XII Physics Rotational Motion
Short Description
An important concept from Mechanics section of Physics, dealing with fundamentals of rotational motion of rigid bodies....
Description
Rotational Motion Prof. Sameer Sawarkar
Contents • • • • • • • • • •
Rigid Body Rotational Motion Cause & Consequence Moment of Inertia Kinetic Energy Angular Momentum Conservation Principle Parallel & Perpendicular Axes Theorems Radius of Gyration Rolling Motion Prof. Sameer Sawarkar
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Rigid Body: A body which does not undergo any appreciable deformation under the action of external forces, i.e. the intermolecular distances remain constant when subjected to external forces.
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Rigid Body: A body which does not undergo any appreciable deformation under the action of external forces, i.e. the intermolecular distances remain constant when subjected to external forces.
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B A C
Rigid Body: A body which does not undergo any appreciable deformation under the action of external forces, i.e. the intermolecular distances remain constant when subjected to external forces.
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B A C
Rigid Body: A body which does not undergo any appreciable deformation under the action of external forces, i.e. the intermolecular distances remain constant when subjected to external forces.
No body is truly rigid nor elastic or plastic. The state is always referred to as rigid/elastic/plastic in context with the magnitude and range of external forces.
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,
Rotational Motion: A body is said to be purely rotating when all the constituents of the body are moving in circular motions, with centers of their paths lying on a fixed straight line called axis of rotation.
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,
A
Rotational Motion: A body is said to be purely rotating when all the constituents of the body are moving in circular motions, with centers of their B paths lying on a fixed straight line called axis of rotation.
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,
The axis of rotation may lie within the body or without the body
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Examples: Motion of table/ceiling fan blades Motion of Turbine rotor Motion of gear wheels Spinning Motion of planets Opening of doors/window panels Motion of hands of clock etc.
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CAUSE & CONSEQUENCE in Rotational Motion
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Force produces translation i.e. linear acceleration, ‘a’
Couple Moment produces rotation i.e. angular acceleration, ‘’
F
d F a
F
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• Rigid body subjected to
torque • Rotating about a fixed axis with angular acceleration
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• Consider ‘n’ particles of the
body in circular motion with
R1 2
1
masses m1, m2, … , mn.
R2
• R1, R2, … , Rn are the radii.
Rn n
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a1
• Using aT = R
R1 R2 an
a2
Rn
• Linear tangential accelerations of constituents; a1, a2, … , an
a1 = R 1 a2 = R 2 ………… ………… an = R n
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(1)
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a1 = R1 , a2 = R2 , … , an = Rn _(1) • Using Newton’s II Law; F = ma
a1
R1
F2 R2
an
a2
F1
Rn
F1 = m1a1 = m1R1 F2 = m2a2 = m2R2 …………………… …………………… Fn = m nan = m nR n
(2)
Fn
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F1 = m1R1 , F2 = m2R2 , … … … Fn = m nR n _(2)
1
F2
• Using definition of torque; = d*F 1 = R1F1 = R1(m1R1)
a1 R1
R2 an
a2
F1
Rn n Fn
1 = m1R12 2 = m2R22 …………… …………… n = mnRn2
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(3)
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1 = m1R12, 2 = m2R22, … … … n = mnRn2 _(3)
1
F2
• Sum of all individual constituent torques must be equal to the externally applied original torque.
a1 R1
R2 an
a2
F1
Rn n Fn
= 1 + 2 + … + n = m1R12 + m2R22 + … + mnRn2
= (miRi2) i = 1, 2, … , n.
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Translational Motion
Rotational Motion
F
F = m*a
a
= (miRi2)*
m miRi2 Quantity miRi2 is called as Moment of Inertia of rotating body about the defined axis of rotation. Prof. Sameer Sawarkar
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Moment of Inertia (miRi2 ) about a given axis of rotation is defined as the sum of product of mass of each constituent and square of its distance from the axis of rotation. Moment of Inertia (abbreviated as MI, denoted by I) represents inertia in rotational motion i.e. reluctance of a rigid body to undergo angular acceleration. Larger the MI, more difficult it is to change the state of the body (to accelerate/decelerate). Prof. Sameer Sawarkar
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With regular geometric boundaries, where division in discrete shapes is possible, MI is expressed as; I = miRi2
With irregular geometric boundaries, where division in elemental shapes is necessary, MI is expressed as; I = R2 dm
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• I = (mi, Ri2) • MI represents mass distribution of the rotating rigid body. • Rotational motion depends not just upon total mass but upon mass distribution!!
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Moment of Inertia I = miRi2 or I = R2 dm Unit: kg-m2 Dimensions: [L2 M1 T0]
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KINETIC ENERGY in Rotational Motion
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• Rigid body rotating about a
fixed axis with angular velocity
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• Consider ‘n’ particles of the
body in circular motion with R1 2
1
masses m1, m2, … , mn.
R2
• R1, R2, … , Rn are the radii.
Rn n
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V1
• Using V = R
R1 R2 Vn
V2
Rn
• Linear tangential velocities of constituents; V1, V2, … , Vn
V1 = R1 V2 = R2 ………… ………… Vn = Rn
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V1 = R1 , V2 = R2 , … , Vn = Rn _(1)
V1 R1
R2 Vn
V2
Rn
• KE = ½ mV2 = ½ m(R22) of each constituent. U1 = ½ m1R122 U2 = ½ m2R222 ……………… ……………… Un = ½ mnRn22
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V1
• Total KE of the rotating rigid body; U = U1 + U2 + … + Un
R1 R2 Vn
V2
Rn
U1 = ½ m1R122, U2 = ½ m2R222, … … Un = ½ mnRn22 _(2)
U = ½ m1R122 + ½ m2R222 + … + ½ mnRn22 U = ½ (miRi2)2
U = ½ I2
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ANGULAR MOMENTUM in Rotational Motion
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Angular Momentum: Property possessed by a rotating body by virtue of its
angular velocity.
L V, mV
Defined as; moment of linear momentum.
R
i.e. L = R*P = R*(mV)
m
Like linear momentum, angular momentum is a vector. Unit: kg-m2/s, Dimensions: [L2 M1 T-1]
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Vector relation between linear momentum and angular momentum: From scalar relation; L = R*P and using Right-hand rule;
L
P
L RP
R m R
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• Rigid body rotating about a
fixed axis with angular velocity
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• Consider ‘n’ particles of the
body in circular motion with R1 2
1
masses m1, m2, … , mn.
R2
• R1, R2, … , Rn are the radii.
Rn n
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V1
• Using V = R
R1 R2 Vn
V2
Rn
• Linear tangential velocities of constituents; V1, V2, … , Vn
V1 = R1 V2 = R2 ………… ………… Vn = Rn
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(1)
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V1 = R1 , V2 = R2 , … , Vn = Rn _(1)
, L
V1 R1
R2 Vn
V2
Rn
• Linear momentum P = mV for each constituent. • Angular momentum for each constituent; L = R*P = RmV = Rm(R) = mR2 L1 = m1R12 L2 = m2R22 ……………… ……………… Ln = mnRn2 Prof. Sameer Sawarkar
(2)
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, L
V1 R1
R2 Vn
V2
Rn
L1 = m1R12, L2 = m2R22, … … Ln = mnRn2
_(2)
• Total angular momentum of the rotating rigid body; L = Li, i = 1, 2, … , n. L = m1R12 + m2R22 + … + mnRn2 L = (miRi2)
L = I
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PRINCIPLE OF CONSERVATION OF ANGULAR MOMENTUM
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d I I dt d d I L dt dt If
0,
then
L
is constant.
In absence of an external torque, the angular momentum of the system remains constant
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Applications of Principle of Conservation of Angular Momentum
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PARALLEL AXES THEOREM PERPENDICULAR AXES THEOREM
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IG
• Rigid body with mass M
• Purely rotating about an axis through C.M. • MI = IG (known)
G
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IP
IG
• It is desired that MI
about a parallel axis at a distance ‘h’ through P i.e. IP be found.
G
P h
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IP
IG
• Assume elemental mass Q
dm at an arbitrary point Q.
PP
G h
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IP
IG
• Construction Q (dm)
P
h
G
S
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IP
IG
IG = QG2dm 2dm I = QP P Q (dm)
QP2 = PS2 + SQ2 = (PG + GS)2 + SQ2 P
h
G
S
= PG2 + 2PG*GS + (GS2 + SQ2) QP2
Prof. Sameer Sawarkar
= PG2 + 2PG*GS + QG2
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QP2 = PG2 + 2PG*GS + QG2 IP
Multiplying throughout by dm and
IG
integrating; Q (dm)
QP2 dm = PG2dm + 2PG GSdm + QG2dm
P
h
G
S
QP2 dm = IP QG2dm = IG PG2dm = PG2dm = Mh2
GSdm = 0, G being the center of mass of the body. Prof. Sameer Sawarkar
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Substituting; IP
IG
IP = IG + Mh2 Q (dm)
MI of a rigid body about any
axis is equal to sum of its MI about a parallel axis through P
h
G
S
center of mass and product of mass of body and square of the distance between two parallel axes.
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• Rigid with mass M
• Laminar body (thickness very small compared to surface area)
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Z
• System of 3 mutually perpendicular axes through any point O. O
• X and Y in the plane of
Y
the lamina, Z being
perpendicular to the
X
plane.
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Z
• Imagine elemental mass dm at a distance ‘r’ from Z axis. O r X
Y dm
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Z
• Moment of inertia of IZ
the lamina @ Z axis; IZ = r2dm
O r X
Y dm
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Z
• Construction – IZ
perpendiculars on X and Y axes from elemental
O
mass. r
X
y
x
Y
dm
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Z
• MI of lamina about X IZ
axis; IX = y2dm
O IX X
• MI of lamina about Y r
y
x dm
Y IY
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axis;
IX = x2dm
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Z
r2 = x 2 + y 2 Multiplying throughout by
IZ
dm and integrating;
r2dm = x2dm + y2dm
O IX X
r y
x dm
Y IY
Substituting;
IZ = IX + IY
Moment of inertia of a lamina about an axis perpendicular to its plane is equal to sum of its moments of inertia about two mutually perpendicular axes in the plane of lamina and concurrent with that axis. Prof. Sameer Sawarkar
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RADIUS OF GYRATION
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Radius of Gyration (K) w.r.t. the given axis of rotation is the theoretical distance at which, when entire mass of the body is assumed to be concentrated, gives same MI (of idealized point mass system) as that of the original rigid body. If MK2 = R2dm, then K is the radius of gyration. I = R2dm
I = MK2
K
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M
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IG = ½MR2
IG = MK2
K
REAL SYSTEMS IG = 2MR2/5
M
MK2 = ½MR2
K = R/2
IDEALIZED SYSTEMS
IG = MK2
MK2 = 2MR2/5 K
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M
K = R*(2/5)
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Thank You!
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