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1. Coordinates of the centre of mass

x =

n

m_{i}x_{i}

å

i = 1

m_{i}

where x_{i} y_{i} and z_{i} are the x y and z co-ordinates respectively, of the i^{th} particle and m_{i} is its mass.

y =

m_{i}y_{i}

O

z =

m_{i}z_{i}

2. Position Vector of the Centre of Mass

=

m_{i}_{i}

M

where is the position vector of the i^{th} particle and M is the mass of the body.

3. Position Vector for a Rigid Body

1

ò dm

where is the position vector of an element of mass dm.

4.

Moment of inertia, I =

m_{i}r^{2}_{i}

.

5.

Radius of gyration (K) is given by the relation I = MK^{2}

\ K =

where I is the moment of inertia and M is the mass of the body.

6.

Kinetic energy of a rotating body, E =

2

Iw^{2}

where I is the moment of inertia and w is the angular velocity of the rotating body.

7.

Torque acting on a rotating body, T = Ia where a is the angular acceleration of the body.

8.

Principle of parallel axes states that, I_{0} = I_{c} + Mh^{2} where I_{0} is the M.I. about an axis through 0, I_{c} is the M.I. about an axis through the centre of mass, M is the mass of the body and h is the distance between the two parallel axes.

9.

Principle of perpendicular axes states that I_{z} = I_{x} + I_{y} where I_{x}, I_{y} and I_{z} are the moments of inertia about the X, Y and Z axes of a plane lamina such that X and Y axes lie in the plane of the lamina, and Z axis is perpendicular to the plane.

10.

M.I. of a thin uniform rod rotating about a perpendicular axis through its middle

I =

Ml ^{2}

12

,

K =

l

11.

M.I. of a thin uniform rod rotating about a perpendicular axis through one end

3

12.

M.I. of a thin uniform disc rotating about an axis passing through its centre and perpendicular to its plane,

MR^{2}

R

(Note: This formula can also be applied to a cylinder or coin)

13.

M.I. of a thin uniform disc rotating about a diameter,

4

14

Angular momentum L = Iw where I is the moment of inertia and w is the angular velocity.

15

Equations of rotational motion of a body are

where q is the angular displacement, w_{0} and w_{t} are the angular velocities at t= 0 and t =t, respectively, and a is the angular acceleration

a) w_{t} =w_{0} + at

b) q = w_{0}t +

at^{2}

c) w_{t}^{2} = w^{2}_{0} + 2 aq

16

Moment of inertia of a ring = MR^{2} R ® radius of ring

17

M.I. of a hollow sphere =

18

M.I. of a solid sphere =

5

19

Total K.E. of a rolling body =

mv ^{2 }+

For a disc: total K.E. =

mv ^{2}+

mv ^{2}=

mv ^{2}

For a ring: total K.E. =

For a sphere: total K.E.=

10

7