Chapter 10 - Rotation - Pennsylvania State University 10 - Rotation Rotational Variables Torque...

Chapter 10 - Rotation

Rotational Variables

Torque

Moment of Inertia

Rotational Energy

“To know that we know whatwe know, and to know that wedo not know what we do notknow, that is true knowledge.”

-Nicolas Copernicus

David J. StarlingPenn State Hazleton

PHYS 211

Torque

Moment of Inertia

Rotational Energy

We want to describe the rotation of a rigid bodyabout a fixed axis.

The object does not deform, and the axis stays put.

Torque

Moment of Inertia

Rotational Energy

We want to describe the rotation of a rigid bodyabout a fixed axis.

The object does not deform, and the axis stays put.

Torque

Moment of Inertia

Rotational Energy

The kinematic and dynamic variables we have

used so far have their rotational counterparts:

How far has the rigid body rotated?

θ =sr

and θ ↔ x

[note: 1 revolution = 360◦ = 2π radians]

Torque

Moment of Inertia

Rotational Energy

θ =sr

and θ ↔ x

Torque

Moment of Inertia

Rotational Energy

θ =sr

and θ ↔ x

Torque

Moment of Inertia

Rotational Energy

If θ is like position, then ∆θ is like displacement.

This is the reference line of a rigid body.

∆θ = θ2 − θ1

[note: counterclockwise is positive]

Torque

Moment of Inertia

Rotational Energy

∆θ = θ2 − θ1

Torque

Moment of Inertia

Rotational Energy

∆θ = θ2 − θ1

Torque

Moment of Inertia

Rotational Energy

Angular velocity ω is like linear velocity, except

x→ θ.

ωavg =∆θ

∆tand ω =

Torque

Moment of Inertia

Rotational Energy

Angular velocity ω is like linear velocity, except

x→ θ.

ωavg =∆θ

∆tand ω =

Torque

Moment of Inertia

Rotational Energy

Angular acceleration α is like linear acceleration,

except v→ ω.

αavg =∆ω

∆tand α =

Torque

Moment of Inertia

Rotational Energy

Angular acceleration α is like linear acceleration,

except v→ ω.

αavg =∆ω

∆tand α =

Torque

Moment of Inertia

Rotational Energy

The linear variables have their rotational

counterparts:

linear angular unitsx θ radv ω dθ/dt rad/sa α dω/dt rad/s2

We use the right-hand rule to assign direction:

Torque

Moment of Inertia

Rotational Energy

The linear variables have their rotational

counterparts:

linear angular unitsx θ radv ω dθ/dt rad/sa α dω/dt rad/s2

We use the right-hand rule to assign direction:

Torque

Moment of Inertia

Rotational Energy

On a rigid body,

I all points trace out a circle;I all points have the same angular displacement

∆θ = ∆s/r

I all points have the same angular velocity

ω =dθdt

=d(s/r)

Torque

Moment of Inertia

Rotational Energy

Taking another derivative,

α =dωdt

=d(vt/r)

s = rθ

vt = rω

at = rα

ar = v2/r = ω2r

Torque

Moment of Inertia

Rotational Energy

Taking another derivative,

α =dωdt

=d(vt/r)

s = rθ

vt = rω

at = rα

ar = v2/r = ω2r

Torque

Moment of Inertia

Rotational Energy

If the angular acceleration of a rotating object is

constant, we can derive the same constantacceleration equations as before.

linear angularv(t) = v0 + at ω(t) = ω0 + αt

x(t) = x0 + v0t + 0.5at2 θ(t) = θ0 + ω0t + 0.5αt2

v2 = v20 + 2a(x− x0) ω2 = ω2

0 + 2α(θ − θ0)

Torque

Moment of Inertia

Rotational Energy

Lecture Question 10.1The Earth, which has an equatorial radius of 6380 km,

makes one revolution on its axis every 23.93 hours. What is

the tangential speed of Nairobi, Kenya, a city near the

equator?

(a) 37.0 m/s

(b) 74.0 m/s

(c) 148 m/s

(d) 232 m/s

(e) 465 m/s

Torque

Moment of Inertia

Rotational Energy

Torque

Rotational motion is generated by torque:

τ = rF sinφ.

I τ = (r)(F sinφ) = rFt

I τ = (r sinφ)(F) = r⊥F

I r⊥ = r sin(φ) is the moment arm

Torque

Moment of Inertia

Rotational Energy

Torque

τ = rF sinφ.

Torque

Moment of Inertia

Rotational Energy

Torque

τ = rF sinφ.

Torque

Moment of Inertia

Rotational Energy

Torque

τ = rF sinφ.

Torque

Moment of Inertia

Rotational Energy

Torque

τ = rF sinφ.

Torque

Moment of Inertia

Rotational Energy

Torque

When a torque is applied to an object, it

accelerates angularly.

Ft = mat

τ = Ftr = matr

τ = m(rα)r

τ = (mr2)α

τ = Iα

Torque

Moment of Inertia

Rotational Energy

Torque

Ft = mat

τ = Ftr = matr

τ = m(rα)r

τ = (mr2)α

τ = Iα

Torque

Moment of Inertia

Rotational Energy

Torque

Ft = mat

τ = Ftr = matr

τ = m(rα)r

τ = (mr2)α

τ = Iα

Torque

Moment of Inertia

Rotational Energy

Torque

Ft = mat

τ = Ftr = matr

τ = m(rα)r

τ = (mr2)α

τ = Iα

Torque

Moment of Inertia

Rotational Energy

Torque

Ft = mat

τ = Ftr = matr

τ = m(rα)r

τ = (mr2)α

τ = Iα

Torque

Moment of Inertia

Rotational Energy

Moment of Inertia

When torque is applied to an object, it resists this

motion. For a point particle, we found

τ = (mr2)α = Iα.

For many point particles in a rigid body, we sum them up:

I =∑

mir2i →

∫r2dm.

Torque

Moment of Inertia

Rotational Energy

Moment of Inertia

When torque is applied to an object, it resists this

motion. For a point particle, we found

τ = (mr2)α = Iα.

For many point particles in a rigid body, we sum them up:

I =∑

mir2i →

∫r2dm.

Torque

Moment of Inertia

Rotational Energy

Moment of Inertia

Moment of Inertia for common shapes:

Torque

Moment of Inertia

Rotational Energy

Moment of Inertia

If Icom is known for an axis through the center of

mass, then any parallel axis has

I = Icom + Md2

a distance d away.

Torque

Moment of Inertia

Rotational Energy

To find the kinetic energy of a rotating object, split

it up into small masses:

m1v21 +

m2v22 + . . .

mi(riω)2

mir2i ω

Torque

Moment of Inertia

Rotational Energy

m1v21 +

m2v22 + . . .

mi(riω)2

mir2i ω

Torque

Moment of Inertia

Rotational Energy

m1v21 +

m2v22 + . . .

mi(riω)2

mir2i ω

Torque

Moment of Inertia

Rotational Energy

m1v21 +

m2v22 + . . .

mi(riω)2

mir2i ω

Torque

Moment of Inertia

Rotational Energy

m1v21 +

m2v22 + . . .

mi(riω)2

mir2i ω

Torque

Moment of Inertia

Rotational Energy

m1v21 +

m2v22 + . . .

mi(riω)2

mir2i ω

Torque

Moment of Inertia

Rotational Energy

We can change the kinetic energy by doing work

(W = ∆K, right?):

∫ xf

F dx→ W =

∫ θf

τ dθ

[Constant torque: W = τ(θf − θi).]

Power is just the derivative of work, so

P =dWdt

= τω.

[compare: P = Fv]

Torque

Moment of Inertia

Rotational Energy

(W = ∆K, right?):

∫ xf

F dx→ W =

∫ θf

τ dθ

P =dWdt

= τω.

[compare: P = Fv]

Torque

Moment of Inertia

Rotational Energy

(W = ∆K, right?):

∫ xf

F dx→ W =

∫ θf

τ dθ

P =dWdt

= τω.

[compare: P = Fv]

Torque

Moment of Inertia

Rotational Energy

(W = ∆K, right?):

∫ xf

F dx→ W =

∫ θf

τ dθ

P =dWdt

= τω.

[compare: P = Fv]

Torque

Moment of Inertia

Rotational Energy

Torque

Moment of Inertia

Rotational Energy

Lecture 10.4Two solid cylinders are rotating about an axis that passes

through the center of both ends of each cylinder. Cylinder A

has three times the mass and twice the radius of cylinder B,

but they have the same rotational kinetic energy. What is the

ratio of the angular velocities, ωA/ωB, for these two

cylinders?

(a) 0.29

(b) 0.50

(c) 1.0

(d) 2.0

(e) 4.0

Chapter 10 - Rotation - Pennsylvania State University 10 - Rotation Rotational Variables Torque...

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