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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
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.
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.
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