Chapter 10

Rotation

Johan van Lierop

PHYS1050

Johan van Lierop PHYS1050 Chapter 10 Rotation

Rotation: How to describe it

I To describe the rotation of a rigid body about a fixed

axis, we need to define a axis of rotation
I In pure rotation (i.e. angular motion) every point of a
body moves in a circle whose centre lies on the axis of
rotation, with every point moving through the same
angle during an interval of time (remember, for linear
motion every point of a body moves in a straight line and
all points move through the same linear distance during a
time interval).
I The angular position is given by θ = s/r (radian measure)

rotation variables (top view)

N.B. we do not set θ = 0 with each complete rotation of
the reference line; two revolutions
⇒ θ = 2(2π) = 4π rad.

Johan van Lierop PHYS1050 Chapter 10 Rotation

Rotation: How to describe it (cont’d)

I The angular displacement is given by ∆θ = θ2 − θ1 .

+∆θ →CCW displacement and −∆θ → CW
displacement.
I The average (ωavg ) and instantaneous (ω) angular
velocities are

∆θ ∆θ dθ
ωavg = ω = lim =
∆t ∆t→0 ∆t dt

[ω] = rad/s = rev/s

I the average (αavg ) and instantaneous (α) angular
accelerations are

∆ω ∆ω dω
αavg = α = lim =
∆t ∆t→0 ∆t dt

[α]= rad/s2 = rev/s2

Johan van Lierop PHYS1050 Chapter 10 Rotation

Rotation: How to describe it (cont’d)

paraboloid ofrevolution1 paraboloid ofrevolution2

rotatingwatertroughs1 rotatingwatertroughs2
Johan van Lierop PHYS1050 Chapter 10 Rotation
Are θ, ω and α vector quantities?
The short answer is “sort of”:
X ω and α are clearly vectors, but they’re odd since they describe how a
body is rotating around the direction of the vector, not moving along the
vector’s direction.
X ∆θ (unless ∆θ → dθ (e.g. ∆θ is very small )) is NOT a vector, since the
order of rotations (e.g. additions) does matter.

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

The wheel has eight equally spaced spokes and

a radius of 30 cm. It is mounted on a fixed
axle and is spinning at 2.5 rev/s. You want to
shoot a 20 cm long arrow parallel to this axle
and through the shell without hitting any of
the spokes. Assume that the arrow and the
spokes are very thin. (a) What minimum
speed must the arrow have? (b) Does it
matter where between the axle and rim of the
wheel you aim? If so, what is the best
location?

Johan van Lierop PHYS1050 Chapter 10 Rotation

Rotation with constant angular acceleration

angular accelerationmachine bike wheel angular ac eleration

air rotatorwith deflectors

The video encyclopedia of physics demonstrations, R. Berg et al., Education Group & Associates (Los Angeles, CA), 2000.

Johan van Lierop PHYS1050 Chapter 10 Rotation

Rotation with constant angular acceleration (cont’d)

linear equation unknown variable unknown variable angular equation

transform
v = v0 + at x − x0 θ − θ0 ω = ω0 + αt
x − x0 = v0 t + 12 at 2 v ω θ − θ0 = ω0 t + 12 αt 2
v 2 = v02 + 2a(x − x0 ) t t ω 2 = ω02 + 2α(θ − θ0 )
x − x0 = 21 (v0 + v )t a α θ − θ0 = 12 (ω0 + ω)t
x − x0 = vt − 12 at 2 v0 ω0 θ − θ0 = ωt − 12 αt 2

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

A merry-go-round rotates from rest with an angular acceleration of 1.50 rad/s2 .

How long does it take to rotate through (a) the first 2.00 rev and (b) the next
2.00 rev?

Johan van Lierop PHYS1050 Chapter 10 Rotation

Linear and angular variable relations

Position: s = θr (radian measurement)

Velocity & Speed: ~ ~ ×~

v =ω r

ds dθ
= r ⇒ v = ωr (radian measurement)
dt dt
2πr 2π
T = ⇒ T = (radian measurement)
v ω
Acceleration: ~ ~ ×~
a=α r

dv dω
= r ⇒ at = αr tangential component (radian measurement)
dt dt
v2
ar = = ω 2 r radial component (radian measurement)
r

Johan van Lierop PHYS1050 Chapter 10 Rotation

Linear and angular variable relations (cont’d)

penny drop stick hingedstickandball

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

The flywheel of a steam engine runs with a constant angular velocity of

150 rev/min. When steam is shut off, the friction of the bearings and of the air
stops the wheel in 2.2 h. (a) What is the constant angular acceleration, in
revolutions per minute-squared, of the wheel during the slowdown? (b) How
many revolutions does the wheel make before stopping? (c) At the instant the
flywheel is turning at 75 rev/min, what is the tangential component of the
linear acceleration of a flywheel particle that is 50 cm from the axis of rotation?
(d) What is the magnitude of the net linear acceleration of the particle in (c)?

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

An early method of measuring the speed of light makes use of a rotating slotted wheel. A beam of
light passes through one of the slots at the outside edge of the wheel and travels to a distant
mirror, and returns to the wheel just in time to pass through the next slot in the wheel. One such
slotted wheel has a radius of 5.0 cm and 500 slots around its edge. Measurements taken when the
mirror is L = 500 m from the wheel indicate a speed of light of 3.0 × 105 km/s. (a) What is the
(constant) angular speed of the wheel? (b) What is the linear speed of a point on the edge of the
wheel?

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

A pulsar is a rapidly rotation neutron star that emits a radio beam the way a
lighthouse emits a light beam. We receive a radio pulse for each rotation of the
star. The period T of rotation is found by measuring the time between pulses.
The pulsar in the Crab nebula has a period of rotation of T = 0.033 s that is
increasing at the rate of 1.26 × 10−5 s/y. (a) What is the pulsar’s angular
acceleration α? (b) if α is constant, how many years from now will the pulsar
stop rotating? (c) The pulsar originated in a supernova explosion seen in the
year 1054. Assuming constant α, find the initial T .

Johan van Lierop PHYS1050 Chapter 10 Rotation

Kinetic energy of rotation

I Treating rotating rigid body as a collection of

particles with different speeds to avoid expressing
1 2
the energy
P 1 of 2the CoM (i.e. with K = 2 mv ), i.e.
K = m v with vi different for different mi .
2 i i
I But, v = ωr , so
P1 P
K = m (ωri )2 = 21 ( mi ri2 )ω 2
2 i
P
I I = ( mi ri2 ) is the rotational inertia (or moment
of inertia) of the body with respect to is axis of
rotation.
I K = 21 I ω 2 (radian measure)

Johan van Lierop PHYS1050 Chapter 10 Rotation

Moments of inertia
I For a continuous body, the moment of
R
inertia is defined by I = r 2 dm
I If we know what Icom of the body about a
parallel axis that extends through the
body’s centre of mass. h is the
perpendicular distance between the given
axis and the axis through the centre of
mass, so that

I = Icom + Mh2 (parallel-axis theorm)

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

In the figure, two particles, each with mass m = 0.85 kg, are fastened to each
other, and to a rotating axis at O, by two thin rods, each with length
d = 5.6 cm and mass M = 1.2 kg. The combination rotates around the
rotation axis with the angular speed ω = 0.30 rad/s. Measured about O, what
are the combination’s (a) rotational inertia and (b) kinetic energy?

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

The uniform solid block in the figure has mass 0.172 kg and edge lengths
a = 3.5 cm, b = 8.4 cm and c = 1.4 cm. Calculate its rotational inertia about
an axis through one corner and perpendicular to the larger faces.

Johan van Lierop PHYS1050 Chapter 10 Rotation

Torque

torque
By Yawe - Own work, Public Domain, https:
//commons.wikimedia.org/w/index.php?curid=1149798

I To determine how F~ results in rotation of a body about its axis of rotation, we

have to resolve F~ into a radial (Fr ) and tangential (Ft ) component.
I Fr points along ~r and F⊥ is perpendicular to ~ r.
I Fr does not cause rotation, and F⊥ = F sin φ does cause rotation.
I The ability of F~ to rotate the body depends on F⊥ and how far from the axis of
rotation (O) F~ is.
I Torque, τ , combines these factors: ~ τ =~r × F~ ; |~
τ | = τ = (r )(F sin φ)
Johan van Lierop PHYS1050 Chapter 10 Rotation

Torque (cont’d)

torque
By Yawe - Own work, Public Domain, https:
//commons.wikimedia.org/w/index.php?curid=1149798
I τ = (r )(F sin θ) = rFt = (r sin θ)(F ) = r⊥ F
I r⊥ is the moment arm of F~ and is defined by the line of action of F~ .
I [τ ] = N m

[work = W ] = N m = J too, but torque is never described in J (Joules);

that’s an energy, not a “force”, like torque.

Johan van Lierop PHYS1050 Chapter 10 Rotation

Torque (cont’d)

torque bar hinge board 1 hinge board 2

torque wrench torque wheel 1 torque wheel 2

The video encyclopedia of physics demonstrations, R. Berg et al., Education Group & Associates (Los Angeles, CA), 2000.

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

A small ball of mass 0.75 kg is attached to one end of a 1.25 m long massless
rod, and the other end of the rod is hung from a pivot. What the resulting
pendulum is 30◦ from the vertical, what is the magnitude of the gravitational
torque calculated about the pivot?

Johan van Lierop PHYS1050 Chapter 10 Rotation

Newton’s 2nd law for rotation
X Consider the simple situation where a rigid body
consists of a particle of mass m on one end of a
massless rigid cord (i.e. rod) of length r that can
move only by rotating about its other end (rotation
axis or axle) ⇒ particle moves in a circular path.
X F
~ acts on the particle, but only F⊥ can accelerate
the particle. Thus F⊥ = ma⊥ .
X The torque acting on the particle is then
τ = F⊥ r = ma⊥ r = m(αr )r = (mr 2 )α
X We know that I = mr 2 for the particle,
so τ = I α (radian measure).
X When more than one force is applied to the particle:

τnet = I α (radian measure)

An object is supported by a horizontal frictionless

table and is attached to a pivot point by a cord that
supplies centripetal force. A force is applied to the

centre of percussion
object perpendicular to the radius r , causing it to
accelerate about the pivot point. The force is
perpendicular to r .

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

In the figure, block 1 has mass m1 = 460 g, block 2 has mass m2 = 500 g, and
the pulley, which is mounted on a horizontal axle with negligible friction, has
radius R = 5.00 cm. When released from rest, block 2 falls 75.0 cm in 5.00 s
without the cord slipping on the pulley. (a) What is the magnitude of the
acceleration of the blocks? What are (b) tension T2 and (c) tension T1 ? (d)
What is the magnitude of the pulley’s angular acceleration? (e) What is its
rotational inertia?

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

The figure shows particles 1 and 2, each of mass m, attached to the ends of a
rigid massless rod of length L1 + L2 , with L1 = 20 cm and L2 = 80 cm. The
rod is held horizontally on the fulcrum and then released. What are the
magnitudes of the initial acceleration of (a) particle 1 and (b) particle 2?

Johan van Lierop PHYS1050 Chapter 10 Rotation

Work and rotational kinetic energy

X During the rotation of the particle from A to B (or

body around its fixed axis) by F~ , the force does work.
Assuming that only K.E. is changed by F~ , from the
work – kinetic energy theorem, ∆K = Kf − Ki = W
X K = 12 mv 2 and v = ωr , so we can write
∆K = 12 mr 2 ωf2 − 12 mr 2 ωi2 = W .
X Since I = mr 2 , ∆K = 12 I ωf2 − 21 I ωi2 = W .
X W is done by Fs (since only the tangential component
of F~ accelerates the particle along the path ds).
X So, dW = Fs ds = Fs (rdθ) = (Fs r )dθ = τ dθ.
X Z θf
W = τ dθ
θi
and if the work is done by a constant torque,
W = τ (θf − θi )
X Also, as the power, P = dW /dt we find
P = τ (dθ/dt) = τ ω (power, rotation about a fixed axis)

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

A meter stick is held vertically with one end on the floor and is then allowed to
fall. Find the speed of the other end just before it hits the floor, assuming that
the end on the floor does not slip. (Hint: Consider the stick to be a thin rod
and use the conservation of energy principle.)

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

A tall, cylindrical chimney falls over when its base is ruptured. Treat the
chimney as a thin rod of length 55.0 m. At the instant it makes an angle of
35.0◦ with the vertical as it falls, what are (a) the radial acceleration of the
top, and (b) the tangential acceleration of the top. (Hint: Use energy
considerations, not a torque). (c) At what angle θ is the tangential acceleration
equal to g ?

Johan van Lierop PHYS1050 Chapter 10 Rotation

Problem

A uniform spherical shell of mass M = 4.5 kg and radius R = 8.5 cm can

rotate about a vertical axis on frictionless bearings over a pulley of rotational
inertia I = 3.0 × 10−3 kg m2 and radius r = 5.0 cm, and is attached to a small
object of mass m = 0.60 kg. There is no friction on the pull’s axle; the cord
does not slip on the pulley. What is the speed of the object when it has fallen
82 cm after being released from rest? Use energy considerations.

Johan van Lierop PHYS1050 Chapter 10 Rotation

Summary and forecast

I Angular position, displacement, velocity and speed: θ = s/r (w/ 1 rev

= 360◦ = 2π rad), ∆θ = θ2 − θ1 , ω = dθ/dt, α = dω/dt.
I Linear and angular variable relations: s = r θ, v = ωr , at = αr ,
ar = v 2 /r = ω 2 r and T = (2πr )/v = (2π)/ω.
P R
I Rotational K.E. and inertia: K = 21 I ω 2 with I = mi ri2 = r 2 dm
– parallel-axis theorem: I = Icom + Mh2
I Torque: τ = rFt = r⊥ F = rF sin φ ⇒ ~τ = ~r × F~
I Newton’s 2nd law in angular form: τnet = I α.
R
I Work and rotational K.E.: W = τ dθ, P = τ ω and
∆K = 21 I ωf2 − 12 I ωi2 = W .
I Read 11-1 to find out how a yo-yo works!

Johan van Lierop PHYS1050 Chapter 10 Rotation