MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Rotational Motion Formulae Page 1

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1. Coordinates of the centre of mass

x =

 n mixi å i = 1

 n mi å i = 1

where xi  yi and zi are the x  y and z co-ordinates
respectively, of the ith particle and m
i is its mass.

y =

 n miyi å i = 1

 n mi å i = 1

O

z =

 n mizi å i = 1

 n mi å i = 1

2. Position Vector of the Centre of Mass

=

 n mii å i = 1

M

where is the position vector of the ith particle
and M is the mass of the body.

3. Position Vector for a Rigid Body

=

 1 M

ò dm

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

4.

Moment of inertia, I =

 n mir2i å i = 1

.

5.

Radius of gyration (K) is given by the relation I = MK2

 \ K =

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

6.

Kinetic energy of a rotating body, E =

 1 2

Iw2

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, I0 = Ic + Mh2 where I0 is the M.I. about an axis through 0, Ic 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 Iz = Ix + Iy where Ix, Iy and Iz 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

I =

 Ml 2 3

,

K =

 l

12.

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

I =

 MR2 2

,

K =

 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,

I =

 MR2 4

,

K =

 R 2

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, w0 and wt are the angular velocities at t= 0 and t =t, respectively, and a is the angular acceleration

a) wt =w0 + at

b) q = w0t +

 1 2

at2

c) wt2 = w20 + 2 aq

16

 Moment of inertia of a ring = MR2              R ® radius of ring

17

M.I. of a hollow sphere =

 2 3

MR2

18

M.I. of a solid sphere =

 2 5

MR2

19

Total K.E. of a rolling body =

 1 2

mv 2 +

 1 2

Iw2

For a disc: total K.E. =

 1 2

mv 2+

 1 4

mv 2=

 3 4

mv 2

For a ring: total K.E. =

 1 2

mv 2+

 1 2

mv 2=

mv 2

For a sphere: total K.E.=

 1 2

mv 2+

 2 10

mv 2=

 7 10

mv 2

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