, a
UNITS AND MEASUREMENTS Percentage error 100%
amean
The SI system : It is the international system of units.
At present internationally accepted for measurement. MOTION IN A STRAIGHT LINE
In this system there are seven fundamental and two
supplementary quantities and their corresponding s1 + s2 + s3
units are: Average speed, Vav =
t1 + t 2 + t 3
Quantity Unit Symbol a t +a t
1. Length metre m Average acceleration, aav = 1 1 2 2
t1 + t 2
2. Mass kilogram kg
3. Time second s The area under the velocity-time curve is equal to the
4. Electric current ampere A displacement and slope gives acceleration.
5. Temperature kelvin K If a body falls freely, the distance covered by it in each
6. Luminous intensity candela cd subsequent second starting from first second will be in
7. Amount of substance mole mol the ratio 1 : 3 : 5 : 7 etc.
Supplementary If a body is thrown vertically up with an initial velocity
1. Plane angle radian rad u, it takes u/g second to reach maximum height and
2. Solid angle steradian sr u/g second to return, if air resistance is negligible.
If air resistance acting on a body is considered, the time
Dimensions : These are the powers to which the taken by the body to reach maximum height is less than
fundamental units are raised to get the unit of a
physical quantity. the time to fall back the same height.
For a particle having zero initial velocity if s t ,
Uses of dimensions
(i) To check the correctness of a physical relation. where 2 , then particle’s acceleration increases with
(ii) To derive relationship between different physical time.
quantities. For a particle having zero initial velocity if s t ,
(iii) To convert one system of unit into another.
n 1u 1 = n 2 u 2 where 0 , then particle’s acceleration decreases with
a b c
time.
n1[M1a Lb1T1c ] = n2 [M 2 L 2T2 ]
Kinematic equations :
Significant figures : In any measurement, the reliable v = u + at (t) ; v2 = u2 + 2at (s)
digits plus the first uncertain digit are known as
1 a
significant figures. S = ut + at (t)2; Sn u (2 n 1)
2 2
Error : It is the difference between the measured value
applicable only when | a t | a t is constant.
and true value of a physical quantity or the uncertainty
at = magnitude of tangential acceleration, S = distance
in the measurements.
If acceleration is variable use calculus approach.
Absolute error : The magnitude of the difference
between the true value and the measured value is Relative velocity : v BA vB vA
called absolute error.
a1 a a1 , a2 a a2 , an a an MOTION IN A PLANE
Mean absolute error
| a1 | | a2 | ..... | an | 1
n If T is the time of flight, h maximum height, R horizontal
a = | ai |
range of a projectile, its angle of projection, then the
n ni 1
relations among these quantities.
Relative error : It is the ratio of the mean absolute error
gT 2
a h ...... (1);
to its true value or relative error = 8
a
gT2 = 2R tan ....... (2);
Percentage error : It is the relative error in per cent. R tan = 4h ....... (3)
, 2
2u sin u 2 sin 2 On a banked road, the maximum permissible speed
T ;h 1/2
g 2g æ u + tan q ö ÷
ççR s ÷
Vmax = çç g ÷
÷
u 2 sin 2 u2 è 1- u s tan qø
R ; Rmax when 45
g g
For a given initial velocity, to get the same horizontal WORK, ENERGY AND POWER
range, there are two angles of projection and 90° – .
The equation to the parabola traced by a body projected Work done W = FS cos
horizontally from the top of a tower of height y, with a Relation between kinetic energy E and momentum
velocity u is y = gx2/2u2, where x is the horizontal P 2mE
distance covered by it from the foot of the tower.
K.E. = 1/2 mV2; P.E. = mgh
gx 2 If a body moves with constant power, its velocity (v) is
Equation of trajectory is y x tan , which related to distance travelled (x) by the formula v x3/2
2u cos 2
2
is parabola. W
Power P = = F.V
Equation of trajectory of an oblique projectile in terms t
x Work due to kinetic force of friction between two contact
of range (R) is y x tan 1 surfaces is always negative. It depends on relative
R displacement between contact surfaces
Maximum height is equal to n times the range when WFK FK (Srel ) .
the projectile is launched at an angle = tan–1(4n).
W K, W total work due to all kinds of
In a uniform circular motion, velocity and acceleration
are constants only in magnitude. Their directions forces, K total change in kinetic energy.
change. Wconservative U ; Wconservative Total work
In a uniform circular motion, the kinetic energy of the due to all kinds of conservative forces.
body is a constant. W = 0, a 0, P constant, L=constant SDu Þ Total change in all kinds of potential energy.
velocity of separation
2 v2 Coefficient of restitution e
Centripetal acceleration, a r r v (always velocity of approach
r
applicable) a r v The total momentum of a system of particles is a
constant in the absence of external forces.
LAWS OF MOTION
SYSTEM OF PARTICLES & ROTATIOAL
Newton’s first law of motion or law of inertia : It is MOTION
resistance to change.
The centre of mass of a system of particles is defined as
Newton’s second law : F ma, F dp / dt
mi ri
2 the point whose position vector is R
Impulse : p F t, p2 p1 F dt M
The angular momentum of a system of n particles about
1
Newton’s third law : F12 F21 n
the origin is L ri p i ; L = mvr = I
Frictional force fs (fs ) max sR ; fk kR i 1
Circular motion with variable speed. For complete The torque or moment of force on a system of n particles
circles, the string must be taut in the highest position,
u 2 ³ 5ga . about the origin is ri Fi
Circular motion ceases at the instant when the string i
becomes slack, i.e., when T = 0, range of values of u for The moment of inertia of a rigid body about an axis is
which the string does go slack is 2ga u 5ga . defined by the formula I mi ri 2
Conical pendulum : w = g / h where h is height of a 1
The kinetic energy of rotation is K I 2
point of suspension from the centre of circular motion. 2
The acceleration of a lift
The theorem of parallel axes : I'z = Iz + Ma2
actual weight apparent weight
a= Theorem of perpendicular axes : Iz = Ix + Iy
mass For rolling motion without slipping vcm = R . The
If ‘a’ is positive lift is moving down, and if it is negative
the lift is moving up. kinetic energy of such a rolling body is the sum of kinetic
energies of translation and rotation :
, 3
1 2 1 2 Kepler’s 3rd law of planetary motion.
K mvcm I
2 2 T12 a13
A rigid body is in mechanical equilibrium if T2 a3 ;
T22 a 32
(a) It is translational equilibrium i.e., the total external
force on it is zero : Fi = 0.
(b) It is rotational equilibrium i.e., the total external MECHANICAL PROPERTIES
torque on it is zero : i = ri × Fi = 0. OF SOLIDS
If a body is released from rest on rough inclined plane,
Hooke’s law : stress strain
n
then for pure rolling r tan (Ic = nmr2) F
n 1 Young’s modulus of elasticity Y
n A
Rolling with sliding 0 s tan ; 1
n 1 Compressibility =
Bulk modulus
g sin
a g sin Y = 3k (1 – 2 )
n 1
Y = 2n (1 + )
GRAVITATION If S is the stress and Y is Young’s modulus, the energy
density of the wire E is equal to S2/2Y.
Newton’s universal law of gravitation If is the longitudinal strain and E is the energy density
Gm1m2 of a stretched wire, Y Young’s modulus of wire, then E
Gravitational force F =
r2 is equal to Y
1 2
Nm 2 2
G = 6.67 × 10 –11 F
kg 2 Thermal stress = = Y Dq
A
The acceleration due to gravity.
(a) at a height h above the Earth’s surface
GM E 2h
MECHANICAL PROPERTIES
g(h) g 1 for h << RE OF FLUIDS
(R E h)2 RE
Pascal’s law : A change in pressure applied to an
2h GM E
g(h) g(0) 1 where g(0) enclosed fluid is transmitted undiminished to every
RE R 2E point of the fluid and the walls of the containing vessel.
(b) at depth d below the Earth’s surface is Pressure exerted by a liquid column P = hrg
GM E d d Bernoulli’s principle P + v2/2 + gh = constant
g(d) 1 g (0) 1 Surface tension is a force per unit length (or surface
R E2 RE RE
energy per unit area) acting in the plane of interface.
(c) with latitude g1 = g – R 2 cos2
Stokes’ law states that the viscous drag force F on a
GM sphere of radius a moving with velocity v through a
Gravitational potential Vg = –
r fluid of viscosity F = – 6 av.
GM 2 r 2 ( – )g
Intensity of gravitational field I = 2 Terminal velocity VT =
r 9
The gravitational potential energy The surface tension of a liquid is zero at boiling point.
The surface tension is zero at critical temperature.
Gm1m 2
V constant If a drop of water of radius R is broken into n identical
r
drops, the work done in the process is 4 R2S(n1/3 – 1)
The escape speed from the surface of the Earth is
3T 1 1
2GM E and fall in temperature Dq = -
ve 2gR E and has a value of 11.2 km s–1. J r R
RE Two capillary tubes each of radius r are joined in
GM E
parallel. The rate of flow is Q. If they are replaced by
Orbital velocity, vorbi = gR E single capillary tube of radius R for the same rate of
RE flow, then R = 21/4 r.
A geostationary (geosynchronous communication) Ascent of a liquid column in a capillary tube
satellite moves in a circular orbit in the equatorial plane 2s cos f
at a approximate distance of 4.22 × 104 km from the h=
rr g
Earth’s centre.
UNITS AND MEASUREMENTS Percentage error 100%
amean
The SI system : It is the international system of units.
At present internationally accepted for measurement. MOTION IN A STRAIGHT LINE
In this system there are seven fundamental and two
supplementary quantities and their corresponding s1 + s2 + s3
units are: Average speed, Vav =
t1 + t 2 + t 3
Quantity Unit Symbol a t +a t
1. Length metre m Average acceleration, aav = 1 1 2 2
t1 + t 2
2. Mass kilogram kg
3. Time second s The area under the velocity-time curve is equal to the
4. Electric current ampere A displacement and slope gives acceleration.
5. Temperature kelvin K If a body falls freely, the distance covered by it in each
6. Luminous intensity candela cd subsequent second starting from first second will be in
7. Amount of substance mole mol the ratio 1 : 3 : 5 : 7 etc.
Supplementary If a body is thrown vertically up with an initial velocity
1. Plane angle radian rad u, it takes u/g second to reach maximum height and
2. Solid angle steradian sr u/g second to return, if air resistance is negligible.
If air resistance acting on a body is considered, the time
Dimensions : These are the powers to which the taken by the body to reach maximum height is less than
fundamental units are raised to get the unit of a
physical quantity. the time to fall back the same height.
For a particle having zero initial velocity if s t ,
Uses of dimensions
(i) To check the correctness of a physical relation. where 2 , then particle’s acceleration increases with
(ii) To derive relationship between different physical time.
quantities. For a particle having zero initial velocity if s t ,
(iii) To convert one system of unit into another.
n 1u 1 = n 2 u 2 where 0 , then particle’s acceleration decreases with
a b c
time.
n1[M1a Lb1T1c ] = n2 [M 2 L 2T2 ]
Kinematic equations :
Significant figures : In any measurement, the reliable v = u + at (t) ; v2 = u2 + 2at (s)
digits plus the first uncertain digit are known as
1 a
significant figures. S = ut + at (t)2; Sn u (2 n 1)
2 2
Error : It is the difference between the measured value
applicable only when | a t | a t is constant.
and true value of a physical quantity or the uncertainty
at = magnitude of tangential acceleration, S = distance
in the measurements.
If acceleration is variable use calculus approach.
Absolute error : The magnitude of the difference
between the true value and the measured value is Relative velocity : v BA vB vA
called absolute error.
a1 a a1 , a2 a a2 , an a an MOTION IN A PLANE
Mean absolute error
| a1 | | a2 | ..... | an | 1
n If T is the time of flight, h maximum height, R horizontal
a = | ai |
range of a projectile, its angle of projection, then the
n ni 1
relations among these quantities.
Relative error : It is the ratio of the mean absolute error
gT 2
a h ...... (1);
to its true value or relative error = 8
a
gT2 = 2R tan ....... (2);
Percentage error : It is the relative error in per cent. R tan = 4h ....... (3)
, 2
2u sin u 2 sin 2 On a banked road, the maximum permissible speed
T ;h 1/2
g 2g æ u + tan q ö ÷
ççR s ÷
Vmax = çç g ÷
÷
u 2 sin 2 u2 è 1- u s tan qø
R ; Rmax when 45
g g
For a given initial velocity, to get the same horizontal WORK, ENERGY AND POWER
range, there are two angles of projection and 90° – .
The equation to the parabola traced by a body projected Work done W = FS cos
horizontally from the top of a tower of height y, with a Relation between kinetic energy E and momentum
velocity u is y = gx2/2u2, where x is the horizontal P 2mE
distance covered by it from the foot of the tower.
K.E. = 1/2 mV2; P.E. = mgh
gx 2 If a body moves with constant power, its velocity (v) is
Equation of trajectory is y x tan , which related to distance travelled (x) by the formula v x3/2
2u cos 2
2
is parabola. W
Power P = = F.V
Equation of trajectory of an oblique projectile in terms t
x Work due to kinetic force of friction between two contact
of range (R) is y x tan 1 surfaces is always negative. It depends on relative
R displacement between contact surfaces
Maximum height is equal to n times the range when WFK FK (Srel ) .
the projectile is launched at an angle = tan–1(4n).
W K, W total work due to all kinds of
In a uniform circular motion, velocity and acceleration
are constants only in magnitude. Their directions forces, K total change in kinetic energy.
change. Wconservative U ; Wconservative Total work
In a uniform circular motion, the kinetic energy of the due to all kinds of conservative forces.
body is a constant. W = 0, a 0, P constant, L=constant SDu Þ Total change in all kinds of potential energy.
velocity of separation
2 v2 Coefficient of restitution e
Centripetal acceleration, a r r v (always velocity of approach
r
applicable) a r v The total momentum of a system of particles is a
constant in the absence of external forces.
LAWS OF MOTION
SYSTEM OF PARTICLES & ROTATIOAL
Newton’s first law of motion or law of inertia : It is MOTION
resistance to change.
The centre of mass of a system of particles is defined as
Newton’s second law : F ma, F dp / dt
mi ri
2 the point whose position vector is R
Impulse : p F t, p2 p1 F dt M
The angular momentum of a system of n particles about
1
Newton’s third law : F12 F21 n
the origin is L ri p i ; L = mvr = I
Frictional force fs (fs ) max sR ; fk kR i 1
Circular motion with variable speed. For complete The torque or moment of force on a system of n particles
circles, the string must be taut in the highest position,
u 2 ³ 5ga . about the origin is ri Fi
Circular motion ceases at the instant when the string i
becomes slack, i.e., when T = 0, range of values of u for The moment of inertia of a rigid body about an axis is
which the string does go slack is 2ga u 5ga . defined by the formula I mi ri 2
Conical pendulum : w = g / h where h is height of a 1
The kinetic energy of rotation is K I 2
point of suspension from the centre of circular motion. 2
The acceleration of a lift
The theorem of parallel axes : I'z = Iz + Ma2
actual weight apparent weight
a= Theorem of perpendicular axes : Iz = Ix + Iy
mass For rolling motion without slipping vcm = R . The
If ‘a’ is positive lift is moving down, and if it is negative
the lift is moving up. kinetic energy of such a rolling body is the sum of kinetic
energies of translation and rotation :
, 3
1 2 1 2 Kepler’s 3rd law of planetary motion.
K mvcm I
2 2 T12 a13
A rigid body is in mechanical equilibrium if T2 a3 ;
T22 a 32
(a) It is translational equilibrium i.e., the total external
force on it is zero : Fi = 0.
(b) It is rotational equilibrium i.e., the total external MECHANICAL PROPERTIES
torque on it is zero : i = ri × Fi = 0. OF SOLIDS
If a body is released from rest on rough inclined plane,
Hooke’s law : stress strain
n
then for pure rolling r tan (Ic = nmr2) F
n 1 Young’s modulus of elasticity Y
n A
Rolling with sliding 0 s tan ; 1
n 1 Compressibility =
Bulk modulus
g sin
a g sin Y = 3k (1 – 2 )
n 1
Y = 2n (1 + )
GRAVITATION If S is the stress and Y is Young’s modulus, the energy
density of the wire E is equal to S2/2Y.
Newton’s universal law of gravitation If is the longitudinal strain and E is the energy density
Gm1m2 of a stretched wire, Y Young’s modulus of wire, then E
Gravitational force F =
r2 is equal to Y
1 2
Nm 2 2
G = 6.67 × 10 –11 F
kg 2 Thermal stress = = Y Dq
A
The acceleration due to gravity.
(a) at a height h above the Earth’s surface
GM E 2h
MECHANICAL PROPERTIES
g(h) g 1 for h << RE OF FLUIDS
(R E h)2 RE
Pascal’s law : A change in pressure applied to an
2h GM E
g(h) g(0) 1 where g(0) enclosed fluid is transmitted undiminished to every
RE R 2E point of the fluid and the walls of the containing vessel.
(b) at depth d below the Earth’s surface is Pressure exerted by a liquid column P = hrg
GM E d d Bernoulli’s principle P + v2/2 + gh = constant
g(d) 1 g (0) 1 Surface tension is a force per unit length (or surface
R E2 RE RE
energy per unit area) acting in the plane of interface.
(c) with latitude g1 = g – R 2 cos2
Stokes’ law states that the viscous drag force F on a
GM sphere of radius a moving with velocity v through a
Gravitational potential Vg = –
r fluid of viscosity F = – 6 av.
GM 2 r 2 ( – )g
Intensity of gravitational field I = 2 Terminal velocity VT =
r 9
The gravitational potential energy The surface tension of a liquid is zero at boiling point.
The surface tension is zero at critical temperature.
Gm1m 2
V constant If a drop of water of radius R is broken into n identical
r
drops, the work done in the process is 4 R2S(n1/3 – 1)
The escape speed from the surface of the Earth is
3T 1 1
2GM E and fall in temperature Dq = -
ve 2gR E and has a value of 11.2 km s–1. J r R
RE Two capillary tubes each of radius r are joined in
GM E
parallel. The rate of flow is Q. If they are replaced by
Orbital velocity, vorbi = gR E single capillary tube of radius R for the same rate of
RE flow, then R = 21/4 r.
A geostationary (geosynchronous communication) Ascent of a liquid column in a capillary tube
satellite moves in a circular orbit in the equatorial plane 2s cos f
at a approximate distance of 4.22 × 104 km from the h=
rr g
Earth’s centre.
