Download PROPERTIES OF MATTER Stress is defined as Restoring force per

PROPERTIES OF MATTER
Stress is defined as Restoring force per unit area.
A stress may be (i) tensile, (ii) compressible, (iii) shearing or tangential, and (iv) hydrostatic. These are illustrated in the figures below :
All stresses are measured in units of Nm-2. The dimensions are the same as those of
pressure viz., ML-1T-2.
See the picture
Strain =
Linear strain = l / l
Shear strain = ߠ = x / l
Volume strain = - V / V
All’s strain’s are dimensionless and are pure numbers without any units.
= Constant = Modulus of electricity
The relation is known as Hooks Law
Young’s Modulus Y =
Bulk Modulus K = -V
Shear Modulus or Modulus or Rigidity
=
Poisson’s Ratio
=
=
Pascal’s Law : The pressure exerted at any point in an enclosed fluid is transmitted
equally in all directions.
Archimede’s Principle : When a solid body is wholly or partially immersed in a fluid, it
experiences an upward buoyant force which acts through its centre of buoyancy.
Law of Flotation : When a body is just floating in a fluid, the weight of the fluid
displaced by it equal the total weight of the body and the centre of gravity of the body
and the centre of buoyancy lie along the same vertical line.
Motion in a Vertical circle : Let a body tied to one end of a string be whirled in a
vertical circle of radius r whose other end O is fixed. Let at any time t, the body be at
position P where the string makes an angle ߠ with the vertical.
If T is the tension in the string.
Tension in string T =
+ Mg cosߠ
At the lowest point A, ߠ = 0 tension at point A will be
TA =
+ Mg
At the highest point B, ߠ = , for highest point B it will be
TB =
– Mg
If TB is greater than zero, the string remains tight, but when TB < 0 or negative, the string
becomes lose and the body no longer remains on circular path.
For just completing the vertical circle TB = 0, then the speed at highest point B is called
the critical speed Vc.
Vc =
This equation gives critical velocity at highest point. If the velocity VB < VC, the string
becomes slack and the body falls downward instead of moving along the circular path.
Case (i) Condition for performing complete revolution :
The condition for the body to perform complete revolution is that tension in the string
must be always greater than zero at the highest point.
For this the velocity at highest point > critical velocity, VC =
and the velocity at
lowest point VA > (5rg).
Case (ii) Condition of Oscillation
If velocity of lowest point VA =
,
The particle will oscillate about the lowest point A.
Case (iii) Condition for the body to leave the circular path.
The body leaves the circular path when the tension in string becomes zero earlier than
the velocity
VA >
Thus if the speed of body at lowest point VA lies between
and
, the
body leaves the circular path somewhere between A and B.
Concial Pendulum
It consists of a string OA, whose upper end O is fixed and a bob is tied at the other free
end. The bob is given a horizontal push a little through angular displacement ߠ arranged
such that the bob describes a horizontal circle with uniform angular velocity in such a
way that the string always makes an angle ߠ with the vertical. As the string traces the
surface of the cone the arrangement is called a conical pendulum.
Surface Tension : It is measured as the tangential force per unit length acting along
any line drawn on the liquid surface. It also equals the free energy per unit area.
T=
=
The dimensions of ‘T’ are [MT-2] and its units are Nm-1 or Jm-2.
Excess pressure : for a drop : p = 2T/r
Excess pressure : for a Soap bubble = p = 4T/r.
(Because it contains two surfaces)
When a tube of radius (r) is immersed in a liquid of surface tension (T) and height of rise
of liquid of density ( ) and intact angle (ߠ).
Then for
<< h, we have
T=
For tubes of ‘insufficient length’
=
=
= a constant
Reynold’s Number
N=
= a dimensionless quantity
Equation of Continuity : For liquids flowing in tubes of area of cross section A &
velocity v, then A = constant.
Bernoulli’s Theorem. For a liquid undergoing stream-lined flow.
Kinetic energy + Gravitational potential energy + Pressure energy = Constant.
or
+ gh + P = constant
Torricelli’s Theorem. The liquid placed in a container to a height having a hole at the
bottom then Velocity of efflux = gh. It is independent of liquid.
Newton’s Law of Viscosity. The force required to maintain flow is
F=- A
where
is coefficient of viscosity, A area of cross-section of the tube,
is the velocity
gradient between two layers of a liquid.
are ML-1T-1 and its SI unit is Tor. This is related to its more often
The dimensions of
used unit, the poise, by the relation
1 Tor = 10 poise
When a particle of radius r placed in a liquid of coefficient of vislosity v moving with a
constant velocity after some time. Then the force experienced by the particle is
F=
r
The volume of liquid flowing in tubes is given by
Volume rate of flow = V =
(Poiseullie’s law)