RotationalMotion

ASUPAN, NhaninaFELICIO, CarmelleLICONG, CindyLISONDRA, DeniseCANANGCA-AN, EricTINIO, Dime

In science, angles are often

measured in radians (rad).

When the arc length s is equal to the radius r, the angle θ swept by r is equal to 1 rad.

r

s

Any angle θ measured in radians is defined as,

Angles in Radians

360

2

or

180

)(deg180

)( reesrad

Convert 110° into radians.

rad

18

1111

18110

180

In translational kinematics, the position of the body is defined as the displacement from a certain reference point.

X

In rotational kinematics, the position of a point on a rotating body is defined by the angular displacement from some reference line that connects this point to the axis of rotation.

Angular Displaceme

nt

The body has rotated

through the angular

displacement if the

point which was

originally at P1 is now at

the point P2.

0 1P

s2P

r

s

This angular

displacement is

a vector that is

perpendicular

to the plane of

the motion.

The magnitude of

this angular

displacement

is the angle θ

itself.

If it is positive, the rotation of the body is counterclockwise and the angular displacement vector points upward.

If it is negative, the rotation is clockwise and the vector points downward.

r

s

which means that

s r

A boy rides on a merry-go-round at a distance of 1.25 m from the center. If the boy moves through an arc length of 2.25 m, through what angular displacement does he move?

Given:

sr = 1.25 m

= 2.25 mFind:

Solution:

r

s

m

m

25.1

25.2

rad8.1

Angular Velocity

It is denoted by the lowercase of the Greek letter omega (ω) and is defined as the ratio of the angular displacement to the time interval, the time it takes an object to undergo that displacement.

rt

s

tave

1

In the limit that the time interval approaches zero, becomes the instantaneous speed, v.

t

s

Angular velocity is expressed in:

• radians per second (rad/s)• revolutions per second (rps)• revolutions per minute (rpm)

1 rev = 2π rad

Linear velocity of a point on the rotating

body and angular velocity of the body are

linked by the equation, s = rθ divided by t.t

r

t

s but we

know that v

t

s an

d t

And so,

rv The farther the distance r that

the body is from the axis of rotation, the greater is its linear

or tangential velocity.

A merry-go-round is rotating at a constant angular velocity of 5.4 rad/s. What is the frequency of the merry-go-round in revolutions per minute?

Given:

sradsrad /4.5)/(

Find:)(rpm

Solution: 1 rev = 6.28 rad/s

rpmm

s

rad

rev

s

rad6.51

1

60

28.6

1

1

4.5

Angular Accelerati

onAngular acceleration occurs when angular velocity

changes with time. It is denoted by the symbol alpha, α.t

ave

if

ifave

tt

There is a

connection

between the

instantaneous

tangential

acceleration (linear

motion) and

angular

acceleration

(rotational motion). rta

A figure skater

begins spinning

counterclockwise

at an angular

speed of 5.0π

rad/s. She slowly

pulls her arms

inward and finally

spins at 9.0π rad/s

for 3.0 s. What is

her average

angular

acceleration

during this time

interval?

Given:

st

srad

srad

i

f

0.3

/0.5

/0.9

Find:

ave

Solution:

t

ifave

s

sradsrad

0.3

/0.5/0.9

sradave /19.4 2

Torque

To make an object start rotating, a force is needed; the position and direction of the force matter as well.The perpendicular distance from the axis of the rotation to the line along which the force acts is called the lever arm.

Here, the lever arm for FA is the distance from the knob to the hinge; the lever arm for FD is zero; and the lever arm for FC is as shown.

The torque is defined as:

sinrF

Rotational Inertia An object rotating

about an axis tends to

continue rotating about

that axis unless an

unbalanced external

torque (the quantity

measuring how

effectively a force

causes rotation) tries

to stop it.

The resistance of an object to changes in its rotational motion is called rotational inertia which is also termed as

moment of inertia.

Torque is required to change the rotational state of motion of an object.

If there is

no net torque, a

rotating object

continues to

rotate at a

constant

velocity.

Rotational inertia depends on the distribution of the mass.

A large mass which is at a greater distance from the axis of rotation, has a greater moment of inertia than a large mass which is near

the axis of rotation.

The larger the moment of inertia of a body, the more difficult it is to put that body into

rotational motion or, the larger the moment of inertia of a body, the more difficult it is to stop

its rotational motion.

The moment of inertia of a single mass m, rotating about an axis, a

distance r from m, we haveI = mr2

The unit for the moment of inertia is kg•m2 and has no special name.

Find the moment of inertia of a solid cylinder of mass 3.0 kg and radius 0.50 m, which is free to rotate about an axis

through its center.

Given:m = 3.0 kgr = 0.50 m

Find:

I

Find:

Solution:

I = ½mr2

= ½(3.0 kg)(0.50 m)2

= ½(3.0 kg)(0.25 m2)I = 0.38 kg • m2

First law for rotational motion:A body in motion at a constant

angular velocity will continue in motion at that same angular velocity, unless acted upon by some unbalanced external torque.

Second law for rotational motion:When an unbalance external torque acts

on a body with moment of inertia I, it gives that body an angular acceleration α, which is directly proportional to the torque τ and inversely to the moment of inertia.

Third law of rotational motion:If body A and body B have the same

axis of rotation, and if body A exerts a torque on body B, then body B exerts an equal but opposite torque on body A.

I

Angular Momentu

m

If the rotational

equivalent of force

is torque, which is the

moment of the force,

the rotational

equivalent of linear

momentum (p) is

angular momentum (L),

which is the moment of

momentum.

Product of the moment of inertia of a rotating body and its angular velocity

IL Unit is kg•m2/s

If an object is small compared with the radial distance to its axis of rotation, the angular momentum is equal to the magnitude of its linear momentum mv, multiplied by the radial distance r.

rvmL

vmrprL

What is the angular momentum of a 250-g

stone being whirled by a slingshot at a

tangential velocity of 6 m/s, if the length of the

slingshot is 30 cm?

Given:

m=250 g=0.25kg

v = 6 m/s

r = 30 cm=0.30 m

Find:L

Solution:

rvmL )30.0)(/6)(25.0( msmkg

smkgL /45.0 2

Conservation of Angular

MomentumLaw of conservation of angular momentum states that:

in the absence of an unbalanced external torque, the angular momentum of a system remains constant.

Key

Concept

s