Class 11 physics --Mechanical properties of solids

 Mechanical properties of solids

Lecture 1 

What is elasticity and plasticity define elastic limit

Lecture 2

What is stress its formula and definition

Lecture 3

Numerical on stress

Lecture 4

What is strain

Numerical on strain

Modulus of Elasticity

Waves Sound Beats

Waves 

Sound 

It is a form of energy which makes us hear

Sound is a longitudinal wave

Wave 

It is a form of disturbance which carries energy from one point to another waves are of two types

1) Longitudinal

2) Transverse

Longitudinal wave 

A wave in which particles of the medium vibrate back and forth in the same direction in which the wave is moving is called longitudinal wave eg sound wave

Longitudinal wave consist of compression and rarefaction

Compression -- It is that part of longitudinal waves in which the particles of medium are closer to one another then they normally are It results in reduction in the volume of the medium

Rarefraction--  where fraction is that part of longitudinal wave in which particles of medium are farther apart than the normal there is increase in volume of the medium

Transverse wave 

Abhi in which the particles of the medium vibrate up and down at right angles to the direction in which wave is moving is called a transverse wave

Eg light wave 

Transverse wave consist of crest and trough

Crest -- the elevation in the transverse wave is called crossed that is crust is that part of transverse waves which is above the line of zero disturbance of the medium

Trough-- the depression in the transverse wave is called rough trust is that part of transverse wave which is below the line of zero disturbance

Watch the lecture 

Characteristics of a sound wave ---

Sound wave can be described by five characteristics

Wavelength -- the minimum distance in which a sound wave repeats itself is called its wavelength

 in other words the distance between the centre of two consecutive compressions or two consecutive their fractions is called its wavelength unit of the wavelength is m it is denoted by lambda

Amplitude -- the maximum displacement of a particle of the medium from their original the undisturbed positions when a wave passes through the medium is called amplitude of the wave 

It is denoted by A Its SI unit is m 

Frequency --  The number of of vibrations per second is called frequency SI unit of frequency is hertzs 

Time period -- The time taken to complete one vibration is called time period its SI unit is second It is denoted by T 

Speed wave --  distance travelled by a wave in one second is called velocity of the wave the SI unit for the speed wave is metre per second

Relation ships

Relationship between frequency and time period

Frequency = 1/Time period

Relationship between velocity frequency and wavelength

Velocity = frequency × wavelength

Beats 

 Mutiple choice questions

Ncert Solutions for physics -Gravitation

 NCERT Solutions for Gravitation

Important Points to Remember

Gravitational force between two point masses

F=GMm/r^2

Gravitational field strength

Navigational field distance at a point in gravitational field is defined as gravitational force per unit mass

E=F/m

Gravitational potential

Gravitational potential at a point in a gravitational field is defined as negative of work done by gravitational force in moving a unit mass from infinity to that point

V=GM/r

Gravitational Potential energy

Rotational potential energy is the negative of work done by gravitational forces in making the system from infinite separation to the present position

U=-GMm/r

Escape velocity

What is the velocity at which an object is thrown so that it can cross out the gravitational attraction of earth and reaches into the vacuum

Escape Velocity= √(2gR)

                           = √(2GM/R)=11.2 km/s

Orbital Velocity

The velocity with which an object is revolving around a planet in space

Orbital velocity= √GM/r

Kepler's Law

He gave three empirical laws which describes the motion of planets

First law

Each planet moves in an elliptical orbit with the sun at one focus of the ellipse

Second Law

The radius vector drawn from the sun to the planet sweeps out equal areas in equal interval of time that is aerial velocity is constant

dA/dt=L/2m= constant

Where L is angular momentum and m is the mass of the planet

Third Law

T^2=r^3

Square of the period of revolution of a planet around the sun is always proportional to the cube of the distance of a planet from the sun

Recorded Lectures

Newtons Law of Gravitation

 Topic discussed in given video

 Newton's law of gravitation  along with  question

Acceleration due to gravity

Relation between acceleration due to gravity and gravitational constant G

Difference between Mass and weight

Lecture given below explain the relationship between mass and weight what is the difference between mass and weight how the weight of a body is affected when it is on the earth and when it is on the moon

Multiple choice questions

1) The radius of the earth where to shrink by 1% its mass remaining the same that solution to gravity on the earth surface

a) increase

b) decrease

c) remains same

d) zero 

Ans (a)

2) If g is the acceleration due to gravity on the Earth's surface the gain in the potential energy of an object of mass m is raised from the surface of the earth to height equal to the radius R of the earth

a) 1/2mgR

b) 2mgR

c) mgR

d) 1/4 mgR

Ans (a)

3) A  satellite is moving with a constant speed v in a circular orbit around the earth an object of mass m is projected from the satellite such that it just escape from the gravitational pull of the earth at the time of its ejection the kinetic energy of the object is

a)1/2 mv^2

b) mv^2

c) 3/4 mv^2

d) mv^2

Ans (d)

4) A simple pendulum has a time period T1 when on the earth surface and T2 when taken to a height h above the Earth surface where R is the radius of the earth the value of T2/T1 is

a) 1

b)√2

c) 4

d) 2

Ans (d)

5) A planet of radius r is one tenth of the radius of earth has the same mass density as Earth scientist dig a well of depth   R/ 5 on it and lower wire of same length and of linear mass density (10 )^-3 kg per metre into it if the wire is not touching anywhere a force applied at the top of the wire by a person holding it in a place is 

(take the radius of earth is 6 ×10^6 and acceleration due to gravity is 10m/s^2)

a) 96N

b) 108N

c) 120 N

d) 150 N

Ans (b)

6)  A rocket is launched normal to the surface of the earth away from the sun along the line joining the sun and the earth the sun is 3 ×10 ^5 times heavier than the earth and is at a distance 2.5x 10^4 times larger than the radius of earth The escape velocity  from the earth's gravitational field 11.2 km/ s  The minimum initial velocity required for the rocket to be able to leave the sun earth system is

a) 72km/s

b )22km/s

c) 42km/s

d) 62km/s

Ans (c)

7) A change in the value of g at a height h above the surface of the Earth is same as at a depth d below the surface of the earth when both d and h are much smaller than the radius of the earth then which one of the following is correct

a) d=h/2

b) d= 3h/2

c) d= 2h

d) d=h

Ans (c)

8) What is the minimum energy required to launch a satellite of mass M   from the surface of a planet of mass m and radius R in a circular orbit at an altitude of 2 R

a) 5GMm/6R

b) 2GMm/3R

c) GMm/R

d) GMm/3R

Ans (a)

9) From a solid sphere of mass M and radius R a spherical portion of radius R/2  is removed as taking gravitational potential

 V =0 zero at r=♾️ the potential at the centre of the cavity is

a) -GM/2R

b) -GM/R

c) -2GM/R

d ) -2GM/3R

Ans (b)

10) A satellite is revolving in a circular orbit at a height h from the Earth surface of radius R the minimum increase in its orbital velocity so that the satellite could escape from the Earth gravitational field

a) √(2gr)

b) √(gr)

C) √(gr/2)

d) √(gr)(√2-1)

Ans (d)

Recorded lectures

1) 

2). 

Ncert solutions chemistry -States of matter

 Ncert solutions chemistry -Kinetic Theory of Gases

Multiple choice Questions

Que 1

At what temperature the RMS velocity of Sulphur dioxide be same as that of oxygen at 303 K

a) 273K

b) 606K

c) 303 K

d) 403K

Ans (b)

Que 2

The ratio of root mean square velocity to average velocity of a gas molecule at a particular temperature is

a) 1:1.086

b) 1086:1

c) 2:1.86

d)1.86:2

Ans ( b )

Que 3

Average velocity of an ideal gas molecule at303 K  is 2m/s  the average velocity at 1200 K will be 

a) 6m/s

b) 4m/s

c) 2m/s

d) 8 m/s

Ans (b)

Que 4

What is the temperature at which the kinetic energy of 0.3  mole of helium is equal to kinetic energy of 0.4  mole of   argon at 400 K

a) 400 K

b) 873 K 

c) 533 K

d) 300 K

Ans ( c) 

Que 5

Root mean square velocity of a gas is double when temperature is

a) increased 4 times

b) increased to 2 times

c) reduced to half

d) reduce to one fourth

Ans (a)

Que 6

Sample of gas has a volume of of 0.2 litres  at  1 atm pressure and 0°C At the same pressure but at 273 °C its volume will become 

 Numericals on Gaseous law

Boyles Law 

Boyles law gave a  relationship between pressure and volume of a gas at a constant temperature

P~1/V

PV = constant

Question which I discussed in the given lecture is

Gas at 298 K shifted from a vessel of 250 cm² capacity to that of 1 litre capacity find  change in pressure

Charles law

Charles law gives a relationship between volume and temperature of a gas at a constant pressure according to Charles law as we increase the temperature the volume of the gas increases and vice versa

V~T

Question which is explained in given lecture is 

300 ml of gas at 27° C is cooled to 3° C at constant pressure find final volume of the gas 

More Numericals on Kinetic Theory if gases

Que1 

A gas occupies 15 L at a pressure of 40mm Hg what us volume when pressure increased to 75mm Hg

Que 2

A gas occupies 12 L at 0.860 atm what is pressure if volume becomes 18 L

Que 3

Given 300 ml of a gas at 25°C what is its volume at 14°C

Que 4

At 250°C a gas has volume of 7.50 L what is its volume of gas at -24° C

Que 5 

If a gas is pressured from 20 atm to 32 atm and critical temperature is 45°C what would be the final temperature in degree celcius

Que 6 

Temperature of a sample of gas in a steel container at 30kPa is increased from -10° C to 1×10³° C what is final pressure

Problems discussed in attached lecture are 

Type 1

A gas at 298 K is shifted from a vessel of 250 cm² capacity to that of 1litre capacity .Find change in pressure

Type 2 

300 ml of a gas at 27° C is cooled to 3°C  at constant pressure Find final volume

Gay Lussac Law

Numericals on combined gas Law

A gas occupies 7.6 litres at 27 ° C and 800 mm Hg  what is its volume at STP 

Concept of average root mean square and most probable velocity

Formula and Theory 

Average velocity

Vav= √8KT/πM. =√8RT/πM

Root Mean Square velocity = √3RT/M

Most Probable Velocity = √2RT/M

What is ratio between most probable average velocity and rootmean square velocity

√2:√8/π:√3

For numericals watch the lecture

Ncert Solutions for physics - Laws of Motion

 Ncert Solutions for physics - Laws of Motion 

Part 1

Part 2

What is force

Force is push or pull on a body Mathematically it is equal to product of mass and acceleration

F=mass×acceleration

It is a vector quantity  Its unit is Newton 

What are different laws of motion

Newton gave three different laws  known as Laws of Motion  to decribe motion of different objects

Newtons First Law

Every body  if at rest or motion remains at rest or in motion until an external force is applied on it

This law is also known as law of inertia of Galileo law of inertia

Newtons second law

This law gave a relation ship between  force applied on a body with acceleration developed in it

According to this law

Rate of change of momentum is directly proportional to applied force

dp/dt= F

mdv/dt=F

m×a=F

Newtons Third Law

To every action there is equal and opposite reaction

Day to day example like walking on floor and rocket propolsion is based on this law

Different questions based on Newtons laws of motion are discussed in attached lecture

 

What is Linear Momentum

Linear momentum is equal to product of mass and velocity 

It is denoted by p Its unit is Kgm/s

            p = mv

Impulse = F×dt

 It is product of  force with time 

Its unit is N/s

It is a vector quantity

when a force acts on abody for a very short interval of time it is impulse

Numerical on Linear Momentum

Laws of motion Tension in string 

In laws of motion if more than one blocks are attached each other with a string and a force is applied on it multiple Tension and acceleration is developed on different blocks in particular video  I explained  some tricks and formula how to solve acceleration produced in blocks and different problems of physics NCERT book of class 11

Difference between mass and weight

Mass is measure of quantity of an object its unit is kg It is always a constant value

Weight = mg

Where value of g =9.8 m^2

Its value  for an object is different on earth and other planets as it depends on gravitational force of attraction

Unit of weight is Newton

In attached lecture given below  definition of mass and weight , difference of mass and weight and questions 

How to find weight of object if mass is given etc are explained in detail

Friction Theory

Friction

It is  a resistant force which comes into play after application of applied force It always act in opposite direction

Friction is of different types 

Static friction

Kinetic friction

Rolling friction

Note

Limiting friction is that friction  at that point static friction is converted into kinetic friction

Value of cofficient of kinetic friction is always less than value of static friction

Cofficient of friction=

 Force of friction/Restoring force

Cofficient of friction has no unit

Angle of Repose

The angle made by inclined plane with horizontal when a body kept o inclined plane start sliding on it

More information can be understood by watching attached lectures

Numericals on friction

How to find coefficient of friction and angle of friction whwn a block is kept on table at rest

And more are discussed in given lecture 

Next lecture

Ncert solutions for physics - Heat and Thermodynamics

 Ncert solutions for physics - Heat and Thermodynamics

Heat --

Heat is a 

Energy which is transferred between system and the surroundings to the temperature difference

Unit of heat is called cal it is defined as the amount of heat required to raise the temperature of 1 gram of water through 1°C

Specific Heat Capacity

National certificate of a substance is the quantity of heat in cal required to raise the temperature of 1 gram of substance by 1 degree Celsius

The quantity of heat required to change the temperature of a body of mass m by∆T is proportional to the product of mass and change in temperature

Q=ms∆T

The product of the mass of the body and the specific heat capacity is called heat capacity it is defined as the amount of heat required to raise the temperature of a body by 1 degree Celsius

Watch the lecture

Principle of calorimetry

Two bodies of masses M1 and M2 specific heat S1 and S2 and at the temperature t1 and t2 are brought in contact with each other as you t1 is greater than t2 and heat will flow from the body 1 to body 2  if T  is the common temperature of the two bodies at the state of thermal equilibrium

Heat lost by body = Heat gained by body 

MS(t1-T)= ms(T-t2)

Watch the lecture 

Kinetic theory of gases state

An ideal gas or perfect gas is that gas which is strictly always a gas laws such as Boyles law , Charles law  at all the values of temperature and pressure

Boyles law --

It states that for a given mass of an ideal gas at constant temperature the volume of gas is inversely proportional to its pressure

P1V1=p2V2

Charles Law -- 

It is stated that for a given mass of an ideal gas at constant pressure the volume of a gas is directly proportional to its absolute temperature

V1/T1=V2/T2

Gay -Lussac Law --

It states that for a given mass of an ideal gas at constant volume pressure of a gas is directly proportional to its absolute temperature

P1/T1=p2/T2

Ideal Gas Equation--

PV=nRT 

Watch the video lecture to understand how to solve numericals--

Kinetic theory of gases

Kinetic theory of gases is based on the following assumptions

1) The molecules of a gas are a small particles and they are very far apart in comparison to their sizes

2) The total volume of the molecules is negligible as compared to the size of a gas

3) The molecules collide elastically with each other

4) The molecules  exert no force on each other except during collisions

Relationship between

#--

 pressure and the kinetic energy of the gas

Pressure = 2/3KE 

#

Absolute temperature and mean square velocity

Absolute temperature of an ideal gas is directly proportional to mean square velocity of its molecule

    v= √T

Various speeds of gas molecules

RMS speed of a gas molecule

Square root of mean of the squares of the speed of a gas molecule is called their root mean square speed

v = ✓ 3RT/m

Average speed

It is the arithmetic mean of the speed of the molecule in a gas

v = ✓8kT/πm

Most probable speed

It is defined as the speed which is possessed by maximum fraction of total number of molecules of a gas

v = ✓2kT/ m

Watch the lecture 

Degree of freedom--

It represents the number of independent possible ways in which the system can have the energy due to its motion or the configuration

1) for ideal monoatomic gas degree of freedom is 3 due to translational motion in three directions

2) for ideal diatomic gas degree of freedom is five due to three translational and two rotational motion

Ncert Solutions for physics- Simple Harmonic Motion

 Ncert Solutions for physics- Simple Harmonic Motion

Simple harmonic motion

It is a special kind of oscillatory motion in which particle moves to and fro about a main position under a restoring force which is directed towards mean position and its magnitude is directly proportional to the displacement of the particle

Equation of simple harmonic motion

Simple harmonic motion me mathematically expressed by single sinusoidal  function of the time

y= A sinwt

And x= A coswt

It is of two types--

 Linear simple harmonic motion 

Motion of a block connected to a spring on a smooth surface is an example of linear harmonic motion 

restoring force is directly proportional to the displacement

F~-x

a~-x

a=-w²x

Where is a acceleration. w  is angular frequency negative sign indicates the direction of restoring force and acceleration towards equilibrium position but in opposite direction of displacement

Angular simple harmonic motion

The restoring torque acting on a particle is proportional to the angular displacement of the particle and directed towards the equilibrium position

Restoring torque~Angular displacement

Force constant--

a= -w²x.    (a=F/m)

F/m= - w²x

F= -mw²x

F=- kx

k is called the spring constant of force constant

k= mw²

Velocity --

It is defined as the time rate of change of its displacement at that instant

Velocity = w√(A²-y²)

Maximum  velocity  at mean position = Aw

Maximum velocity at extreme position= 0

Acceleration --

The acceleration of a particle executing simple harmonic motion at an instant is defined as the time rate of change of velocity at that instant

a= -w²y

Acceleration at mean position a= 0

Acceleration at  extreme position a= Aw²

Time period 

The time taken by a particle to complete one oscillation is called time period

T= 2Π/ w

   = 2π✓|y|/|a|

= 2π ✓ displacement/Acceleration

How to find Total energy in case of simple harmonic motion

A particle executes simple harmonic motion  its kinetic energy changes into potential energy  and vice versa keeping total energy constant

1) Kinetic energy-- particle executing SHM causes kinetic energy due to an account of velocity of the particle

        KE = 1/2 mw²(A²-x²) 

Kinetic energy is maximum at the mean position

2) Potential Energy -- particle executing SHM causes potential energy due to displacement of the particle from its mean position

U= 1/2 mw²x²

Potential energy is minimum at the mean position

Mechanical Energy---

The total sum of kinetic energy and potential energy of the particle executing the SHM is called mechanical energy

E = 1/2 mw²A²

Watch the video lecture for concepts in detail --

A particle executing simple harmonic motion with a frequency f then the frequency with which the potential energy oscillate is f/2

Spring Block System 

Spring pendulum

Point mass suspended from a massless spring constitutes a spring pendulum time period of a spring pendulum is

T=2π✓m/k

Where k is  the force constant of the spring and m is the mass of the spring

Series combination of springs 

If two springs of spring constant k1 and k2 are joined in series then their equivalent time period is given by 

T= 2π ✓m(k1+k2)/k1k2

Parallel combination of springs

If two springs of spring constant k1 and k2 are joined in parallel then their equivalent time period is given by

T= 2π✓m/(k1+k2)

Watch the lecture  to understand concept of spring block system 

Simple pendulum

Simple pendulum consists of heavy metal Bob suspended by a light inextensible and flexible string

Time period 

T= 2π✓l/g

Note 

1) how does the time period of a simple pendulum changes with increase in temperature

∆T/T = 1/2 a∆t

a. is temperature coefficient ∆t is temperature difference

2) second pendulum is a pendulum whose time period 2 second length of the second pendulum is 1 metre

Free vibrations --

If a body is once set into vibration and then let free to vibrate with its own natural frequency the vibrations are called free vibrations 

Forced vibrations ---

The vibrations in which a body oscillates under the effect of an external periodic force is frequency is different from the natural frequency of the oscillating body are called force vibrations

Damped vibration

When a  body is set in free vibrations there is a dissipation of energy due to dissipative causes as a result the amplitude of vibration regularly decreases with the time such vibrations of continuously falling amplitudes are called damped vibrations

Resonant Vibrations

It is a special case of forced vibrations in which frequency of external force is exactly same as the natural frequency of the oscillator

Ncert solutions for physics - Centre of mass and Rotational Motion

 Ncert solutions for physics - Centre of mass and Rotational Motion

Rotational motion

A body said to be in rotational motion if all of its particles move along circles in parallel Planes the centre of the circles lie on a fixed line perpendicular to the parallel planes and is called axis of rotation

Equations of Rotational motion

w= wo+at

¢=wot+1/2at^2

w^2-wo^2=2a¢

a = (alpha) angular acceleration

wo = initial angular velocity

w = final angular velocity

¢=( theta ) angular displacement after time t

Centre of mass

It is the point at which entire mass of a system is supposed to be concentrated

R1 and R2 are the position vectors of the two particles of masses M1 and M2 then their centre of mass is

Rcm= (M1R1+M2R2)/M1+M2

Torque or .Moment of Couple 

The turning effect of a force about axis of rotation is called moment of force or torque due to a force

Torque = force × perpendicular distance from axis of rotation

SI unit is Nm

Principle of moments of rotational equilibrium

When a body is in rotational equilibrium the sum of clockwise moment about any point is equal to sum of anticlockwise moment about that point

Work done by a Torque  and power of Torque 

If a torque applied on a body rotates it through an angle the work done by the torque is equal to

Workdone

 = Torque × Angular displacement

∆W=T∆¢

Power = work done /Time

P= Torque×Angular displacement/Time

= Torque ×Angular Velocity

Angular Momentum

It is moment of linear momentum of particle about axis of rotation

Angular Momentum

 = Linear momentum × perpendicular distance from axis of rotation

L=pd

SI unit of angular momentum is kgm^2/s

A particle of mass m moves with uniform speed v along a circle of radius r then

Angular momentum (L) = mvr

Relation between areal velocity (∆A/∆t)and the angular momentum

Areal velocity is half of its angular momentum per unit mass

∆A/∆t=L/2m

Relation between torque and angular momentum

Rate of change of angular momentum is equal to total external torque acting on the system

dL/dt=Torque

Moment of Inertia 

Moment of Inertia is the sum of product of the masses of radius particles and squares of their perpendicular distances from axis of rotation

Formula

I= m1r1^2+ m2r2^2.…...

SI unit of moment of inertia is kg m^2

How to find moment of inertia for a Ring 

Relation between moment of inertia and angular momentum

Angular momentum =

 moment of inertia × angular velocity

L= Iw

Relation between moment of inertia and torque

Torque =

 moment of inertia ×angular acceleration(alpha)

T=Ia

Factors on which moment of inertia depends

Mass of the body

Size and shape of the body

Distribution of mass about axis of rotation

Position and orientation of the axis of rotation with respect to body

Radius of gyration(K)

The distance from the axis of rotation at which the whole mass of a body were supposed to be concentrated

I=MK^2or 

K=√I/M

Radius of Gyration is equal to the root mean square distance of particles from the axis of rotation as a unit of radius of gyration is m

Moment of Inertia of Rectangular Slab

HOW TO DERIVE FORMULA FOR Moment of Inertia for a Rectangular Slab when axis of perpendicular rotation passes through centre of slab

Theorem of perpendicular axes

States that moment of inertia of a plane lamina about an axes perpendicular to its plane is equal to the sum of the moments of inertia of the lamina about any two mutually perpendicular axis and its plane and intersect each other at the point where the perpendicular axes  passes through lamina

Iz=Ix+Iy

Theorem  of parallel axes

It is states that the moment of inertia of a rigid body about any axis is equal to the moment of inertia of the body about the parallel axis through its centre of mass + the product of mass of the body and the square of the perpendicular distance between the parallel axis

I= Icm+Md^2

How to find moment of Inertia for a Rectangular Slab 

When axis of rotation passes through centre of slab 

When acis of rotation passes through one end of slab

How to find moment of Inertia for a Ring 

Tricks to learn formula of Moment of

 Inertia

Rotational kinetic energy

If a body of mass M and moment of inertia I  rotates about an axis of rotation with an angular velocity w

Rotational kinetic energy =1/2×moment of inertia×(angular velocity)^2

 KE =1/2( Iw^2)

Mock Test Series

Que 1

Four particles of mass 2kg and 3kg 4kg and 8 kg are situated at the corners of a square of side length 2 metre the centre of mass of the given as

a) 30/18,28/18

b) 20/18,24/18

c) 14/17,24/17

d) 34/18,34/18

Ans (c)

Solution

Que 2

Two bodies of masses 2kg and 4kg are moving with velocities to metre per second and 10 metre per second what is the velocity of their centre of mass

a) 5.3 m/ s

b) 7.3 m/s 

c) 6.4 m/ s

d) 8.1 m/ s

Que 3 

Three masses of 2 kg 4 kg and 4kg are placed at the three (1,0,0) (1,1,0)(0,1,0) 

Find the position vector of centre of mass

a) 3/5 i+ 4/5 j

b) 3i+j

c) 2/5i+4/5j

d) 1/5i+3/5j

Ans (a) 

Que 4

Angular acceleration of a body is given by relation 4at^3- 3bt^2

If initial angular velocity of the body is wo then its velocity at time t will be 

a) wo+at^4-bt^3

b) wo-at^4-bt^3

c)wo+at^4+ bt^3

d)-wo+at^4-bt^3

Ans ( a) 

Solution

Que 5

Find the torque of a force  F = -3i +j+5k acting at a  point r= 7i+3j+k

a) 14i-38j+16k

b) 4i+4j+6k

c) -14i+38j-16k

d) -21 i+3j +5k

Ans  ( a )

Solution 

Que 6 

Two wheels having radii in the ratio 1:3 are connected by a common belt if the smaller wheel is accelerated from rest at the rate 1.5rad/s^2 for 10 sec find the velocity of the bigger wheel

a) 15

b) 5

c) 45

d) none of these

Ans  ( b )

Solution

Que 7

Moment of inertia of a body is 1 kg per metre  square if a body makes 2 revolutions per second find its angular momentum

a) 2π

b) 4π

c) π /2

d) 6π

Ans ( b )

Solution

Que 8

An automobile engine develops 100 KW when rotating at a speed of 1800 rev/ min what torque does it deliver?

a) 350 Nm

b) 531 Nm

c) 440 Nm

d) 628 Nm

Ans ( b ) 

Solution

NCERT Solutions -physics Class11 - Motion in one plane

NCERT Solutions  -physics - Motion in one plane

Introduction

Watch the lecture to revise whole lesson in one shot specially for  competitive exam 

Multiple choice questions

Que 1

What is unit of speed?

a) m

b) m/s

c) m/s^2

d) kg m/s

Ans (b)

Que 2

Displacement  a body when revolving around a circle of radius r   is

a) 2πr

b) 0

c) 2r

d) πr^2

Ans (b)

Que 3

If a man goes to office at a distance of 10 km and  comes back to his home  in 2 hours  find  velocity of man

a) 5m/s

b) 10m/s

c) 0 m/s

d) 20 m/s

Ans ( c )

Que 4

The direction of the angular velocity vector is along

a) a tangent to the circular path

b) the inward radius

c) the outward radius

d) the axis of rotation

Ans (d) 

Que 5 

At the upper  most point of a projectile its velocity and acceleration are at an angle of

a) 0°

b)45°

c) 90°

d)180°

Ans (c) 

Que 6

Two equal vectors have a resultant equal to either the angle between them is

a) 60°

b) 90°

c) 100°

d) 120°

Ans (d) 

Que 7 

Two projectiles of same mass and with the same velocity are thrown at an angle of 60 degree and 30 degree with the horizontal then which will remain the same

a) time of flight

b) range of projectile

c) maximum height acquired

d) all of them

Ans (b)

Que 8

The width of a river is 1 km vlocity of boat is 5 km per hour the boat cover the width of river in the shortest time 15  minute then the velocity of river is stream is

a) 3km/h

b) 4 km/h

c) √29 km/h

d) √41 km/h

Ans (a)

Hint 

 v(resultant)= width of river /time 

V(river)= ✓(vres)²-(vboat)²

Que 9 

Two particles having mass M and m are moving in a circular path having radiusR and r if their time periods are same then the ratio of angular velocity will be

a) r/R

b) R/r

c) 1

d) ✓R/r

Ans (c)

Que 10 

Aeroplane flying horizontally with a speed of 360 km/h releases a bomb at a height of 492 m from the ground if gravity is equal to 9.8 m/s² it will  strike the ground at

a) 10 km

b) 100 km

c) 1 km

d) 16 km

Ans (c)

Hint

Time = ✓2h/g

Distance = horizontal velocity×time 

Motion

When a body changes its position with respect to time in refrence to surrounding body is said to be in motion

Distance and Displacement

Distance 

Length of total path covered by a body in moving from one place to another is called distance covered It is a scalar quantity Its unit is meter

Displacement

The difference in initial and final positionof a body when moves from one place to other is called displacement

It is a vector quantity

Its unit is also meter

Note 

Distance can never be zero but displacement can be zero in some cases

Speed

Distance covered  per unit time is called speed It is a scalar quantity 

Formula

Speed = Distance/Time

Unit m/s

Velocity

Rate of change of displacement per unit time in specified direction is called velocity

Velocity=Displacement/Time

Unit m/s .It is a vector quantity

Average Speed

Average speed is Total distance covered by a body in total time elapsed

Let a body covers x km  in time T and y km in time t then 

Total distance covered = x+y

Total time elapsed = T+t

Formula for Average Speed

(x+y)/(T+t)

Its unit is also m/s

Conversion of units

Speed =distance /time
to convert from one unit to other
if distance is in kilometer then time is in hour ie km/hr
if distance is in meter then time is in seconds ie m/s
if distance is in centimeter then time is in seconds ie cm/s

now if we have to convert km/hour to m/s then

multiply that number with 5/18
eg if speed is 36km/h then to convert it into m/s
=36*5/18
=2*5=10m/s

if we have to convert m/s to km/h multiplythat number with 18/5
eg 20 m/s
20*18/5=4*18
=72km/h

Watch the lecture to solve questions based on it

Acceleration

When velocity of a body changes in due course of time or when a body starts from rest or it comes to rest after motion then motion is termed as accelerated 

So 

Change in velocity per unit time is called acceleration 

accaleration =

(final velocity-initial velocity)/time

a= ∆v/∆t

Unit of acceleration is m/s^2

It is a vector quantity

When velocity of body increases with time it is termed as positive acceleration

When velocity of body decreases with time it is termed as negative acceleration or retardation

More questions are explained 

A driver of a car travelling at 52 km/h applies the brake The car stops in 5seconds A driver going in other car at 3 km /h appling brakes and stops in 10 sec which travels  faster and plot the graph

How to find distance ,velocity and acceleration using differentiation and integration

 A car is moving along x-axis if it moves from 0 to p in 18 seconds and returned from p to q in 6 seconds what are the average velocity and average speed of the car in going from o topi and from OTP and back to you

How to derive equation of motion graphically

There are three equations of motion

v= u+at

s=ut+1/2at²

v²=u²  -- 2as

The lecture attached given below how we derive  different Newton laws of equation graphically is explained in detail

How  to solve numericals based on graph 

Body covers one third of its journey with speed u and next one third with speed v and the last one third with speed w what is the formula for the average speed of the body during the entire journey

Average speed 

= 3uvw/(uv+vw+uw)

Relative velocity 

How to solve questions when a object falls freely under gravity

When body is released into the surface of the earth it is accelerated downward under the influence of force of gravity in the absence of air resistance all the bodies fall with the same acceleration this motion of a body of falling towards Earth from a small height is called free fall and acceleration with which a body falls is called acceleration due to gravity and it is denoted by g the value of g near the surface of earth is equal to 9.8m/s²

When a body falls freely under the action of gravity it velocity increases and the value of g is taken as positive

When body is thrown vertically upward its velocity decreases and the value of g is taken as negative

Watch the lecture in order to understand different types of questions based on the topic of motion under gravity

Projectile motion

Any body projected into space such that it moves under the effect of gravity alone is called a projectile the path followed by a projectile is called its trajectory which is always a parabola

 The projectile executes two independent motions simultaneously

 uniform horizontal motion

uniform accelerated downward motion

Time of flight

It is the total time for which the projectile remains in its flight

 

When  projectile fired horizontally

T= √2h/g

A projectile fired at an angle with the horizontal

T= 2usin¢/g

Horizontal Range 

What is the horizontal distance covered by the projectile during its time of flight

Projectile fired horizontally

R= u√(2h/g)

Projectile fired at an angle with the horizontal

R = u²sin2¢/g

Maximum height of a projectile

It is the maximum vertical distance attained by the projectile above the horizontal plane of projection

H= u²sin²¢/2g

Watch the lecture to understand concept of projectile 

Points to be noted

The horizontal range is maximum when angle is 45 degree R=u²/g

Highest point of parabolic path the velocity and acceleration of a projectile are perpendicular to each other

The maximum horizontal range is Ho times the maximum height attained by the projectile when they are fired at an angle of 45 degree

H=R/4=u²/4g

Numerical on projectile motion

Circular Motion

A particle moves along a circular path with constant speed then its motion is said to be uniform circular motion

Angular displacement

Angular displacement of a particle moving along a circular path is angle swept out by its radius vector in the given time interval

Angle =Arc/radius

¢=∆s/r=

Unit of angular displacement is radian so it is the dimensionless quantity

Angular Velocity

The time rate of change of angular displacement of a particle is called its angular velocity it is denoted by  w (Omega) it is measured in radian per second

w= ∆¢/∆t

Time period

The time taken by a particle to complete one revolution along its circular path it is denoted by T and it is measured in second

Frequency

The frequency of an object in a circular motion is the number of revolutions completed per unit time it is denoted by (nu) v

Relation between time period and frequency

Time period=1/frequency 

Angular velocity

 = angular displacement/time 

w=¢/t=2π/T=2πv

What is the relation between linear velocity and angular velocity

Linear velocity = angular velocity × radius

v=rw

What is angular acceleration and what is the relation between angular acceleration and linear acceleration

The time rate of change of angular velocity of a particle is define as angular acceleration the unit of angular acceleration is radian per second square

Relation 

Linear acceleration = angular acceleration× radius

Centripetal acceleration

Body undergoing uniform circular motion is acted upon by an acceleration which is directed along the radius towards the centre of the circular path is acceleration is called centripetal acceleration

Its magnitude remains constant (v²/r) but its direction continuously changes and remain perpendicular to the velocity vector at all the positions

a=v²/r= w²r

Vector 

The physical quantities which have both magnitude and direction are called vectors

Representation of a vector a vector is represented by a straight line with an arrow head over it the length of the line give the magnitude and the arrowhead gives the direction of the vector

Types of vector 

Equal vectors two vectors are said to be equal if they have the same magnitude and direction

Negative factor negative bacteria is defined as another vector having the same magnitude but having an opposite direction

Zero vector vector having zero magnitude and an arbitrary direction is called as zero or null vector

Collinear vector vector which either act along the same line or along the parallel line are called collinear vector

Coplanar vector character which act in the same plane are called coplanar vectors

Polar vectors these are the factors which have a starting point for a point of application force 

Axial vector directors which represent rotational effect and act along the axis of rotation are coaxial vector for example torque 

Coterminous directors directors which have the common terminal point are called coterminous vectors

Addition and subtraction

Vectors addition is commutative and associative

A+B=B+A

Subtraction of two vectors

A-B= A+(-B)

Watch the attach lecture in order to understand how will you add and subtract two vectors

Unit vector

Unit vector is a vector of unit magnitude drawn in the direction of a given vector a unit vector in the direction of A is given as

A=A/|A|

Watch the lecture to understand how to find the unit vector of a given vector

Resolution of vector 

The blouse is off his bedding elevator into two or more vectors is known as the resolution of the vector 

The unit vectors i j k are the factors of unit magnitude and point in the direction of the x axis y axis and y axis respectively in a co-ordinate system they are collectively known as orthogonal trade of unit vectors and their schedule is always equal to 1

When a vector is resolved along two mutually perpendicular directions components of dinda called rectangular components of the given vector

Watch the attached lecture in order to understand how will we resolve the two vectors into direction and how will we find their magnitude and the angle between them

Numericals

Attached lecture  contains a collection of numericals based on vectors

Thank you for watching the post if you have any doubts or if you want to take the live classes you can contact by sending us email on on the WhatsApp number

Ncert Solutions -Units,Dimensions and Vectors

Units and Measurement

Physical quantities

All those quantities which can be measured directly or indirectly are called physical quantities

Fundamental and Derived units

Fundamental quantity

Physical quantities which can not be defined in terms of other physical quantities are called fundamental quantities there are seven fundamental quantities mass length time electric current temperature luminous intensity and amount of substance

Derived Quantity

Physical quantities which can be expressed in terms of other physical quantities are called derived quantities for example speed

Watch video lecture to understand 

System of units

There are three system of units

1) CGS -- in this system the fundamental units of length mass and time are centimetre gram and second

2) FPS -- In this system of fundamental units of length mass and time foot pound and second

3) MKS -- In the system the fundamental units of length mass and time are metre kilograms and second

4) SI system -- In this system is adopted internationally the fundamental units of length mass time electric current temperature luminous intensity and amount of substance are metre kilogram second ampere kelvin candela  and mol respectively

Significant figures

Rules

All non zero digits are significant

All zeros between two non zero digits are significant

All zeros to the right of a non zero digit but to the left of an understood decimal point are not significant but such zeros are significant if they come from a measurement

All zeros to the right of a non zero digit but to the left of a decimal point are significant

All zeros to the right of a decimal point are significant

All zeros to the right of a decimal point but to the left of a non zero digit are not significant

The number of significant figure does not depend on the system of units

Watch the lecture to find number of significant figures using questions

Rules for rounding off

The digit  to be dropped is less than 5 then preceding digit is left unchanged

If digit  is greater than 5 then the preceding digit is increased by 1

The digit  to be dropped is 5 followed by non zero digits then the preceding digit is increased by 1

If the digit to be dropped is 5 then the preceding digit is left unchanged if it is even

If the digit  to be dropped is 5 then the preceding digit is increased by 1 if it is odd

Watch the lecture to understand how questions in competitive exams are asked on this topic 

Least count 

What is least count how it is calculated how to calculate the value of significant figures after adding and subtracting two given digits all these questions are explained in the lecture attached given below

Error Analysis

Error --It  is the difference between the measured value and the true value of a physical quantity

Random Error -- The errors which occur regularly and at random in magnitude and direction are called random error when there are a lot of readings we find the arithmetic mean

Absolute Error-- The magnitude of the difference between true value and measured value is called  absolute error

Mean Absolute Error --- automatic mean of the positive magnitude of all the absolute error is called mean absolute error

Relative Error -- It  is the ratio of the mean absolute error to the true value

Percentage Error -- the relative error expressed in the percentage is called percentage error

Error Combination 

Sum or Difference 

when two quantities are added or subtracted the absolute error in the final result is the sum of the absolute errors associated with the individual quantities

∆Z= ∆A+∆B 

Product or Quotient

When two quantities are multiplied or divided the fractional error in the final result is a sum of the fractional errors of the two quantities

∆Z/Z= ∆A/A+∆B/B

Watch the lecture in order to understand concept and how to solve numerical

Error due to power of a measured quantity 

The fractional error in the power of a quantity is equal to the end times the fractional error in the quantity  itself

Z=A^n then ∆Z/Z=n∆A/A

General Rule 

If Z=A²B³/C⁴

Then 

Relative error in Z will be

∆Z/Z = 2(∆A/A)+ 3(∆B/B)+4(∆C/C)

Percentage error in Z will be

∆Z/Z ×100= 2(∆A/A)×100+ 3(∆B/B)×100+4(∆C/C)×100

Dimensions

Dimensions are the powers to which the fundamental units of mass length and time must be raised in order to represent a derived quantity completely

Dimensional Formula 

It is an expression which shows how and which of the fundamental units of mass length and time occur in the derived unit of a physical quantity

Dimensional Equation

Equation which express the physical quantity in terms of the fundamental units of mass length and time is called dimensional equation

Uses of dimensional equations

1) to convert a physical quantity from one system of units to the other

2 )to check accuracy of a formula ie check the correctness of a given physical relation

3) to drive a relationship between different physical quantities