Rotational Motion and Angular Momentum

Rotational Motion and Angular Momentum

Unit 6

A particle at point P at a fixed

distance r from origin is rotating

about axis O.

Lesson 1 : Angular Position, Velocity, and Acceleration

When a rigid object rotates about its axis, at any given time different parts of the object have different linear velocities and linear

accelerations.

P is at polar coordinate (r, )

is measured counterclockwise from some reference line ( = 0)

s = r

= s

r

is measured in radians (rad)

One radian is the angle subtended by an arc length equal to the radius of the arc.

= s

r

360o = 2r

rrad = 2 rad

1 rad =360o

2= 57.3o

Converting from Degrees to Radians

(rad) =

180o (deg)

90o = /2 rad

60o = /3 rad

270o = 3/2 rad

45o = /4 rad

As particle moves from A to B in time

interval t, the reference line of

length r sweeps out an angle

= f - i

Angular Displacement ()

Average Angular Speed ()

Ratio of the angular displacement of a rigid object to the time interval t.

= f - i

tf - ti

=t

The rad/s is the unit for angular speed.

is positive when increases (counterclockwise motion)

is negative when decreases (clockwise motion)

Instantaneous Angular Speed ()

= limt 0

t

=dt

d

Average Angular Acceleration ()

= f - i

tf - ti

=t

Instantaneous Angular Acceleration ()

= limt 0

t

=dt

d

The rad/s2 is the unit for angular speed.

is positive when object rotates counterclockwise and speeds up

OR

when object rotates clockwise and slows down

Direction of Angular Velocity and Acceleration Vectors

The directions of and are along the axis of rotation.

and are vector quantities with magnitude and direction

Right-Hand Rule

Wrap four fingers of the right hand in

the direction of rotation.

Thumb will point in the direction of angular velocity

vector ().

Example 1

A rigid object is rotating with an angular speed < 0. The angular velocity vector and the angular acceleration vector

are antiparallel. The angular speed of the rigid object is

a) clockwise and increasing

b) clockwise and decreasing

c) counterclockwise and increasing

d) counterclockwise and decreasing

Example 2

During a certain period of time, the angular position of a swinging door is described by = 5.00 + 10.0 t + 2.00 t2, where is in

radians and t is in seconds. Determine the angular position, angular speed, and angular acceleration of the door at

a) at t = 0 b) at t = 3.00 s

Lesson 2 : Rotational Kinematics with Constant Angular Acceleration

=ddt

d = dt

Since =ddt

ddt

i + t =

(by integrating)f = i + t

f = i + it + ½ t2 (by integrating)

Eliminating t from previous two

equations,

f2 = i

2 + 2(f – i)

Eliminating from previous two

equations,

f = i + ½ (i + f) t

xPosition

vVelocity

aAcceleration

Example 1

A wheel rotates with a constant angular acceleration of 3.50 rad/s2.

a) If the angular speed of the wheel is 2.00 rad/s at ti = 0, through what

angular displacement does the wheel rotate in 2.00 s ?

b) Through how many revolutions has the wheel turned during this time interval ?

c) What is the angular speed of the wheel at t = 2.00 s ?

Example 2

A rotating wheel requires 3.00 s to rotate through 37.0 revolutions. Its angular

speed at the end of the 3.00 s interval is 98.0 rad/s. What is the constant angular

acceleration of the wheel ?

Lesson 3 : Angular and Linear Quantities

Every particle of a rigid rotating object moves in a circle whose center is the

axis of rotation.

Tangential speed v = ds

dt

Since s = r

v = ddt

r

v = r

v = r

Tangential Speed

perpendicular distance of a point

from the axis of rotation

angular speed

Tangential speed depends

on distance from axis of

rotation

Angular speed is the same for

all points

Tangential Acceleration

at =dv

dt

Since v = r

at = ddt

r

at = r

Centripetal Acceleration in terms of Angular Speed

ac =v2

r

Since v = r

ac = r2

Total Linear Acceleration

a = at2 + ar

2

a = r22 + r24

a = r 2 + 4

In order to keep the tangential speed of the disc surface at the location of the lens constant, the

disc’s angular speed must vary as the lens moves radially along the disc. In a typical compact disc player, the constant speed of the surface at the

point of the laser-lens system is 1.3 m/s.

Example 1

a) Find the angular speed of the disc in rev/min when information is being read from the innermost first track (r = 23 mm) and the outermost final track (r = 58 mm).

b) The maximum playing time of a standard music CD is 74 min 33 s. How many revolutions

does the disc make during that time ?

c) What total length of track moves past the lens during this time ?

d) What is the angular acceleration of the CD over the 4,473 s time interval ? [Assume that is constant.]

The drive train of a bicycle is shown to the right. The

wheels are 67.3 cm in diameter and pedal cranks 17.5 cm long. The cyclist

pedals at a steady angular rate of 76.0 rev/min. The

chain engages with a front sprocket 15.2 cm in diameter and a rear sprocket 7.00 cm in

diameter.

Example 2

a) Calculate the speed of a link of the chain relative to the bicycle frame.

b) Calculate the angular speed of the bicycle wheels.

c) Calculate the speed of the bicycle relative to the road.

d) What pieces of data, if any, are not necessary for the calculations ?

Lesson 4 : Rotational Kinetic Energy

The total kinetic energy of a rotating rigid object is the sum of the kinetic energies of

its individual particles.

KErot = KEi = ½mivi2

Since v = r,

KErot = ½ miri2i

2

Factoring out 2,

KErot = ½ ( miri2) 2

KErot = ½ ( miri2) 2

Moment of Inertia (I)

I = miri2

The kg . m2 is the SI unit for moment of inertia.

Moment of inertia is a measure of the resistance of an object to changes in

its rotational motion.

Rotational Motion Linear Motion

I m

v

Substituting I,

KErot = ½ I2

KErot = ½ I2 KE = ½ mv2

Example 1

Consider an oxygen molecule (O2) rotating in the x-y plane about the z-axis. The rotation axis passes through the center of the molecule,

perpendicular to its length. The mass of each oxygen atom is 2.66 x 10-26 kg, and at room

temperature the average separation between the two atoms is d = 1.21 x 10-10 m. (The atoms are

modeled as particles.)

a) Calculate the moment of inertia of the molecule about the z-axis.

b) If the angular speed of the molecule about the z-axis is 4.60 x 1012 rad/s, what is its

rotational kinetic energy ?

Lesson 5 : Calculation of Moments of Inertia

Moment of inertia of a rigid object is evaluated by dividing the object into many small volume elements, each

with mass = mi.

Since = m/V,

dm = dV

I = r2 dV

I = lim ri2mi = r2 dm

mi 0

Find the moment of inertia of a uniform thin hoop of mass M and radius R about an axis perpendicular to the plane of the

hoop and passing through its center.

Example 1

Calculate the moment of inertia of a uniform rigid rod of length L and mass M about an axis perpendicular to the

rod (the y-axis) and passing through its center of mass.

Example 2

A uniform solid cylinder has a radius R, mass M, and length L. Calculate its

moment of inertia about its central axis (the z-axis).

Example 3

Parallel-Axis Theorem

The moment of inertia about any axis parallel to and a distance D away from

this axis is

I = ICM + MD2

Consider once again the uniform rigid rod of mass M and length L. Find the moment

of inertia of the rod about an axis perpendicular to the rod through one end

(the y’ axis).

Example 4

Moment of Inertia of a Thin Cylindrical Shell (Hoop)

Moment of Inertia of a Hollow Cylinder

Moment of Inertia of a Solid Cylinder (Disk)

Moment of Inertia of a Rectangular Plate

Moment of Inertia of a Long Thin Rod

(Axis Through Center)

Moment of Inertia of a Long Thin Rod

(Axis Through End)

Moment of Inertia of a Solid Sphere

Moment of Inertia of a Thin Spherical Shell

The component Fsin tends to

rotate the wrench about

point O.

Lesson 6 : Torque

The tendency of a force to rotate an object about some axis is measured by a

vector quantity called torque.

= r Fsin = Fd

distance between pivot point and point of application of F

perpendicular distance from pivot point to the

line of action of F

Torque should not be confused with force.

Torque has units of force x length or N.m. (Same as work but not called Joules.)

F1 tends to rotate counterclockwise (+)

F2 tends to rotate clockwise (-)

= 1 + 2 = F1d1 – F2d2

The net torque about axis 0 is

A one-piece cylinder is shaped as shown with a core section

protruding from the larger drum. The cylinder is free to rotate about

the central axis shown in the drawing. A rope wrapped around the drum, which has radius R1, exerts a force T1 to the right on the cylinder. A rope wrapped

around the core, which has radius R2, exerts a force T2 downward on

the cylinder.

Example 1

a) What is the net torque acting on the cylinder about the rotation axis (z-axis) ?

b) Suppose T1 = 5.0 N, R1 = 1.0 m, T2 = 15.0 N, and R2 = 0.50 m. What is the net torque about the rotation axis, and which way does the cylinder rotate starting from rest ?

Lesson 7 : Relationship between Torque and Angular Acceleration

Ft = mat

= Ftr = (mat)r

= (mr)r = (mr2)

Since I = mr2,

= I

(rotational analog to Newton’s Second Law)

A uniform rod of length L and mass M is attached at one end to a frictionless pivot and is free to rotate about the pivot in the vertical

plane. The rod is released from rest in the horizontal position. What is the initial angular

acceleration of the rod and the initial linear acceleration of its right end ?

Example 1

A wheel of radius R, mass M, and moment of inertia I

is mounted on a frictionless horizontal

axis. A light cord wrapped around the wheel supports

an object of mass m.

Example 2

a) Calculate the angular acceleration of the wheel.

b) Calculate the linear acceleration of the object.

c) Calculate the tension in the cord.

Work done by F on an object as it rotates

through an infinitesimal distance ds = r d

Lesson 8 : Work, Power, and Energy in Rotational Motion

dW = F.ds = (Fsin)r d

(The radial component of F does no work because it is perpendicular to the

displacement.)

Since = rFsin,

Rate at which work is being done as an object rotates through angle d in

time interval dt is

dW

dt=

ddt

dW = d

(Power delivered to a rotating rigid object.)

P =dW

dt=

analogous to P = Fv in

linear motion

analogous to dW = Fx dx in linear motion

= I= Iddt

= Idd

= Idd

ddt

(chain rule)

d = I d

Since dW = d,

dW = I d

Integrating to find total work done,

W = i

f

I d = ½ If2 – ½ Ii

2

Work–KE Theorem for Rotational Motion

The net work done by external forces in rotating a symmetric rigid object about a

fixed axis equals the change in the object’s rotational energy.

A uniform rod of length L and mass M is free to rotate on a

frictionless pin passing through one

end. The rod is released from rest in

the horizontal position.

Example 1

a) What is its angular speed when it reaches its lowest position ?

b) Determine the tangential speed of the center of mass and the tangential speed of the lowest point on the rod when it is in the vertical position.

Consider two cylinders having different masses m1 and m2,

connected by a string passing over a pulley. The pulley has a

radius R and moment of inertia I about its axis of

rotation. The string does not slip on the pulley, and the

system is released from rest. Find the linear speeds of the

cylinders after cylinder 2 descends through a distance h, and the angular speed of

the pulley at this time.

Example 2

A light string that is attached to a large block of mass 4m passes over a pulley with negligible rotational inertia and is wrapped around a vertical pole of radius r, as shown in

Experiment A above. The system is released from rest, and as the block descends the string unwinds and the vertical pole with its attached apparatus rotates. The

apparatus consists of a horizontal rod of length 2L, with a small block of mass m attached at each end. The

rotational inertia of the pole and the rod are negligible.

Example 3 : AP 2001 #3

a) Determine the rotational inertia of the rod-and-block apparatus attached to the top of the

pole.

b) Determine the downward acceleration of the large block.

c) When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare with the value 4mgD ? Check the appropriate space below.

____ Greater than 4mgD

____ Equal to 4mgD

____ Less than 4mgD

Justify your answer.

The system is now reset. The string is rewound around the pole to bring the large block back to its

original location. The small blocks are detached from the rod and then suspended from each end of the

rod, using strings of length l. The system is again released from rest so that as the large block

descends and the apparatus rotates, the small blocks swing outward, as shown in Experiment B above. This time the downward acceleration of the block decreases with time after the system is released.

d) When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare to that in part c) ? Check the appropriate space below.

____ Greater

____ Equal

____ Less

Justify your answer.

Cylinder rolling on a straight path.

Lesson 9 : Rolling Motion of a Rigid Object

Center moves in a straight line (green line).

A point on the rim moves in a path called a cycloid (red curve).

Speed of CM of Cylinder Rolling without Slipping

vCM =ds

dt

Since s = R,

vCM =ddt

R

vCM = R

for pure rolling motion only

Acceleration of CM of Cylinder Rolling without Slipping

aCM =dvCM

dt

aCM =ddt

R

aCM = R

Total Kinetic Energy of a Rolling Cylinder

KErot = ½ IP2

Since IP = ICM + MR2, (parallel-axis theorem)

KErot = ½ (ICM + MR2)2

OR

KErot = ½ ICM2 + ½ MR22

Since vCM = R,

KErot = ½ ICM2 + ½ MvCM2

KEtotal = ½ ICM2 + ½ MvCM2

rotational KE translational KE

The total kinetic energy of a rolling object is the sum of the rotational kinetic energy about the center of mass and the translational kinetic

energy of the center of mass.

Three objects of uniform density – a solid sphere, a solid cylinder, and a hollow cylinder – are placed at the top of an incline. They are

all released from rest at the same elevation and roll without slipping. Which object reaches the

bottom first ? Which reaches it last ?

Example 1

For the solid sphere shown above, calculate the linear speed of the center of mass at the bottom of the incline and the magnitude of the linear acceleration

of the center of mass.

Example 2

An inclined plane makes an angle of with the horizontal, as shown above. A solid sphere of radius

R and mass M is initially at rest in the position shown, such that the lowest point of the sphere is a

vertical height h above the base of the plane. The sphere is released and rolls down the plane without slipping. The moment of inertia of the sphere about an axis through its center is 2/5 MR2. Express your

answers in terms of M, R, h, g, and .


a) Determine the following for the sphere when it is at the bottom of the plane.

i. Its translational kinetic energy

ii. Its rotational kinetic energy

b) Determine the following for the sphere when it is on the plane.

i. Its linear acceleration

ii. The magnitude of the frictional force acting on it

The solid sphere is replaced by a hollow sphere of identical radius R and mass M. The hollow sphere, which is released from the same location as the solid sphere, rolls down the incline without slipping.

c) What is the total kinetic energy of the hollow sphere at the bottom of the plane

d) State whether the rotational kinetic energy of the hollow sphere is greater than, less than, or equal to that of the solid sphere at the bottom of the plane. Justify your

answer.

Lesson 10 : Angular Momentum

F =

From linear motion :

dp

dt

Take cross product of each side with r :

r x F = r x dp

dt

Net Torque

= r x dp

dt

x p is zero since = v, dr

dt

dr

dt

and v and p are parallel.

Add x p to right-hand side :dr

dt

= r x dp

dt+

dr

dtx p

=d(r x p)

dt

Analogous to dp in

translational motion angular momentum (L)

L = r x p

Instantaneous Angular Momentum

The instantaneous angular momentum L of a particle relative to the origin O is defined by the cross

product of the particle’s instantaneous position vector r and

its instantaneous linear momentum p.

L = r x p

The SI unit of angular momentum is kg . m2/s.

Since L = r x p,

=dL

dt rotational analog of Newton’s second law

The torque acting on a particle is equal to the time rate of change of the particle’s angular momentum.

The direction of L is always perpendicular to the plane formed by r and p.

Since p = mv,

L = mvr sin

A particle moves in the xy plane in a circular path of radius r, as shown above. Find the magnitude

and direction of its angular momentum relative to O when its linear velocity is v.

Example 1

Angular Momentum of a System of Particles

The total angular momentum of a system of particles about some point is defined as the vector sum of the angular

momenta of the individual particles.

Ltot = L1 + L2 + …. + Ln = Li

Differentiating with respect to time :

dLtot

dt=

dLi

dt

The net external torque acting on a system about some axis passing through an origin in an inertial frame equals the time rate of change of the total angular momentum of

the system about that origin.*

=

dLtot

dt

* This theorem applies even if the center of mass is accelerating, as long as and L are

evaluated relative to the center of mass.

A sphere of mass m1 and a block of mass m2 are connected by a light cord that passes over a pulley, as

shown above. The radius of the pulley is R, and the mass of the rim is M. The spokes of the pulley have negligible mass. The block slides on a frictionless, horizontal surface. Find an expression for the linear

acceleration of the two objects, using the concepts of angular momentum and torque.

Example 2

Each particle of this rigid object rotates in the xy plane about the z axis with angular speed .

Lesson 11 : Angular Momentum of a Rotating Rigid Object

Since v = r

L = mr2

Since I = mr2

L = I

L = I

Differentiating with respect to time :(I is constant for a rigid object)

dL

dt= I

ddt

= I

=dL

dtSince

= Irotational form

of Newton’s second law

Estimate the magnitude of the angular momentum of a bowling ball spinning at

10 rev/s.

Example 1

A father of mass mf and daughter of mass md sit on opposite ends of a

seesaw at equal distances from the

pivot at the center. The seesaw is modeled as a rigid rod of mass M and length l and is pivoted

without friction. At a given moment, the

combination rotates in a vertical plane with an

angular speed .

Example 2

a) Find an expression for the magnitude of the system’s angular momentum.

b) Find an expression for the magnitude of the angular acceleration of the system when the seesaw makes an angle with the horizontal.

A uniform solid cylinder of mass m1 and radius R is mounted on frictionless bearings about a fixed axis

through O. The moment of inertia of the cylinder about the axis is I = ½ m1R2. A block of mass m2,

suspended by a cord wrapped around the cylinder as shown above, is released at time t = 0.


a) On the diagram below draw and identify all of the forces acting on the cylinder and on the block.

b) In terms of m1, m2, R, and g, determine each of the following.

i) The acceleration of the block.

iii) The angular momentum of the disk as a function of t.

ii) The tension in the cord.

A system consists of two small disks, of masses m and 2m, attached to a rod of negligible mass of length 3l as shown above. The rod is free to turn about a vertical axis through

point P. The two disks rest on a rough horizontal surface; the coefficient of friction between the disks and the surface is . At time t = 0, the rod has an initial counterclockwise angular velocity o about P. The system is gradually brought to rest by friction. Develop expressions for the following quantities

in terms of , m, l, g, and o.


a) The initial angular momentum of the system about the axis through P.

b) The frictional torque acting on the system about the axis through P.

c) The time T at which the system will come to rest.


Consider a thin uniform rod of mass M and length l , as shown above.

l

M

a) Show that the rotational inertia of the rod about an axis through its center and

perpendicular to its length is Ml 2/12.

The rod is now glued to a thin hoop of mass M and radius R = l /2 to form a rigid assembly, as shown above. The centers of the rod and hoop coincide at point P. The assembly is mounted

on a horizontal axle through point P and perpendicular to the page.

b) What is the rotational inertia of the rod-hoop assembly about the axle ?

Several turns of string are wrapped tightly around the circumference of the hoop. The system is at rest when a cat, also of mass M, grabs the free end of the string and hangs vertically from it

without swinging as it unwinds, causing the rod-hoop assembly to rotate. Neglect friction and the

mass of the string.

c) Determine the tension T in the string.

d) Determine the angular acceleration of the rod-hoop assembly.

e) Determine the linear acceleration of the cat.

f) After descending a distance H = 5 l /3, the cat lets go of the string. At that instant, what is the

angular momentum of the cat about point P ?

Lesson 12 : Conservation of Angular Momentum

The total angular momentum of a system is constant in both magnitude and

direction if the resultant external torque acting on the system is zero, that is, if the

system is isolated.

=

dLtot

dt= 0,Since Ltot = constant

Lbefore = Lafter

When arms are moved inward, I

decreases.

Since I remains constant, as I

decreases, must increase.

Lbefore = Lafter

Since L = I

Iii = Iff = constant

A horizontal platform in the shape of a circular disk rotates freely in a horizontal plane about a frictionless

vertical axle. The platform has a mass M = 100 kg and a radius R = 2.0 m. A student whose mass is m = 60 kg

walks slowly from the rim of the disk toward its center. If the angular speed of the system is 2.0 rad/s when the student is at the rim, what is the angular speed when he

reaches a point r = 0.50 m from the center ?

Example 1

In a favorite classroom demonstration, a student holds the

axle of a spinning bicycle wheel while seated on a stool that is free to rotate. The student and stool are

initially at rest while the wheel is spinning in a horizontal plane with an initial angular momentum Li that points upward. When the wheel is inverted about its center by 180o,

the student and stool start rotating. In terms of Li, what are the

magnitude and the direction of L for the student plus stool ?

Example 2

A 2.0 kg disk traveling at 3.0 m/s strikes a 1.0 kg stick of length 4.0 m that is lying flat on nearly frictionless ice. Assume that the collision is elastic and that the disk does not deviate from its original line of motion. Find the translational speed of the disk, the translational

speed of the stick, and the angular speed of the stick after the collision. The moment of inertia of the stick

about its center of mass is 1.33 kg.m2.

Example 3

A 1.0 kg object is moving horizontally with a velocity of 10 m/s, as shown above, when it makes a glancing

collision with the lower end of a bar that was hanging vertically at rest before the collision. For the system

consisting of the object and bar, linear momentum is not conserved in this collision, but kinetic energy is

conserved. The bar, which has a length l = 1.2 m and a mass m = 3.0 kg, is pivoted about the upper end.


Immediately after the collision the object moves with speed v at an angle relative to its original direction. The bar swings freely, and after the collision reaches a maximum angle of 90o with respect to the vertical. The moment of inertia of the bar about the pivot is

Ibar = ml 2/3. Ignore all friction.

a) Determine the angular velocity of the bar immediately after the collision.

b) Determine the speed v of the 1 kg object immediately after the collision.

c) Determine the magnitude of the angular momentum of the object about the pivot just before the collision.

d) Determine the angle .

Two identical spheres, each of mass M and negligible radius, are fastened to opposite ends of a rod of negligible mass and length 2l. This system is initially at rest with the rod horizontal, as shown above, and is free to rotate about a frictionless, horizontal axis through the center of the rod and perpendicular to the plane of the page. A bug, of mass 3M, lands gently on the sphere on the left. Assume that the size of the bug is small compared to the length of the rod. Express your answers to all parts of the question in terms

of M, l , and physical constants.


a) Determine the torque about the axis immediately after the bug lands on the sphere.

b) Determine the angular acceleration of the rod-spheres-bug system immediately after the bug lands.

The rod-spheres-bug system swings about the axis. At the instant that the rod is vertical, as

shown above, determine each of the following.

c) The angular speed of the bug.

d) The angular momentum of the system.

e) The magnitude and direction of the force that must be exerted on the bug by the sphere to keep the bug from being thrown off the sphere.

Lesson 13 : Rotational Equilibrium

A system is in rotational equilibrium if the net torque on it is zero about any axis.

= 0

Since = I

= 0

= 0 does not mean an absence of rotational motion. Object can be rotating at a constant

angular speed.

Consider the object subject to the two

forces shown to the right. Choose the

correct statement with regard to this situation.

The object is in

Example 1

____ force equilibrium but not torque equilibrium.

____ torque equilibrium but not force equilibrium.

____ both force and torque equilibrium.

____ neither force nor torque equilibrium.

Consider the object subject to the three forces shown to the

right. Choose the correct statement with regard to this situation.

The object is in

Example 2

____ force equilibrium but not torque equilibrium.

____ torque equilibrium but not force equilibrium.

____ both force and torque equilibrium.

____ neither force nor torque equilibrium.

Center of Gravity

To compute the torque due to the gravitational force on an object of mass M, we need only consider the force Mg acting

at the center of gravity of the object.

Center of gravity = center of mass if g is constant over the object.

A seesaw consisting of a uniform board of mass M and length l supports a father and daughter with masses mf and md, respectively. The support (called the fulcrum) is under the center of gravity of the board, the father is a

distance d from the center, and the daughter is a distance l /2 from the center.

Example 3

a) Determine the magnitude of the upward force n exerted by the support on the board.

b) Determine where the father should sit to balance the system.

A person holds a 50.0 N sphere in his hand. The forearm is horizontal, as shown above. The biceps muscle is attached 3.00 cm from the joint, and the sphere is 35.0 cm from the joint. Find the upward force exerted by the biceps on the

forearm and the downward force exerted by the upper arm on the forearm and acting at the joint. Neglect the weight

of the forearm.

Example 4

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Rotational Motion and Angular Momentum