Chapter 10
Rotation
In this chapter we will study the rotational motion of rigid bodies about
a fixed axis. To describe this type of motion we will introduce the
following new concepts:
-Angular displacement -
Average and instantaneous angular velocity (symbol: ω )
-Average and instantaneous angular acceleration (symbol: α )
-Rotational inertia also known as moment of inertia (symbol
I) -Torque (symbol τ )
We will also calculate the kinetic energy associated with rotation, write
Newton’s second law for rotational motion, and introduce the work-
kinetic energy for rotational motion
(10-1)
The Rotational Variables (10-2)
In this chapter we will study the rotational motion of rigid
bodies about fixed axes. A rigid body is defined as one that
can rotate with all its parts locked together and without any
change of its shape. A fixed axis means that the object
rotates about an axis that does not move. We can describe
the motion of a rigid body rotating about a fixed axis by
specifying just one parameter. Consider the rigid body of
the figure.
We take the the z-axis to be the fixed axis of rotation. We define a reference line
which is fixed in the rigid body and is perpendicular to the rotational axis. A top
view is shown in the lower picture. The angular position of the reference line at
any time t is defined by the angle θ(t) that the reference lines makes with the
position at t = 0. The angle θ(t) also defines the position of all the points on the
rigid body because all the points are locked as they rotate. The angle θ is related
to the arc length s traveled by a point at a distance r from the axis via
s
the equation: θ = Note: The angle θ is measured in radians
r
2 Angular Displacement (10-3)
t In the picture we show the reference line at a time t1 and
at a later time t2 . Between t1 and t2 the body undergoes
1
an angular displacement ∆θ = θ 2 − θ1. All the points of the
t rigid body have the same angular displacement because they
rotate locked together.
Angular Velocity
We define as average angular velocity for the time interval ( t1 , t2 ) the ratio:
θ 2 − θ1 ∆θ
ωavg = = The SI unit for angular velocity is radians/second
t2 − t1 ∆t
∆θ
We define as the instantaneous angular velocity the limit of as ∆t → 0
∆t
∆θ dθ
ω = lim This is the definition of the first derivative with t ω =
∆t →0 ∆t dt
Algerbraic sign of angular frequency: If a rigid body rotates counterclockwise
(CCW) ω has a positive sign. If on the other hand the rotation is clockwise (CW)
ω has a negative sign
2
(10-4)
2
Angular Acceleration
t
ω 1
If the angular velocity of a rotating rigid object changes
with time we can describe the time rate of change of ω
1
ω
t by defining the angular aceleration
In the figure we show the reference line at a time t1 and at a later time t2 .
The angular velocity of the rotating body is equal to ω1 at t1 and ω2 at t2 .
We define as average angular acceleration for the time interval ( t1 , t2 ) the ratio:
ω2 − ω1 ∆ω
α avg = = The SI unit for angular velocity is radians/second 2
t2 − t1 ∆t
∆ω
We define as the instantaneous angular acceleration the limit of as ∆t → 0
∆t
∆ω dω
α = lim This is the definition of the first derivative with t α=
∆t → 0 ∆t
dt
Angular Velocity Vector
For rotations of rigid bodies about a fixed axis
we can describe accurately the angular velocity
by asigning an algebraic sigh. Positive for
counterclockwise rotation and negative for
clockwise rotation
We can actually use the vector notation to describe rotational motion which
is more complicated. The angular velocity vector is defined as follows:
r
The direction of ω is along the rotation axis.
r
The sense of ω is defined by the right hand rule (RHL)
Right hand rule: Curl the right hand so that the fingers point in the direction
r
of the rotation. The thumb of the right hand gives the sense of ω
(10-5)
Rotation with Constant Angular Acceleration
When the angular acceleration α is constant we can derive simple expressions
that give us the angular velocity ω and the angular position θ as function of time
We could derive these equations in the same way we did in chapter 2. Instead we
will simply write the solutions by exploiting the analogy between translational
and rotational motion using the following correspondance between the two motions
Translational Motion Rotational Motion
x ↔ θ
v ↔ ω
a ↔ α
v = v0 + at ↔ ω = ω0 + α t (eqs.1)
at 2 αt2
x = xo + vot + ↔ θ = θ + ωo t + (eqs.2)
2 2
v 2 − vo = 2a ( x − xo )
2
↔ ω 2 − ωo = 2α ( θ − θ o ) (eqs.3)
2
(10-6)
Relating the Linear and Angular Variables (10-7)
Consider a point P on a rigid body rotating about
a fixed axis. At t = 0 the reference line which connects
s the origin O with point P is on the x-axis (point A)
O θ
A
»
During the time interval t point P moves along arc AP
and covers a distance s. At the same time the reference
line OP rotates by an angle θ .
Relation between angular velocity and speed
The arc length s and the angle θ are connected by the equation:
ds d ( rθ ) dθ
s = rθ where r is the distance OP. The speed of point P v = = =r
dt dt dt
v = rω
circumference 2π r 2π r 2π
The period T of revolution is given by: T = = = =
speed v ωr ω
2π 1
T= T= ω = 2π f
ω f
The Acceleration
The acceleration of point P is a vector that has two
components. A "radial" componet along the radius
r
O and pointing towards point O. We have enountered
this component in chapter 4 where we called it
"centripetal" acceleration. Its magnitude is:
v2
ar = = ω 2 r
r
The second component is along the tangent to the circular path of P and is thus
known as the "tangential" component. Its magnitude is:
dv d ( ω r ) dω
at = = =r = rα at = rα
dt dt dt
The magnitude of the acceleration vector is: a = at2 + ar2
(10-8)
r Kinetic Energy of Rotation
vi
Consider the rotating rigid body shown in the figure.
mi
We divide the body into parts of masses m1 , m2 , m3 ,..., mi ,...
ri
O The part (or "element") at P has an index i and mass mi
The kinetic energy of rotation is the sum if the kinetic
1 1 1
energies of the parts K= m1v12 + m2 v2 + m3v3 + ...
2 2
2 2 2
1 1
K = ∑ mi vi2 The speed of the i -th element vi = ω ri → K = ∑ mi ( ω ri )
2
i 2 i 2
1 1
K = ∑ mi ri 2 ω 2 = I ω 2 The term I = ∑ mi ri 2 is known as
2 i 2 i
rotational inertia or moment of inertia about the axis of rotation. The axis of
rotation must be specified because the value of I for a rigid body depends on
its mass, its shape as well as on the position of the rotation axis. The rotational
inertia of an object describes how the mass is distributed about the rotation axis
I = ∑ mi ri 2 I = ∫ r 2 dm 1 2
K = Iω
i 2 (10-9)
In the table below we list the rotational inertias for some rigid bodies
I = ∫ r 2 dm
(10-10)
Calculating the Rotational Inertia
The rotational inertia I = ∑ mi ri 2 This expression is useful for a rigid body that
i
has a discreet disstribution of mass. For a continuous distribution of mass the sum
becomes an integral I = ∫ r 2 dm
Parallel-Axis Theorem
We saw earlier that I depends on the position of the rotation axis
For a new axis we must recalculate the integral for I . A simpler
method takes advantage of the parallel-axis theorem
Consider the rigid body of mass M shown in the figure.
We assume that we know the rotational inertia I com
about a rotation axis that passes through the center
of mass O and is perpendicular to the page.
The rotational inertia I about an axis parallel to the axis through O that passes
through point P, a distance h from O is given by the equation:
I = I com + Mh 2 (10-11)
A Proof of the Parallel-Axis Theorem We take the origin O
to coincide with the center of mass of the rigid body shown
in the figure. We assume that we know the rotational inertia
I com for an axis that is perpendicular to the page and passes
through O.
We wish to calculate the rotational ineria I about a new axis perpendicular
to the page and passes through point P with coodrinates ( a, b ) . Consider
an element of mass dm at point A with coordinates ( x, y ) . The distance r
( x − a) + ( y − b)
2 2
between points A and P is: r =
Rotational Inertia about P: I = ∫ r 2 dm = ∫ ( x − a ) + ( y − b ) dm
2 2
I = ∫ ( x 2 + y 2 ) dm − 2a ∫ xdm −2b ∫ ydm − ∫ ( a 2 + b 2 ) dm The second
and third integrals are zero. The first integral is I com . The term ( a 2 + b 2 ) = h 2
Thus the fourth integral is equal to h 2 ∫ dm = Mh 2 → I = I com + Mh
2
(10-12)
Torque
In fig.a we show a body which can rotate about an axis through
r
point O under the action of a force F applied at point P a distance
r
r from O. In fig.b we resolve F into two componets, radial and
tangential. The radial component Fr cannot cause any rotation
because it acts along a line that passes through O. The tangential
component Ft = F sin φ on the other hand causes the rotation of the
r
object about O. The ability of F to rotate the body depends on the
magnitude Ft and also on the distance r between points P and A.
Thus we define as torque τ = rFt = rF sin φ = r⊥ F
The distance r⊥ is known as the moment arm and it is the
r
perpendicular distance between point O and the vector F
The algebraic sign of the torque is asigned as follows:
r
If a force F tends to rotate an object in the coubterclockwise
r
direction the sign is positive. If a force F tends to rotate an
τ = r⊥ F object in the clockwise direction the sign is negative.
(10-13)
Newton's Second Law for Rotation
For translational motion Newton's second law connects
the force acting on a particle with the resulting acceleration
There is a similar relationship between the torque of a force
applied on a rigid object and the resulting angular acceleration
This equation is known as Newton's second law for rotation. We will explore
this law by studying a simple body which consists of a point mass m at the end
r
of a massless rod of length r. A force F is applied on the particle and rotates
r
the system about an axis at the origin. As we did earlier, we resolve F into a
tangential and a radial component. The tangential component is responsible
for the rotation. We first apply Newton's second law for Ft . Ft = mat (eqs.1)
The torque τ acting on the particle is: τ = Ft r (eqs.2) We eliminate Ft
between equations 1 and 2: τ = mat r = m ( α r ) r = ( mr 2 ) α = Iα
τ = Iα (compare with: F = ma )
(10-14)
1 Newton's Second Law for Rotation
2
3 We have derived Newton's second law for rotation
i
for a special case. A rigid body which consists of a point
i
mass m at the end of a massless rod of length r. We will
r now derive the same equation for a general case.
O
Consider the rod-like object shown in the figure which can rotate about an axis
through point O undet the action of a net torque τ net . We divide the body into
parts or "elements" and label them. The elements have masses m1 ,m2 , m3 ,..., mn
and they are located at distances r1 , r2 , r3 ,..., rn from O. We apply Newton's second
law for rotation to each element: τ 1 = I1α (eqs.1), τ 2 = I 2α (eqs.2),
τ 3 = I 3α (eqs.3), etc. If we add all these equations we get:
τ 1 + τ 2 + τ 3 + ... + τ n = ( I1 + I 2 + I 3 + ... + I n ) α . Here I i = mi ri 2 is the rotational inertia
of the i -th element. The sum τ 1 + τ 2 + τ 3 + ... + τ n is the net torque τ net applied.
The sum I1 + I 2 + I 3 + ... + I n is the rotational inertia I of the body.
Thus we end up with the equation: τ net = Iα
(10-15)
Work and Rotational Kinetic Energy W = ∆K (10-16)
In chapter 7 we saw that if a force does work W
on an object, this results in a change of its kinetic energy
∆K = W . In a similar way, when a torque does work W
on a rotating rigid body, it changes its rotational kinetic
energy by the same amount
Consider the simple rigid body shown in the figure which consists of a mass m
r
at the end of a massless rod of length r. The force F does work dW = Ft rdθ = τ dθ
The radial component Fr does zero work because it is at right angles to the motion.
θf
The work W = ∫ Ft rdθ = ∫ τ dθ . By virtue of the work-kinetic energy theorem we
θi
1 2 1 2 1 2 2 1 2 2
have a change in kinetic energy ∆K = W = mv f − mvi = mr ω f − mr ωi →
2 2 2 2
W = ∆K θf
1 2 1 2 W = ∫ τ dθ
∆K = I ω f − I ωi
2 2 θi
Power
Power has been defined as the rate at which work is done
by a force and in the case of rotational motion by a torque
We saw that a torque τ produces work dW = τ dθ as it
rotates an object by an angle dθ .
dW d dθ
P= = ( τ dθ ) = τ = τω (Compare with P = Fv)
dt dt dt
Below we summarize the results of the work-rotational kinetic energy theorem
θf
W = ∫ τ dθ W = τ ( θ f − θi ) For constant torque
θi
1 2 1 2
W = ∆K = I ω f − I ωi Work-Rotational Kinetic Energy Theorem
2 2
P = τω
(10-17)
Analogies between translational and rotational Motion
Translational Motion Rotational Motion
x ↔ θ
v ↔ ω
a ↔ α
v = v0 + at ↔ ω = ω0 + α t
at 2 αt2
x = xo + vo t + ↔ θ = θ o + ωo t +
2 2
v 2 − vo = 2a ( x − xo )
2
↔ ω 2 − ωo = 2α ( θ − θ o )
2
mv 2 Iω 2
K= ↔ K=
2 2
m ↔ I
F = ma ↔ τ = Iα
F ↔ τ
P = Fv ↔ P = τω
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