Rigid Body: A rigid body consists of a number of particles such that the distance between any pair of particles always remains constant.

Moment of Inertia (Rotational Inertia) I: -

1. Mass of body

2. Distribution of mass about the axis of rotation

3. Moment of inertia of a body should always be referred to as about a given axis, since it depends upon distribution of mass about that axis.

4. It does not depend upon the state of motion of rotating body. It is same whether the body is at rest, rotating slowly or rotating fast about the given axis. I=∑mr2

Rotational Kinetic Energy: -

Radius Gyration: - Radius of gyration of a body about a given axis is that distance, at which if whole of the mass of the body were concentrated, it would have same moment of inertia as that of body.

Again, Radius of gyration of a body about a given axis is defined as the square root of the mean of the squares of distances of various particles of the body from the axis of rotation.

System of mass for many particle system: -

1. Perpendicular axes theorem: -

2. Parallel axes theorem: -

Moments of Inertia of Various Objects: -

Moment of inertia of a ring about an axis passing through its center and perpendicular to its plane: -

(a) About one of its diameters: -

(b) About a tangent

Moment of inertia of a solid disc: -

(a) About an axis passing through its center and perpendicular to its plane: -

(b) About one of its diameters: -

(c) About a tangent: -

     (i) Tangent lying in the plane of disc: -

     (ii) Tangent perpendicular to the plane of disc: -

Moment of inertia of an annular disc: -

(a) About an axis passing through the center and perpendicular to the plane:-

(b) About any of its diameter: -

 (c) About a tangent: -

Moment of Inertia (I) and Angular momentum (L): -

Law of conservation of angular momentum: -The net angular momentum of an isolated system (no external torque), always remains constant.

Motion of a point mass attached to a string would over a cylinder capable of rotating about its axis of symmetry: -

Motion of a body rolling down an inclined plane without slipping: -

The maximum allowed angle for rolling without slipping.

Universal Law of Gravitation

Gravitational Force is independent of the medium. The gravitational force inside a body is zero.

Gravitational Constant

Acceleration due to Gravity

Variation with Height

Variation with Depth

for every percentage decrease in radius, with mass constant; there is 2% increase in acceleration due to gravity.

Variation with Latitude

Relation between Variation of height and depth

If acceleration due to gravity is same at a certain height and depth;

then d = 2h

Gravitational Field Intensity

Gravitational field is the field of attraction around a mass. Gravitational Field Intensity is the amount of Force experienced by an object of unit mass in a gravitational Field.

Gravitational Potential Energy

Gravitational Potential Energy is the work done in bringing a charge of unit mass from infinity to a point against the gravitational field.

Gravitational Potential

Gravitational potential is the amount of work done per unit mass in a gravitational field.

Gravitational Potential is always directed opposite to Gravitational Field.

Cases of Potential:

1.Due to point mass:

The graph will be negative, curved and parallel to V and r axes.

2.Inside a Solid Sphere:

The graph is negative, parabolic.

3.Outside a Solid Sphere:

4.Due to a Spherical Shell:

5.On the Axis of a Ring:

Kepler's Laws

Law of Orbits Kepler's Law of Orbits states that all planets revolve around the sun in elliptical orbits with sun as the focus.

Law of Areas

Kepler's Law of Areas states that a planet sweeps equal Areas in equal intervals of time around the sun.

Law of Periods

Kepler's Law of Periods states that the square of Time Period is directly proportional to the cube of Distance.

Critical Velocity

Time Period

Binding Energy

Escape Velocity

The escape velocity remains independent of angle of projection. Escape Velocity for an object on earth is 11.2 km/s.

GRAVITATIONAL POTENTIAL & FIELD DUE TO VARIOUS OBJECTS

Uniform Circular Motion: - Circular motion is said to the uniform if the speed of the particle (along the circular path) remains constant.

Angular Displacement: -

Angular Velocity: -

Relation between linear velocity (v) and angular velocity (ω) is as 𝑣 = 𝑟𝜔

Angular Acceleration: -

Relation between linear velocity (v) and angular velocity (ω) as as 𝑎=𝑟𝛼

Centripetal Force: -

The force, acting along the radius towards the centre, which is essential to keep the body moving in a circle with uniform speed is called centripetal force. It acts always along the radius towards the centre. A centripetal force does no work.

Centrifugal Force: -

Centrifugal force is the fictitious force which acts on a body, rotating with uniform velocity in a circle, along the radius away from the centre. Magnitude of centrifugal force is,

Centripetal and centrifugal forces are equal in magnitude and opposite in direction. They cannot be termed as action and reaction since action and reaction never act on same body.

Banking of Roads: -

Bending of Cyclist: -

(a) Velocity of the cyclist: -Greater the velocity, greater is his angle of inclination with the vertical. 

(b) Radius of curvature: - Smaller the radius, greater is the angle with the vertical.

Time period of conical pendulum: -

Motion in a vertical circle