Rotational Kinematics of Rigid Bodies

Rotational Kinematics of Rigid Bodies

A rigid body is an idealized collection of particles whose pairwise distances remain constant:

(1)

for every pair of material points i and j.

This constraint allows the motion of a large collection of particles to be described using a small number of translational and rotational variables.

The simplest rotational motion is fixed-axis rotation. Every material point moves in a circle centered on the same fixed axis, and all points share the same angular position, angular velocity, and angular acceleration.

For planar fixed-axis rotation, define an angular coordinate 𝜃(t). Then

(2)

and

(3)

For a point a perpendicular distance r_⊥ from the axis,

(4)

(5)

(6)

and

(7)

These equations connect the angular motion of the rigid body to the linear motion of each material point.

1 What makes a body rigid?

Consider two material points A and B inside a body.

The rigid body assumption requires

(8)

The body may translate through space and it may rotate, but it does not deform.

Real materials are never perfectly rigid. Disturbances propagate through matter at finite wave speeds, and sufficiently large forces produce deformation.

Rigid body mechanics is therefore an approximation.

It is extremely useful when:

• elastic deformation is negligible for the problem,
• dimensions of the body can be treated as constant,
• rotational motion is important.

2 Translation and rotation

A rigid body can undergo several basic types of motion.

2.1 Pure translation

In pure translation, every point has the same instantaneous velocity:

(9)

The body’s orientation does not change.

2.2 Pure rotation about a fixed axis

In pure rotation, one axis is stationary in an inertial frame and every other point moves in a circle around it.

2.3 General rigid body motion

In general, a rigid body can translate and rotate simultaneously.

The present article focuses on fixed-axis rotational kinematics. General plane motion is developed later in the M05 sequence.

3 Angular position

For planar rotation about a fixed axis, choose a reference line attached to the body.

Its orientation can be described by an angle

(10)

A common sign convention is:

• counterclockwise angles are positive,
• clockwise angles are negative.

The zero-angle direction is arbitrary but must be stated or implied consistently.

4 Radians

Angular motion equations are simplest when angles are measured in radians.

A radian is defined geometrically by

(11)

where s is arc length and r is radius.

Therefore

(12)

when 𝜃 is measured in radians.

One full revolution corresponds to

(13)

Thus

(14)

Although the radian is dimensionless in SI, it is often retained explicitly in angular units to make the physical meaning clear.

5 Angular displacement

Suppose the angular position changes from

(15)

to

(16)

The angular displacement is

(17)

Angular displacement is signed.

A positive angular displacement corresponds to the chosen positive rotational direction.

A negative angular displacement corresponds to the opposite direction.

For multiple revolutions, Δ𝜃 need not lie between 0 and 2π.

For example, three complete counterclockwise revolutions give

(18)

6 Arc length and angular displacement

A material point at perpendicular distance r_⊥ from the rotation axis travels an arc length

(19)

For a finite rotation, the distance traveled along the circular path is

(20)

The same angular displacement produces a larger linear path for points farther from the axis.

7 Average angular velocity

Average angular velocity is defined by

(21)

Its SI unit is commonly written

(22)

The sign of ω_avg indicates rotational direction.

8 Instantaneous angular velocity

Take the time interval to zero:

(23)

For fixed-axis motion in a plane:

• ω > 0 means rotation in the positive angular direction,
• ω < 0 means rotation in the negative angular direction.

The magnitude

(24)

is the angular speed.

9 Period and frequency

If a body rotates steadily with period T, one revolution occurs in time T:

(25)

Therefore

(26)

Frequency is

(27)

Thus

(28)

If rotational speed is given in revolutions per minute,

(29)

it should normally be converted to radians per second before substitution into mechanics equations.

10 Example 1: rpm to angular speed

A wheel rotates at

(30)

Convert to revolutions per second:

(31)

Then

(32)

Thus

(33)

11 Angular acceleration

Average angular acceleration is

(34)

Instantaneous angular acceleration is

(35)

The SI unit is

(36)

The sign of α describes how the signed angular velocity changes.

A positive α does not necessarily mean the body is speeding up.

For example:

• ω > 0, α > 0: angular speed increases,
• ω > 0, α < 0: angular speed decreases,
• ω < 0, α < 0: angular speed increases,
• ω < 0, α > 0: angular speed decreases.

12 Constant angular acceleration

If

(37)

then the angular-motion equations have exactly the same mathematical form as one-dimensional constant-acceleration translation.

Integrating

(38)

gives

(39)

Integrating

(40)

gives

(41)

Eliminating time gives

(42)

The average angular velocity under constant angular acceleration is

(43)

(44)

13 Example 2: spin-up from rest

A rotor starts from rest and has constant angular acceleration

(45)

for

(46)

The final angular velocity is

ω	= ω0 + αt	(47)
	= 0 + (4.0)(5.0)	(48)
	= 20 rad∕s.	(49)

The angular displacement is

Δ𝜃	= ω0t + αt2	(50)
	= (4.0)(25)	(51)
	= 50 rad.	(52)

Thus

(53)

The number of revolutions is

(54)

14 All points share the same angular motion

In a rigid body rotating about a fixed axis, every material point sweeps through the same angular displacement during the same time interval.

Therefore all points share the same:

(55)

They do not share the same linear speed.

For a point at perpendicular distance r_⊥ from the axis,

(56)

Thus a point farther from the axis moves faster.

15 Derivation of tangential speed

A point at radius r_⊥ travels

(57)

Divide by dt:

(58)

Thus the tangential speed is

(59)

The velocity direction is tangent to the circular path.

16 Tangential acceleration

Differentiate the tangential speed relation for a fixed material point:

(60)

using signed tangential components.

Then

(61)

Therefore the tangential-acceleration magnitude is

(62)

Tangential acceleration changes the magnitude of the point’s velocity.

17 Normal acceleration

Even if

(63)

a rotating point is accelerated because the direction of its velocity changes.

For circular motion,

(64)

Since

(65)

(66)

Thus

(67)

This component points inward toward the rotation axis.

18 Total acceleration of a point

The tangential and Normal components are perpendicular:

(68)

Therefore the magnitude is

(69)

Equivalently,

(70)

19 Example 3: two points on a rotating disk

A disk rotates with

(71)

and

(72)

Point A is at

(73)

and point B is at

(74)

Their speeds are

vA	= rAω = 1.0 m∕s,	(75)
vB	= rBω = 3.0 m∕s.	(76)

Their tangential accelerations are

at,A	= rAα = 0.30 m∕s2,	(77)
at,B	= rBα = 0.90 m∕s2.	(78)

Their normal accelerations are

an,A	= rAω2 = 10 m∕s2,	(79)
an,B	= rBω2 = 30 m∕s2.	(80)

Both points share the same ω and α, but point B has three times the linear speed and three times each linear acceleration component.

20 Points on the rotation axis

For a point exactly on the fixed axis,

(81)

Therefore

v	= 0,	(82)
at	= 0,	(83)
an	= 0.	(84)

The axis is stationary in fixed-axis rotation.

This fact distinguishes fixed-axis rotation from more general rotational motion in which the axis itself can translate or change direction.

21 Angular graphs

The angular variables have the same derivative hierarchy as translational kinematics:

(85)

Therefore:

• the slope of a 𝜃-versus-t graph is ω,
• the slope of an ω-versus-t graph is α,
• the area under an ω-versus-t graph is angular displacement,
• the area under an α-versus-t graph is change in angular velocity.

These interpretations remain valid when ω and α vary with time.

22 Example 4: variable angular acceleration from a graph

Suppose

(86)

in rad∕s², with

(87)

Then

ω(t) − ω(0)	= ∫ 0t2τ dτ	(88)
	= t2.	(89)

Therefore

(90)

Angular position follows by integrating again:

𝜃(t) − 𝜃(0)	= ∫ 0t(3 + τ2) dτ	(91)
	= 3t + t3.	(92)

Thus

(93)

23 Angular velocity is common, tangential velocity is not

A frequent mistake is to say every point of a rigid body has the same velocity during rotation.

That is false.

All material points share the same angular velocity for fixed-axis rotation:

(94)

But

(95)

Thus linear velocities depend on position.

Their directions also differ because each velocity is tangent to its own circular path.

24 Angular acceleration is common, linear acceleration is not

Similarly,

(96)

for the shared rigid body rotation, but

(97)

and

(98)

Linear acceleration therefore varies across the body.

25 Relation to uniform circular motion

Uniform circular motion is recovered when

(99)

Then

(100)

but

(101)

remains nonzero.

A point can therefore move at constant speed while continuously accelerating.

26 Finite rotations and orientation

In planar motion, a single angle 𝜃 completely specifies orientation.

In three dimensions, a general rigid body orientation requires more information than one scalar angle.

This becomes important because finite rotations about different axes do not generally commute.

The present article avoids that complication by concentrating on fixed-axis motion.

M05-02 develops angular velocity and angular acceleration as vectors and begins the transition to three-dimensional rotational kinematics.

27 Vector bridge

For fixed-axis rotation, introduce the angular-velocity vector

(102)

along the rotation axis according to the right-hand rule.

If ρ is the position of a material point measured from a point on the fixed axis, then

(103)

Its magnitude is

(104)

Similarly, the point acceleration can be written

(105)

The first term is tangential acceleration.

The second term is inward normal acceleration.

28 Example 5: vector velocity check

Let

(106)

and let a point have position

(107)

Then

v	= ω ×ρ	(108)
	= 4ez × (3ex + 2ey)	(109)
	= 12ey − 8ex.	(110)

Thus

(111)

The perpendicular radius is

(112)

The speed predicted by the scalar relation is

(113)

The vector magnitude is

(114)

so the two forms agree.

29 Common mistakes

1. Treating a rigid body as a single particle and ignoring the different linear motions of its material points.
2. Using degrees directly in s = r𝜃 instead of converting to radians.
3. Confusing angular displacement with distance traveled along an arc.
4. Confusing angular velocity with tangential speed.
5. Saying every point of a rotating rigid body has the same linear velocity.
6. Forgetting that r_⊥ is the perpendicular distance to the rotation axis, not necessarily the distance to an arbitrary origin.
7. Forgetting that normal acceleration remains nonzero when α = 0.
8. Assuming positive angular acceleration always means increasing angular speed.
9. Applying constant-angular-acceleration formulas when α varies with time.
10. Forgetting the factor 2π when converting between frequency and angular speed.
11. Confusing rpm with rad/s.
12. Treating a point on the fixed rotation axis as though it had nonzero tangential speed.
13. Using a = r_⊥α as the complete acceleration and omitting the normal term.
14. Treating planar angular position 𝜃 as a complete description of arbitrary three-dimensional rigid body orientation.

30 Practice exercises

1. Explain the rigid body constraint in terms of the distance between any two material points.
2. Convert 900 rpm to rad/s.
3. A wheel turns through 18 revolutions. Find its angular displacement in radians.
4. A disk starts at 𝜃₀ = 0.5 rad and rotates with constant ω = 6 rad∕s for 4 s. Find 𝜃_f.
5. A rotor starts from rest with constant α = 5 rad∕s² for 3 s. Find its final angular velocity and angular displacement.
6. A wheel slows from 20 rad∕s to 5 rad∕s in 3 s with constant angular acceleration. Find α.
7. A point lies 0.25 m from a fixed rotation axis. If ω = 12 rad∕s, find its speed.
8. For the point in the previous problem, find the normal acceleration.
9. If α = 4 rad∕s² in the same motion, find the tangential acceleration and total acceleration magnitude.
10. Two points lie at radii 0.10 m and 0.40 m on the same rotating disk. Compare their angular velocities, tangential speeds, and normal accelerations.
11. Show by differentiation that v = r_⊥ω implies a_t = r_⊥α for a fixed material point.
12. Use v = r_⊥ω and a_n = v²∕r _⊥ to derive a_n = r_⊥ω².
13. If α(t) = 3t² and ω(0) = 2 rad∕s, find ω(t).
14. If ω(t) = 4 + 2t and 𝜃(0) = 1 rad, find 𝜃(t).
15. Verify the magnitude of v = ω ×ρ is |ω|r_⊥.

31 Summary

A rigid body preserves all internal distances:

(115)

For fixed-axis planar rotation,

(116)

When α is constant,

ω = ω0 + αt,		(117)
𝜃 − 𝜃0 = ω0t + αt2,		(118)
ω2 = ω 02 + 2α(𝜃 − 𝜃 0).		(119)

For a material point at perpendicular distance r_⊥ from the axis,

s = r⊥𝜃,		(120)
v = r⊥|ω|,		(121)
at = r⊥|α|,		(122)
an = r⊥ω2.		(123)

In vector form,

(124)

and

(125)

M05-02 develops the vector character of angular velocity and angular acceleration and extends these ideas toward three-dimensional rigid body kinematics.

References

References