## Important Information

When a body moves along a circular path with uniform speed, it is said to perform a uniform circular motion (U.C.M.).
In U.C.M.
1.       Angular speed of a particle,
Where θ is the angular displacement in time‘t’. T is the period of revolution of the particle performing a U.C.M. and n is the frequency of revolution i.e. no .of revolutions completed by the particle per second. S.I. unit of ω is rad/s.

Angular Velocity (ω): It is the rate of change of angular displacement w.r.t. time.

2.       The linear speed of a particle v = rω where r is the radius of the circle.
3.         S.I. unit of α is rad/sec2
Linear acceleration (a) = r α.
4.       In U.C.M. the acceleration is always directed along the radius, towards the centre of the circle. It is called radial or centripetal acceleration. The magnitude of C.P. acceleration is . If m is the mass of the body performing U.C.M., then the centripetal force acting upon it is . It is also given by

5.       If the angular acceleration α is constant then for accelerated circular motion,
(i)
(ii)
(iii)
Where ω0 is the angular speed of the particle at time t = 0 and ω is the angular speed of the particle after time‘t’.
6.       Banking of Roads: For a vehicle moving along a circular track of radius r, banked at an angle θ, the maximum speed limit for the safety of the vehicle is given by

V is also known as critical speed. In the case of a train, the elevation of the outer rail above the inner rail is given by
h = ℓ sin θ.
Where  is the distance between the rails.
For a metre gauge railway,
7.       For a vehicle moving along a horizontal curve of radius r, (unbanked circular road), the C.P. force is provided by the friction. The maximum safe speed (v) is calculated by using the relation

Where μ is the coefficient of friction between the tyres of the car and the surface of the road.
8.      A cyclist, moving with a speed v, along a horizontal curved road of radius r, has to   lean inwards to keep his balance. His angle of inclination (θ) with the vertical is given by

9.
10.
11.     Direction of are given by right hand thumb rule.
are axial vectors
12.     When a particle performs a non uniform circular motion it has a centripetal acceleration  and a tangential acceleration.  directed along the tangent
13.     As  and  are at right angles, the net acceleration

14.     For a U.C.M. aT = 0 or α = 0. No work is done in U.C.M.
15.    Motion in a Vertical circle:

When a body moves in a vertical circle,
·         The tension (if string is used) or normal reaction at any point P is given by

·         Tension at the lowest point (A) is maximum.

·         Tension at the highest point (B) is minimum.

·         The minimum horizontal velocity that should be given to the body at the bottom (A) to complete the vertical loop is given by

·         Minimum speed required at
·         Velocity at C or D =
·         Tension at C or D (ends of the horizontal diameter)

·         K.E. at A – K.E. at B = 2 mgr
·         TA – TB = 6 mg
16.     For looping a loop of radius r, the minimum height from which the body should be released is given by
17.     The expressions, are used in the following cases:
(i)       Bucket full of water rotated in a vertical circle.
(ii)      Motor cyclist riding in a vertical circle in a hollow sphere.
18.     Conical Pendulum:
(i)       If T is the tension in the string, then

(ii)
Where T is the period of the conical pendulum.

(iii)     The period of a conical pendulum is less than the period of a simple pendulum of the same length.
19.     Useful Points:
(i)       Newton’s laws of motion are valid in an inertial frame of reference. In this frame, we consider the real forces.
(ii)      A frame of reference which is moving with an acceleration w.r.t. the system of fixed stars is called a non-inertial frame or accelerated frame of reference. In this frame, Newton’s laws of motion are not valid. To apply Newton’s laws of motion to situations in non inertial frames, we consider the fictitious or pseudo forces. Centrifugal force is a pseudo force.
(iii)     Centripetal force and centrifugal force do not form an action-reaction pair.
(iv)     For a speed breaker,

(v)      In banked roads, the horizontal components R sin θ provides the necessary centripetal force. While the vertical component R cos θ balances the weight (mg) of the car.

## Important Information

When a body moves along a circular path with uniform speed, it is said to perform a uniform circular motion (U.C.M.).
In U.C.M.
1.       Angular speed of a particle,
Where θ is the angular displacement in time‘t’. T is the period of revolution of the particle performing a U.C.M. and n is the frequency of revolution i.e. no .of revolutions completed by the particle per second. S.I. unit of ω is rad/s.

Angular Velocity (ω): It is the rate of change of angular displacement w.r.t. time.

2.       The linear speed of a particle v = rω where r is the radius of the circle.
3.         S.I. unit of α is rad/sec2
Linear acceleration (a) = r α.
4.       In U.C.M. the acceleration is always directed along the radius, towards the centre of the circle. It is called radial or centripetal acceleration. The magnitude of C.P. acceleration is . If m is the mass of the body performing U.C.M., then the centripetal force acting upon it is . It is also given by

5.       If the angular acceleration α is constant then for accelerated circular motion,
(i)
(ii)
(iii)
Where ω0 is the angular speed of the particle at time t = 0 and ω is the angular speed of the particle after time‘t’.
6.       Banking of Roads: For a vehicle moving along a circular track of radius r, banked at an angle θ, the maximum speed limit for the safety of the vehicle is given by

V is also known as critical speed. In the case of a train, the elevation of the outer rail above the inner rail is given by
h = ℓ sin θ.
Where  is the distance between the rails.
For a metre gauge railway,
7.       For a vehicle moving along a horizontal curve of radius r, (unbanked circular road), the C.P. force is provided by the friction. The maximum safe speed (v) is calculated by using the relation

Where μ is the coefficient of friction between the tyres of the car and the surface of the road.
8.      A cyclist, moving with a speed v, along a horizontal curved road of radius r, has to   lean inwards to keep his balance. His angle of inclination (θ) with the vertical is given by

9.
10.
11.     Direction of are given by right hand thumb rule.
are axial vectors
12.     When a particle performs a non uniform circular motion it has a centripetal acceleration  and a tangential acceleration.  directed along the tangent
13.     As  and  are at right angles, the net acceleration

14.     For a U.C.M. aT = 0 or α = 0. No work is done in U.C.M.
15.    Motion in a Vertical circle:

When a body moves in a vertical circle,
·         The tension (if string is used) or normal reaction at any point P is given by

·         Tension at the lowest point (A) is maximum.

·         Tension at the highest point (B) is minimum.

·         The minimum horizontal velocity that should be given to the body at the bottom (A) to complete the vertical loop is given by

·         Minimum speed required at
·         Velocity at C or D =
·         Tension at C or D (ends of the horizontal diameter)

·         K.E. at A – K.E. at B = 2 mgr
·         TA – TB = 6 mg
16.     For looping a loop of radius r, the minimum height from which the body should be released is given by
17.     The expressions, are used in the following cases:
(i)       Bucket full of water rotated in a vertical circle.
(ii)      Motor cyclist riding in a vertical circle in a hollow sphere.
18.     Conical Pendulum:
(i)       If T is the tension in the string, then

(ii)
Where T is the period of the conical pendulum.

(iii)     The period of a conical pendulum is less than the period of a simple pendulum of the same length.
19.     Useful Points:
(i)       Newton’s laws of motion are valid in an inertial frame of reference. In this frame, we consider the real forces.
(ii)      A frame of reference which is moving with an acceleration w.r.t. the system of fixed stars is called a non-inertial frame or accelerated frame of reference. In this frame, Newton’s laws of motion are not valid. To apply Newton’s laws of motion to situations in non inertial frames, we consider the fictitious or pseudo forces. Centrifugal force is a pseudo force.
(iii)     Centripetal force and centrifugal force do not form an action-reaction pair.
(iv)     For a speed breaker,

(v)      In banked roads, the horizontal components R sin θ provides the necessary centripetal force. While the vertical component R cos θ balances the weight (mg) of the car.

## HSC Books

To achieve perfection HSC BOARD EXAMS these are the top scorer books :-

Physics :-
Government Sanctioned Text Book (G.T.B)

Navneet Publications – The latest edition has completely solution of the unsolved part of the Government Sanctioned Textbook (G.T.B). Moreover they have provided with errors in the questions and answers of G.T.B

Sheth Publications – Problems in Physics [For Numerical Problems]
Uttams Papers With Solutions – It gives complete solutions of Latest Previous Years Board Papers . Also It has Specimen Papers too.
Navneet Log Table Book – It is required to solved difficult problems which has lengthy calculations. Logarithm tables and many tables are requires for preparation of HSC

Chemistry :-
Government Sanctioned Text Book (G.T.B)

Navneet Publications – The latest edition has completely solution of the unsolved part of the Government Sanctioned Textbook (G.T.B). Moreover they have provided with errors in the questions and answers of G.T.B
Sheth Publications – Problems in Chemistry
Uttams Papers With Solutions – It gives complete solutions of Latest Previous Years Board Papers . Also It has Specimen Papers too.
Navneet Log Table Book – It is required to solved difficult problems which has lengthy calculations. Logarithm tables and many tables are requires for preparation of HSC
Mathematics :-
Government Sanctioned Text Book (G.T.B)
Navneet Publications – The latest edition has completely solution of the unsolved part of the Government Sanctioned Textbook (G.T.B). Moreover they have provided with errors in the questions and answers of G.T.B
Neelkanth Publications
Uttams Papers With Solutions – It gives complete solutions of Latest Previous Years Board Papers . Also It has Specimen Papers too.
Biology :-
Government Sanctioned Text Book (G.T.B)
Neelkanth Publications – Steps In Biology
Narendra Prakashan  – A Text Book Of Biology
Modern Publishers – Textbook of Modern’s ABC of  Biology
Nirali Prakashan – Fundamental Biology
Balaji Publications – Scholars Biology
Sheth Prakashan – Biological Science
Vision Publicationsm- Biology
Nirmitee Publications – Nirmitee Biology
Uttams Papers With Solutions – It gives complete solutions of Latest Previous Years Board Papers . Also It has Specimen Papers too.
Studying these books will totally complete your preparation for HSC. Buying a G.T.B is a must

## >HSC Books

>

To achieve perfection HSC BOARD EXAMS these are the top scorer books :-

Physics :-
Government Sanctioned Text Book (G.T.B)

Navneet Publications – The latest edition has completely solution of the unsolved part of the Government Sanctioned Textbook (G.T.B). Moreover they have provided with errors in the questions and answers of G.T.B

Sheth Publications – Problems in Physics [For Numerical Problems]
Uttams Papers With Solutions – It gives complete solutions of Latest Previous Years Board Papers . Also It has Specimen Papers too.
Navneet Log Table Book – It is required to solved difficult problems which has lengthy calculations. Logarithm tables and many tables are requires for preparation of HSC

Chemistry :-

Government Sanctioned Text Book (G.T.B)
Navneet Publications – The latest edition has completely solution of the unsolved part of the Government Sanctioned Textbook (G.T.B). Moreover they have provided with errors in the questions and answers of G.T.B
Sheth Publications – Problems in Physics
Uttams Papers With Solutions – It gives complete solutions of Latest Previous Years Board Papers . Also It has Specimen Papers too.
Navneet Log Table Book – It is required to solved difficult problems which has lengthy calculations. Logarithm tables and many tables are requires for preparation of HSC
Mathematics :-
Government Sanctioned Text Book (G.T.B)
Navneet Publications – The latest edition has completely solution of the unsolved part of the Government Sanctioned Textbook (G.T.B). Moreover they have provided with errors in the questions and answers of G.T.B
Neelkanth Publications
Uttams Papers With Solutions – It gives complete solutions of Latest Previous Years Board Papers . Also It has Specimen Papers too.
Biology :-
Government Sanctioned Text Book (G.T.B)
Neelkanth Publications – Steps In Biology
Narendra Prakashan  – A Text Book Of Biology
Modern Publishers – Textbook of Modern’s ABC of  Biology
Nirali Prakashan – Fundamental Biology
Balaji Publications – Scholars Biology
Sheth Prakashan – Biological Science
Vision Publicationsm- Biology
Nirmitee Publications – Nirmitee Biology
Uttams Papers With Solutions – It gives complete solutions of Latest Previous Years Board Papers . Also It has Specimen Papers too.
Studying these books will totally complete your preparation for HSC. Buying a G.T.B is a must

## MCQ Books For MHT-CET

For Offline Study it is necessary to practice some of the books mentioned below :-

Physics :-
Marvel Publications
Nikita Publications ( Famously known  Pradnya )
Target Publications
Navneet Publications
Vidyabhartee Publications
Uttam Publications

Chemistry :-
Himalaya Publishing House
Sheth Publications  ( Famously known Scholars )
Marvel Publications
Nikita Publications ( Famously known  Pradnya )
Vidyabhartee Publications

Uttam Publications
Mathematics :-
Nikita Publications ( Famously known  Pradnya )
Uttam Publications
Marvel Publications
Vidyabhartee Publications
Navneet Publications
Sheth Publications  ( Famously known Scholars )
Biology:-
Marvel Publications
Nikita Publications ( Famously known  Pradnya )
Target Publications
Navneet Publications
Vidyabhartee Publications
Uttam Publications
Himalaya Publishing House
Sheth Publications  ( Famously known Scholars )

You should try solve at least 2-3 completely for a better score in MHT-CET. These are not sanctioned books but these are relevant books for strengthening your preparation