### DYNAMICS

DYNAMICS

1. Dynamics:-
The study of motion of a body under the action of a force is called “dynamics”.

2. Types of motion:-
For any body there are three types of motions. They are:-
1. Translatory motion.
2. Oscillatory motion.
3. Rotatory motion.

Example:-
Running boy, motion of car etc. are some of the examples for translatory motion.
The oscillation of a simple pendulum is an example for oscillatory motion.
The motions of the wheel of a bicycle, the motion of a fly wheel about its axis are some examples for rotatory motion.

3. Rigid body:-
A body which does not undergo any change in its shape and volume by the application of a force is called a rigid body.

4. Angular displacement:-
The angular displacement is the angle described by a radius vector of a particle on rotating body in a given interval of time.

Units of Angular displacement:-
Angular displacement is measured in the units of radians.

5. Angular velocity:-(ω)
The rate of change of angular displacement is defined as angular velocity.
Angular velocity(ω)=Angular displacement/time. The unit of angular velocity is radians/sec.

Note:-
1. The time period of a particle in a circular motion is defined as the time taken by it to complete one revolution.

2. One revolution=2П radians.

3.

6. Uniform Circular motion:-
If the magnitude of the linear velocity ‘v’ of a body remains constant but its direction changes continuously, then the body is said to be in uniform circular motion. In uniform circular motion the angular velocity is constant.

7. Relation between linear velocity and angular velocity:- As shown in the figure,
Let the body be in circular motion of radius ‘r’ with linear velocity ‘v’ and angular velocity ‘ω’.
Let the body move from ‘A’ to ‘B’ in small interval of time ‘t’. During this time interval angular displacement is θ. Now,
The linear speed of the body is given by
V= V=displacement/time.
V= ( AB= θr from eq(1))
Rearranging the formula, we get V=r ω.

Note:-
1. Linear momentum=p=mass x velocity.
P=mv.

2 Angular momentum=L=mvr.
L=m(rω)r.
L=m ωr2.

8. Centripetal acceleration:-
A particle executing uniform circular motion undergoes a continuous change in the direction of its velocity which results in an acceleration directed towards the centre of the circle is called centripetal acceleration.
Centripetal acceleration =a 9. Centripetal force:-
The force which acts continuously on a particle initially moving with a linear velocity and makes it travel along a circular path is called centripetal force. Centripetal force is directed towards the centre of the circle.
Centripetal force is given by
F=ma. F=m ω2r.
F=mr ω2.

10. Examples of Centripetal force:-
1. The gravitational force of attraction of the earth on the moon.

2. The electrostatic force of attraction of nucleus on the revolving electrons.

3. The frictional force applied by rails on the wheels of the train moving in a curved path.

11. Non-inertial frame of reference:-
An imaginary co-ordinate system which is attached to a rotating or accelerated body where Newton’s laws are not valid is called non-inertial frame of reference.

12. Inertial frame of reference:-
An imaginary co-ordinate system which is either at rest or in uniform motion where Newton’s laws are valid is called an “inertial frame of reference”.

13. Differences between inertial frame of reference and non inertial frame of reference:-

 Inertial frame Non-inertial frame 1. An imaginary co-ordinate system either at rest or in uniform motion where Newton’s law are valid is called inertial frame of reference. An imaginary coordinate system which is  attached to a rotating or accelerated body where Newton’s law are not valid is called  non-inertial frame of reference.

14. Centrifugal force:-
A radially outward force on a body in a uniform circular motion, observed only in a rotating frame of reference is called centrifugal force.
Centrifugal force is directed away from the centre of the circle. 15. Examples of centrifugal force:-
1. Centrifuge.

2. Centrifugal pump.

3. Banking of roads and railway tracks.

16. Centrifuge:-
A centrifuge is a machine used to separate particles of higher mass from those of lower mass in a given mixture.

Description:-
A centrifuge consists of a cylindrical vessel rotated about its own axis at high speed with the help of an electric motor.

Working:-
When milk is poured into the cylindrical vessel of the centrifuge and rotated with high speed, the particles of higher mass (skimmed milk) are thrown away from the centre due to greater centrifugal force and lower mass cream particles collected at the centre i.e., near the axle.

17. Working of a Laundry drier:-
1. In a laundry drier, the wet clothes are dropped into a cylindrical vessel containing holes.

2. When the vessel is rotated, the wet clothes get stuck to the walls of the vessel.

3. The centrifugal force pushes the water molecules from the clothes out through the holes. Thus, the clothes are dried.

18. Banking angle:-
The angle made by the line joining the outer raised edge of the road to the inner edge with the horizontal line is called banking angle.

19. Expression for angle of banking:- Let AB and AC be the horizontal and banked paths respectively and BC be the raised edge.
Let θ be the angle of banking. Let a vehicle of mass ‘m’ take a turn on the curved path of radius ‘r’ with the speed ‘v’. Its weight ‘mg’ acts vertically downwards. The normal reaction ‘R’ of the road on the body will be perpendicular to AC.
The normal reaction can be resolved into vertical and horizontal components Rcosθ and Rsinθ respectively.
The vertical component is equal to the weight of the body i.e.,
Rcosθ=mg. (1)
The horizontal component provides the centripetal force. Equating equations (1) and (2) we get,  By knowing the values of ‘v’ and ‘r’ we can determine the angle of banking ‘θ’.

20. Write the differences between centripetal and centrifugal force?

 Centripetal force Centrifugal force 1. It acts normally on a particle executing a uniform circular motion. 1.  It acts normally on the (same) particle executing uniform circular motion. 2.  It is always directed radially towards the centre of the circle. 2.  It is directed radially away from the centre of the circle. 3.  Its magnitude is =mrω2. 3.  Its magnitude is =mrω2. 4.  It is associated with an external agent. 4.  It cannot be associated with any agent. 5.  It is real force in an inertial frame of reference. 5. It is a fictitious force in an inertial frame of reference. 6.  It  is a pull on the body towards the centre of the circle. 6.  It is a tendency of the body to fly away from the centre of the circle. 7.  It is necessary to make a body to travel on a curved path. 7.  It comes into play in a rotating frame of reference. 8.  In a uniform circular motion, its magnitude is constant. 8.  In a uniform circular motion, its magnitude is constant and equal to that of the centripetal force. 9.  Centripetal force depends on mass of a body in a circular motion with speed ‘v’ 9. centrifugal force depends on mass ‘m’.  Hence bodies of higher mass rotate on a circle of higher radius (principle of centrifuge)

21. Why is centrifugal force called a fictitious force?
Since, centrifugal force cannot be associated with any object or agent, it is called fictitious force.

22. What is the direction in which sparks fly when a knife pressed on rotating grindstone to sharpen the knife?
When a knife is pressed on a rotating grinding stone, the sparks fly tangential to the rotating grind stone.

23. What supplies the centripetal force for an electron to revolve around the nucleus?
The centripetal force is provided by electro static force of attraction between the electrons and the nucleus.

24. What is the necessity for the banking of roads?
If there is no banking, the vehicle has to take the help of frictional force between the tyres and the road. Since, the frictional force is limited and is not always dependable, banking of roads is necessary otherwise the vehicle would skid.
25. Why skidding of motor car generally takes place in a curved path on a rainy day?
While moving in a curved path the necessary centripetal force for the car is derived from the friction. During rainy day the frictional force decreases due to the presence of water layer. So the required centripetal force is not supplied. Hence it skids.

26. Satellite:-
A satellite is a natural or artificial body orbiting around another body of larger mass and larger radius.

27. Principle of satellite launching:-
The principle of launching an artificial satellite into a proper space orbit is to impart sufficient initial horizontal speed such that it revolves round the earth at the chosen height.

28. Orbital velocity:-
The horizontal speed which is imparted to an artificial satellite such that it goes round the earth in an orbit is called its orbital speed.

Note:-
Generally the orbital velocity of an artificial satellite is above 8km/sec and below 11km/sec.

29. Uses of artificial Satellites:-
1. Artificial satellites are helpful to develop reliable communication links.

Example:-
Telephone, Fax, messages, internet etc.,.

2. They are used to forecast the weather.

3. they are useful in ‘Remote Sensing’ (finding the natural deposits of resources like coal, gas, oil etc. with out actually drilling the earth).

4. They are helpful for spying defence services.

5. They are helpful in distant education.

30. What is the orbital velocity of an artificial satellite?
The orbital velocity of an artificial satellite is above 8km/sec and below 11km/sec.

31. What happens when an artificial satellite attains the velocity more than 11km/sec?
It escapes from the earth’s gravitational force.

32. Radian:-
The angle subtended by an arc of unit length on a circle of unit radius is defined as radian.

33. Radian:-
The angle subtended by an arc of unit length on a circle of unit radius is defined as radian. 34. Differences between linear velocity and angular velocity?

 Linear velocity Angular velocity 1.  The rate of change of displacement of a body along a straight line is called linear velocity. 1.  The rate of change of angular displacement of a body is called angular velocity.

34. Differences between linear acceleration and centripetal acceleration?

 Linear acceleration Centripetal acceleration 1.  The rate of change of velocity of a body along a straight line is called linear acceleration. 1.  The rate of change of velocity of a body moving along the circumference of a circle is called centripetal acceleration.

35. Differences between linear acceleration and angular acceleration?

 Linear acceleration Angular acceleration 1.  The rate of change of velocity of a body along a straight line is called linear acceleration. 1.  The rate of change of angular velocity of a body is called Angular acceleration.

36. Differences between circular motion and rotatory motion?

 Circular motion Rotatory motion 1. If a force acts such that the magnitude of ‘v’  does not change but direction alone changes then it is called “circular motion”. 1. The motion of a particle under the action of a force always directed towards a fixed point away from its path is called “Rotatory motion”. 2. It is a special case of rotatory motion where the magnitude of ‘r’ remains constant. 2. Rotatory motion is denoted by changing linear velocity vector ‘v’ and radius vector ‘r’.

II. Problems:-

1. A particle undergoes an angular displacement of 90ο . What is the value of the angular displacement in radians and revolutions?
Given,
θ =90ο
θ in radians=?
Number of revolutions=?
We know that, or
x= 1.5714
θ=1.5714 radians.
We know that, 2. A stone moving along the circular path makes 20 revolutions in 10 seconds. What is the angular velocity of the stone?
Given, ω=12.56 radians/sec
Angular velocity=12.56 radians/sec

3. A satellite revolves with a time period of 84.3 minutes at a height of 40 kms from the surface of the earth. Find speed of the satellite.
Given,
Time period of the satellite(T)=84.3 minutes
=84.3×60 sec
=5058 sec
Height of the satellite=40 kms.
V=?
The total height of a satellite from the centre of the earth.
Radius of the earth=6.4×103 km
r=6400+40
r=6440 km (1km=1000m)
r=6440x103m
We know that,
V=r ω
But, V=rω V=7997.0m/s
Speed of the satellite=7997.0m/s

4. The object is moving along a circle of radius 6m with a constant speed of 12m/s. Calculate the angular velocity.
Given,
The radius of a circle(r)=6m.
Velocity(v)=12m/s.
Angular velocity(ω)=?
We know that,
V=r ω ω=2 radians/sec

Angular velocity = 2 radians/sec.

5. The speed of a wheel is 1800 rotations per minute. Find its average angular velocity in radians/sec.
Given,
1800 revolutions  _____________ 1minute(60sec)
1 revolution _____________ ?x ω=3.14×60
ω=188.40 radians/sec

6. Find the acceleration of the moon towards the earth. Given the time taken by the moon to complete one revolution around the earth is 27.3 days. The distance between the earth and the moon is 3.85x105km.
Given,
Time period(T) = 27.3 days
T=27.3x24x60x60 sec=27.3x86400sec
r=3.85x105km
r=3.85x108m
a=?
we know that, 7. The radius of curvature of a road is 60m. It is to be banked so that no frictional force is required for the car traveling on the road at 25m/sec. Find the angle of banking?
Given,
r=60m.
v=25m/sec
g=9.8 m/s2.
θ=?
We know that, 8. A curved road of 100m radius is banked with an angle of 100. Find the safe velocity of the vehicle moving on the road?
Given,
r=100m.
θ=100
v=?
g=9.8 m/s2.

We know that, log v= log(17.63×9.8)
log v= x2.2374
log v=1.1187
v=antilog(1.1187)
v=13.14 m/s
Safe velocity of the vehicle moving on the road=13.14 m/s

9. What is the angular velocity of the earth about its own axis?
T=24 hours.
T=24x60x60sec = 86400 sec.
ω=?
We know that, ω=7.269×10-5 radians/sec

10. Express 0.210 radians in degrees and revolutions?
Given,
0.210 radians in degrees and revolutions. 0.210 radians = 0.033 revolutions

III. Match the following:-
A                                              B

1. Angular displacement           (b) a) centrifugal force.

2. Angular velocity                    (c) b)radian.

3. Fictitious force                       (a) c)radians/sec

4. Banking angle                        (e) d)centripetal force.

5.Centre-seeking force              (d) e)banking of roads.

2.   Group A                                                    Group B

1. (b) a)tanθ.
2. rω                                                         (d) b) Instantaneous angular
velocity.
3. (a) c) L
4. (f) d) V
5. (g) e) f
(f)ω
(g) Centripetal force.

3. A B
1. Linear velocity                                  (d) a) 2. Centripetal acceleration                   (a) b) centrifugal force.
3. Centrifugal force                              (e) c) 4. Center-fleeing force                         (b) d)rω
5. Tangent of the banking angle          (c) e)m rω2

IV Fill in the blanks:-

1. In an uniform circular motion, if the radius is doubled the centripetal force now required is (c)
a) ¼ as great as before. c) twice as great as before.
b) as great as before. d) 4 as great as before.

2. A car of mass 1200 kg takes a turn of a curved road of radius 180m with a speed of 6m/s . The centripetal force acting on the car is (c)
a)48 Newtons b) 147 Newtons.
c)240 Newtons d)1440 Newtons.

Solution:-
Given,
m=1200 kgs
r=180 m
v=6m/s
We know that, 3. A car moves on a curve but level road, the necessary centripetal force on the car is provided by                                                                                               (c)
a) inertia                                                   b) gravity.
c)Friction between tyres and road.           d) Normal reaction of the car.

4. The study of motion of a body under the action of force is called dynamics.

5. The motion of the body suspended from a retort stand by means of thread is known as                                                                                                     (a)
a) Vibratory                                               b)Translatory.
c)rotatory                                                  d)linear.

6. The forces that govern the translatory motion of a body obeys             (b)
a) Kepler’s law                                           b)Newton’s law
c)Faraday’s law                                          d)Coulomb’s law.

7. An imaginary line passing through the centre of the circle and is perpendicular to the plane of the circle when a body is making circular motion is axis.

8. The motion of the particle under the action of a force always directed towards a fixed point away from its centre is called rotatory motion.

9. The simplest form of rotatory motion is circular motion.

10. The angle described by a radius vector of a particle on rotating body in a given interval of time is called Angular displacement.
11. The angle subtended by an arc of unit length on a circle of unit radius is called radians.

12. One radian=                                                                                            (c)
a)900                                                                 b)67.180
c)570181                                                           d)105.350.

13. The rate of change of angular displacement is Angular velocity.

14. The formula for finding the linear velocity if the radius vector and angular velocity are ‘r’ and ‘ω’ respectively is V=rω.

15. A body experiences a centripetal force when the body moves in            (b)
a)straight line                                                   c)in an elliptical orbit
b)in circular path                                             d)in hyperbolic path(curved)

16. The forces that help the electron to revolve around the nucleus of atom is (a)
a)electro static force.                                        b)electro magnetic force.
c)electro valent force                                        d)photo electric effect.

17. The time taken by a particle to complete one revolution along a circular path is called time period.

18. If ‘T’ is the time period, ‘ω’ is the angular velocity then                              (d)
a) rad/sec                                                          b) rad/sec
c) rad/sec.                                                         d) radians/sec

19. In an uniform circular motion, the angular velocity is constant.

20. The force which continuously deflects a particle from its straight line and makes it travel along the circular path is called centripetal force.

21. The expression that gives the magnitude of centripetal force F= (or) mω2r.

22. Centrifugal force appears only in accelerated bodies which are    (a)
a) Non inertial frame of reference.                   b)Inertial frame of reference.
c)along x-axis.                                                  d)along y-axis.

23. The force because of which the body tends to move away from the centre along the radius is called centrifugal force.

24. This is a fictitious force in an inertial frame of reference                      (d)
a) Centripetal force                                           b)electro motive force.
c)Buoyancy                                                       d)centrifugal force.
25. The force that comes into play in a rotating frame of reference is      (b)
a)centripetal force.                                           b)centrifugal force.
c)upper thrust                                                  d)electro motive force.

26. Centrifugal force is more for a body of                                                (c)
a)zero mass                                                     b)lower mass.
c)higher mass                                                  d)Negligible force.

27. A machine used to separate particles of higher mass from those of lower mass in a given mixture is called centrifuge.

28. A machine used to separate sugar crystals from molasses is centrifuge.
29. Honey is separated from bee’s-wax by using centrifuge.
30. Laundry drier is used to dry wet clothes works under the principle of (a)
a)Centrifugal force                                            b)centripetal force
c)gravitational force                                          d) electromotive force.

31. A precipitate from a non-homogenous mixture of a solution can be separated by the use of centrifuge.

32. The rising of the outer edge of the road slightly above the level of inner edge to provide centripetal force necessary to make a vehicle turn a curve is called banking of roads.

33. If θ is the angle between the actual road width and horizontal ‘v’ is the speed of vehicle, ‘r’ is the radius of the curved path and ‘g’ is the acceleration due to gravity. 34. A natural or artificial body orbiting around another body of larger mass and larger radius is called satellite.

35. The natural satellite of the earth is moon.

36. Man-made satellites are called Artificial satellites.
37. The faster the initial horizontal speed of the stone thrown, greater is the radius of                                                                                                              (a)
a)curved path.                                             b)straight path.
c)elliptical path                                            d)hyperbolic path.

38. The horizontal speed which is imparted to the satellite such that it goes round the earth in an orbit is called orbital speed.

39. The value of escape velocity is                                                            (d)
a)120 km/s.                                                  b)6 km/s
c) 9km/s                                                       d)11.2km/s

40. Fax messages are sent by using                                                           (d)
a)remote sensing satellite.                           b) space satellite.
c) space research stations.                           d) communication satellites.

41. Satellites that help us to prevent deforestation and expansion of deserts are (a)
a) remote sensing satellites.                         b)space shuttles.
c) communication satellites.                         d)space stations.

42. Presence of minerals and ores in a region of the land in a country can be detected by remote sensing satellites.

43. Presence of ground water in a region of the country can be detected by remote sensing satellites.

44. space shuttles also behave like satellites.

45. satellites are also been used for spying in defence services of country.

46. One of the artificial satellites that has been launched by India is      (c)
a) sputnik-I                                                     b)voyager-II
c)INSAT-I                                                        d)Soyuz

47. Circular motion is a special case of rotatory motion.

48. If the string of rotating stone is cut, the stone moves in tangential direction.
49. In a uniform circular motion angular velocity is constant.
50. Center-seeking force is called centripetal force.
51. Newton’s laws are valid in inertial frame of reference.
52. Centrifugal force is known as center-fleeing force.
53. Centrifugal force means radially out ward force on a body in an uniform circular motion.
54. If a body moves along a straight line it is said to have linear (or) translatory motion.

55. In translatory motion, the action of force on the body and the resulting displacement are along the same direction.

56. The forces acting on a body subjected to linear motion obeys Newton’s law.

57. A force acting continuously on a bob in a direction perpendicular to its linear motion is necessary to make it travel along a circular path.

58. The axis is an imaginary line passing through the centre of the circle and is perpendicular to the plane of the circle.

59. If the force on a body which is under continuous circular motion is suddenly with drawn, then the body moves along tangential to the circle.

60. Time period of a particle in a circular motion is the time taken by it to complete one revolution.

61. Motion of a particle under the action of force always directed towards a fixed point away from its path is rotatory motion.

62. If force acts such that the magnitude of ‘v’ does not change, but direction alone changes while ‘r’ remains constant, the particle makes Circular motion.

63. The angular displacement is the angle described by a radius vector of a particle on a rotating body in a given interval of time.

64. angular displacement θ along an arc of length ‘s’ on a circle of radius ‘r’ is given by . 66. The rate of change of angular displacement is called angular velocity.
67. Angular velocity ‘ω’ 68. Instantaneous angular velocity ‘ω’ = 69. If ‘T’ is the time period then 70. If ‘f’ is the frequency, ω is the angular velocity, then ω = 71. In a uniform circular motion angular velocity is constant.
72. The magnitude of the radius vector remains constant in uniform circular motion.
73. If ‘r’ is the radius vector and ω is the angular velocity then linear velocity v=.
74. Magnitude of angular momentum L =mvr (or) mωr2.
75. If ‘v’ is the linear velocity and ‘r’ is the radius then centripetal acceleration a 76. The centripetal force is always directed along the radius towards the centre of the circle.

77. The centripetal force acting on a body of ‘m’ and possessing a linear velocity ‘v’ and moving along a circular path of radius ‘r’ is 78. For a uniform circular motion magnitude of centripetal force is constant.

79. In the case of an electron going around the nucleus in an atom, the centripetal force is provided by electrostatic force of attraction between them.

80. A system of co-ordinate axis attached to the rotating platform will be under the same acceleration as that of the plat form in rotating frame of reference (or) non-inertial frame of reference.

81. An imaginary co-ordinate system which is attached to a rotating or accelerated body where Newton’s laws are not valid is called non-inertial frame of reference.

82. An imaginary coordinate system which is either at rest or in uniform motion where Newton’s laws are valid is called an inertial frame of reference.

83. An example for fictitious force is centrifugal force.

84. Radially outward force on a body in uniform circular motion, observed only in a rotating frame of reference is called centrifugal force.

85. Centrifugal force is a force developed with in a rotating body because of which a body tends to move away form the centre along the radius.

86. When a running car takes a sudden left turn, a man sitting in the back seat falls to the right due to inertia of direction.

87. centripetal force is directed towards the centre of the circle.
88. centrifugal force is directed away from the centre of the circle.
89. centrifugal force is necessary to keep a body of higher mass in equilibrium in a rotating frame of reference.

90. There is a natural tendency for all the bodies in rotating frame of reference to move away from the centre.

91. The Tendency for the bodies in rotating frame of reference to fly away from the centre is high (or) more for bodies of higher mass.

92. A centrifuge consists of a cylindrical vessel rotating about its own axis at high speed.

93. An electric motor is used to rotate a cylindrical vessel of a centrifuge.

94. In a centrifuge higher mass particles are thrown away from the centre where as lower mass particles are collected at the centre.

95. A domestic churner works on the principle of centrifuge.
96. With the help of a centrifuge sugar crystals are separated from molasses.
97. By using a centrifuge honey can be separated from a bee’s wax.
98. The centrifugal force pushes the water molecules from the wet clothes in a laundry drier.

99. Laundry drier works on the principle of centrifuge.

100. The gravitational force of attraction between the sun and earth acts as centripetal force and so the earth goes round the sun.

101. The centripetal force required to make the train travel on the curved track is provided by the outer rail slightly raised above the level of inner rail.

102. The outer rail of a curved track is slightly raised above the level of inner rail towards the centre of curvature and this is called banking of railway tracks.

103. If the banking of curved roads is not done then the vehicles would skid.

104. Man-made satellites are called artificial satellites.

105. Faster the initial horizontal speed of an object thrown greater is the radius of curved path.

106. Initially the motion of the satellite is tangential to the earth’s surface.

107. The artificial satellite revolves around the earth because of centripetal force provided by gravitational pull.
108. The principle of launching artificial satellite into a proper space orbit is to impart sufficient initial horizontal speed such that it revolves round the earth at a choosen height.

109. The artificial satellite that helps in telephone conversation is communication satellite.

110. The communication satellites send telephone conversations and fax messages through INTERNET.

111. Weather prediction satellite (INSAT) send information about cyclone.

112. Deforestation and expansion of deserts can be prevented with the help of the pictures taken by remote sensing satellites.

113. Satellites and space stations are used for the study of planets, stars and galaxies in the universe.

114. Spy satellites are being used for spying in defence services.

115. Centripetal force is a pull on the body towards the centre of the circle.