ = 6283.2 radians