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#### Page No 297:

The ratio of stress to strain will decrease.

Beyond the elastic limit, the body loses its ability to restore completely when subjected to stress. Thus, there occurs more strain for a given stress. At some points, however, the body undergoes strain without the application of stress. So, the ratio of stress to strain decreases.

#### Page No 297:

Let the CSA of the wire be A.

The other $\frac{1}{2}Mgl$ is converted into kinetic energy of the mass.

When the mass leaves its initial point on the spring, it acquires a velocity as it moves down. The velocity reaches its maximum at the end point. The spring oscillates. Finally, when the kinetic energy is dissipated into heat, the spring comes to rest.

#### Page No 297:

The elephant has a greater weight than a mouse, but the material that makes their bones is the same. This means that in order to sustain an elephant's weight, one's bones need to suffer less stress. Stress = Force/area. A greater cross-sectional area reduces stress on the bones. This is why an elephant's bones are thicker.

#### Page No 297:

Let my length = L
Let the increase in length = l
Strain

Yes, the yield point is much less than the 1% strain because the human body consists of joints and not one uniform solid structure.

#### Page No 297:

The energy of vibration dissipates as heat from the shock absorber.

#### Page No 297:

When a compressed spring dissolves in an acid, the acid molecules leave the sold lattice of the spring faster than the uncompressed spring. This in turn increases the kinetic energy of the solution. As a result, the temperature of the acid also increases. However, this temperature increase will be very small because the mechanical energy content in the spring is lesser than its chemical energy content.

#### Page No 297:

It floats because of the surface tension of water. The surface of water behaves like a stretched membrane. When a blade is placed on the water surface, it's unable to pierce the stretched membrane of water due to its low weight and remains floating.

However, if the blade is placed below the surface of water, it no longer experiences the surface tension and sinks to the bottom as the density of the blade is greater than that of water.

#### Page No 297:

A liquid wets a surface when the angle of contact of the liquid with the surface is small or zero. Due to its fibrous nature, cloth produces capillary action when in contact with water. This makes clothes have very small contact angles with water. When wax is rubbed over cloth, the water does not wet the cloth because wax has a high contact angle with water.

#### Page No 297:

No, the water will neither rise nor fall in the silver capillary.

Thus, the water level neither rises nor falls.

#### Page No 297:

No, we cannot conclude the surface tension to be zero solely by the fact that the liquid neither rises nor falls in a capillary.

The height of the liquid inside a capillary tube is given by $h=\frac{2T\mathrm{cos}\theta }{r\rho g}\phantom{\rule{0ex}{0ex}}$. From the equation, we see that the height (h) of the liquid may also be zero if the contact angle $\theta$ between the liquid and the capillary tube is

#### Page No 297:

When water is poured in a glass, it reaches the brim and rises further. The edge of the glass lies below the water level. In this case, the force of attraction due to molecules of the glass surface is not perpendicular to the solid. Here, the contact angle can be greater than the standard contact angle for a pair of substances.

#### Page No 297:

As the angle of contact is 0, there is no force between the surface of the tube and the liquid. The diameter of the liquid surface is pulled on both sides by equal and opposite forces of surface tension. This results in no net force remaining on the surface of the liquid. Hence, the liquid stays in equilibrium. #### Page No 297:

No, it does not violate the principle of conservation of energy.

There is a force of attraction between glass and water, which is why the liquid rises in the tube. However, when water and glass are not in contact, there exists a potential energy in the system. When they are brought into contact, this potential energy is first converted into kinetic energy, which lets the liquid rush upwards in the tube, and then into gravitational potential energy. Therefore,  energy is not created in the process.

#### Page No 297:

A mosquito thrown into water has its wings wetted. Now, wet wing surfaces tend to stick together because of the surface tension of water. This does not let the mosquito fly.

#### Page No 297:

The forces act tangentially to the bubble surface on both sides of a given line but they have one component normal to the bubble surface. This component balances the force due to excess pressure inside the bubble.

In the figure, let us consider a small length AB on the surface of the spherical bubble. Let the surface forces act tangentially along A and B. On producing the forces backwards, they meet at a point O. By the parallelogram law of forces, we see that the resultant force acts opposite to the normal. This balances the internal forces due to excess pressure. #### Page No 297:

No. The average intermolecular distances do not increase with an increase in the surface area.

A soap bubble's layer consists of several thousand layers of molecules. An increase in the surface area causes the surface energy to also increase. This in turn allows more and more molecules from the inner liquid layers of the bubble to attain potential energy, enabling them to enter the outer surface of the bubble. Hence, the surface area increases.

#### Page No 298:

No. For a liquid at rest, no viscous forces exist.

Viscous forces oppose relative motion between the layers of a liquid. These layers do not exist in a liquid that is at rest. Therefore, it is obvious that viscous forces are non-existent in a static liquid.

#### Page No 298:

The motion of any liquid is dependent upon the amount of stress acting on it. The motion of one layer of liquid is resisted by the other due to the property of viscosity. A river bed remains in a static state. Therefore, any immediate layer of liquid in contact with the river bed will also remain static due to the frictional force. However, the next layer of liquid above this static layer will have a greater velocity due to lesser resistance offered by the static layer. Moving upwards, subsequent layers provide lesser and lesser resistance to the movement of the layers above it. Finally, the topmost layer acquires the maximum velocity. Therefore, for a river, the surface waters flow the fastest.

#### Page No 298:

Castor oil will come to rest more quickly because it has a greater coefficient of viscosity than water.

Castor oil has a higher viscosity than water. It will therefore, lose kinetic energy and come to rest faster than water.

#### Page No 298:

Correct option: (d) 2000 N

#### Page No 298:

Correct option: (a)

Breaking stress depends upon the intermolecular/ inter-atomic forces of attraction within materials. In other words, it depends upon the material of the wire.

#### Page No 298:

Correct option: (b) 20 kg

As the wire is cut into two equal parts, both have equal cross-sectional areas. Therefore, a weight of 20 kg exerts a force of 20g on both the pieces. Breaking stress depends upon the material of the wire. Since 20g of force is exerted on wires with equal cross-sectional areas, both the wires can sustain a weight of 20 kg.

#### Page No 298:

Correct option: (a)  1/8

#### Page No 298:

Correct option: (b) 1.0 mm

#### Page No 298:

Correct option: (a) s​mallest at the top and gradually increases down the rod

As the rod is of uniform mass distribution and stretched by its own weight, the topmost part of the rod experiences maximum stress due to the weight of the entire rod. This stress leads to lateral strain and the rod becomes thinner. Moving down along the length of the rod, the stress decreases because the lower parts bear lesser weight of the rod. With reduced stress, the lateral strain also reduces. Hence, the diameter of the rod gradually increases from top to bottom.

#### Page No 298:

Correct option: (a)  ll

#### Page No 298:

Correct option: (c)

)

#### Page No 298:

Correct option: (b)

If the velocity of the mass is a maximum at the bottom, then the string experiences tension due to both the weight of the mass and the high centrifugal force. Both these factors weigh the mass downwards. The tension is therefore, maximum at the lowest point, causing the string to most likely break at the bottom.

#### Page No 298:

Correct option: (d)

None of these is the correct option. The decreased gravitational potential energy transforms partly as elastic energy, partly as kinetic energy and also in the form of dissipated heat energy.

#### Page No 298:

The correct option is (c).

The surface  of a liquid refers to the layer of molecules that have higher potential energy than the bulk of the liquid. This layer is typically 10 to 15 times the diameter of the molecule. Now, the size of an average molecule is around 1 nm = ${10}^{-9}$m, so a diameter of 10 to 15 times would be of order ​$10×{10}^{-9}={10}^{-8}$ m.

#### Page No 298:

Correct option: (b)
As the ice cube melts completely, the water thus formed will have minimum surface area due to its surface tension. Any state of matter that has a minimum surface area to its volume takes the shape of a sphere. Therefore, as the ice melts, it will take the shape of a sphere.

#### Page No 298:

Correct option: (a)

As the water droplets merge to form a single droplet, the surface area decreases. With this decrease in surface area, the surface energy of the resulting drop also decreases. Therefore, extra energy must be liberated from the drop in accordance with the conservation of energy.

#### Page No 298:

Correct option: (c)
​Dimension of modulus of elasticity: $\frac{F}{A}}{∆l}{l}}=\frac{\left[ML{T}^{-2}\right]}{{L}^{2}}=\left[M{L}^{-1}{T}^{-2}\right]$
Dimension of moment of force: ​$FL=\left[ML{T}^{-2}\right]\left[L\right]=\left[M{L}^{2}{T}^{-2}\right]$
Dimension of surface tension: ​$\frac{F}{L}=\frac{\left[ML{T}^{-2}\right]}{L}=\left[M{T}^{-2}\right]$
Dimension of coefficient of viscosity: ​$\frac{FL}{Av}=\frac{\left[ML{T}^{-2}\right]\left[L\right]}{\left[{L}^{2}\right]\left[L{T}^{-1}\right]}=\left[M{L}^{-1}{T}^{-1}\right]$

#### Page No 298:

Correct option: (d)

#### Page No 298:

Correct option: (a)

Excess pressure inside a bubble is given by: $P=\frac{4T}{r}$.
When air is pushed into the bubble, it grows in size. Therefore, its radius increases. An increase in size causes the pressure inside the soap bubble to decrease as pressure is inversely proportional to the radius.

#### Page No 298:

Correct option: (c)

The smaller bubble has a greater inner pressure than the bigger bubble. Air moves from a region of high pressure to a region of low pressure. Therefore, air moves from the smaller to the bigger bubble.

#### Page No 298:

Correct option: (c) #### Page No 299:

Correct option: (d)
Let the excess pressure inside the second bubble be P.
​∴ Excess pressure inside the first bubble = 2P
Let the radius of the second bubble be R.
Let the radius of the first bubble be x.

#### Page No 299:

Correct option: (c)

This equation has the characteristic of a rectangular hyperbola. Therefore, curve (c) is a rectangular hyperbola.

#### Page No 299:

Correct option: (b) #### Page No 299:

Correct option: (d)

#### Page No 299:

Correct option: (d)

Viscosity is one property of fluids. Fluids include both liquids and gases.

#### Page No 299:

Correct option: (a)

The force of viscosity arises from molecular interaction between different layers of fluids that are in motion. Molecular forces are electromagnetic in nature. Therefore, viscosity must also be electromagnetic.

#### Page No 299:

Correct option: (c)
The viscous force acts tangentially between two parallel layers of a liquid. In terms of force on a material, it is analogous to a shearing force.

#### Page No 299:

Correct option: (c)
Air has viscosity. During rainfall, the raindrops acquire acceleration due to gravity. However, the increase in velocity is hindered by the viscous force acting upwards. A gradual balance between the two opposing forces causes the raindrops to attain a terminal velocity, thus, falling with a uniform velocity.

#### Page No 299:

Correct option: (b)

The density of wood is less than that of water.When a piece of wood is immersed deep inside a long column of water and released, it experiences a buoyant force that gives it an upward acceleration. The velocity of wood increases as its motion is accelerated by the buoyant force. However, the viscous drag force acts simultaneously to oppose its upward motion. As a result, the initial acceleration decreases and the wood rises with a decreasing upward acceleration.

#### Page No 299:

Correct option: (d)

In vacuum, no viscous force exists. The sphere therefore, will have constant acceleration because of gravity. An accelerated motion implies that it won't have uniform velocity throughout its motion. In other words, there will be no terminal velocity.

#### Page No 299:

Correct option: (b)
Initially, when the ball starts moving, its velocity is small. Gradually, the velocity of the ball increases due to acceleration caused by gravity. However, as the velocity increases, the viscous force acting on the ball also increases. This force tends to decelerate the ball. Therefore, after reaching a certain maximum velocity, the ball slows down.

#### Page No 299:

Correct option: (a), (b), (c), (d)

All options are correct.
(a) When a weight is loaded on a wire, the length of the wire increases. The relationship between weight and length is linear.
(b) When a weight is loaded, it produces stress on the wire. The relationship between stress and increase in length is also linear.
(c) When stress is applied, strain develops. Therefore, both are linearly related.
(d) Since the value of Y  for the wire is unknown, X may also be the increase in its length. Nevertheless, they still show the same linear relationship.

#### Page No 299:

Correct option: (c) & (d)

(c) The surface molecules acquire air and liquid molecules in their sphere of influence.
(d) The surface molecules have different magnitudes of forces pulling them from the top and the bulk. So, they are affected by a net finite force in one direction.

#### Page No 299:

Correct option: (a), (b), (d)

#### Page No 299:

Correct option: (a), (b)

The angle of contact between a solid and a liquid depends upon the molecular forces of both the substances. Therefore,  it depends upon the material of the solid and the liquid.

#### Page No 299:

Correct option: (c) Let the height of the liquid-filled column be L.
Let the radius be denoted by R.

#### Page No 299:

Correct option: (c), (d)

If the liquid level does not rise, it may be assumed that the surface tension is zero or the contact angle is 90°, or both. However, we cannot tell for sure whether the surface tension of the liquid is zero or the contact angle is 0°.

#### Page No 299:

Correct option: (b), (c), (d)

(b) There is no gravitational force acting downwards. However, when the starting velocity is 20 m/s, the viscous force, which is directly proportional to velocity, becomes maximum and tends to accelerate the ball upwards.

Thus, we see that the initial acceleration of the ball will be 9.8 ms${}^{-2}$.

(c) The velocity of the ball will decrease with time because of the upward viscous drag. As the force of viscosity is directly proportional to the velocity of the ball, the acceleration due to the viscous force will also decrease.

(d) When all the kinetic energy of the ball is radiated as heat due to the viscous force, the ball comes to rest.

#### Page No 300:

Given:
Mass of the load (m) = 10 kg
Length of wire (L) = 3 m
Area of cross-section of the wire (A) = 4 mm2 = 4.0 × 10−6 m2
Young's modulus of the metal = 2.0 × 1011 N m−2

(a) Stress = F/A

F = mg
= $10×10$
= 100 N    (= 10 m/s2)

(b) Strain = $\frac{\Delta L}{L}$

Or,

(c) Let the elongation in the wire be $∆L$.

#### Page No 300:

Given:
Radius of cylinder (r) = 2 cm =
Length of cylinder (L) = 2 m
Mass of the load = 100 kg
Young's modulus of the metal =

(a) Stress(ρ) is given by: $\frac{F}{A}$
Here, F is the force given by mg = ( Taking g = 10 m/s2)
A is the area of cross-section = πr2  = 4π

(b) Strain is given by:

(c) Compression of the cylinder:
ΔL = strain × L
= 4 × 10−6 × 2 = 8 × 10−6 m

#### Page No 300:

Given:

The elastic limit of steel indicates the maximum pressure that steel can bear.
Let the maximum elongation of steel wire be $∆L$.

Hence, the required elongation of steel wire is 2 mm.

#### Page No 300:

Given:
Young's modulus of steel = 2 × 1011 N m−2
Young's modulus of copper = 1.3 × 10 11 N m−2
Both wires are of equal length and equal cross-sectional area. Also, equal tension is applied on them.
As per the question:

Here: Lsteel and LCu denote the lengths of steel and copper wires, respectively.
Asteel and ACu denote the cross-sectional areas of steel and copper wires, respectively.
Fsteel and FCu  denote the tension of steel and cooper wires, respectively.

(b)

Hence, the required ratio is 20 : 13.

#### Page No 300:

Given that both wires are of equal length and equal cross-sectional area,
the block applies equal tension on both of them.

∴

Hence, the required ratio of the longitudinal strains is 20 : 13.

#### Page No 300:

(a) Given:

Tension in lower wire ${T}_{l}={m}_{1}g+w$
Here: g is the acceleration due to gravity
∴ Stress in lower wire$=\frac{{\mathrm{T}}_{l}}{{\mathrm{A}}_{l}}=\frac{{m}_{1}g+w}{{\mathrm{A}}_{l}}$

Now, tension in upper wire ${T}_{2}={m}_{1}g+{m}_{2}g+w$

∴ Stress in upper wire$=\frac{{\mathrm{T}}_{u}}{{\mathrm{A}}_{u}}=\frac{{m}_{\mathit{2}}g+{m}_{1}g+w}{{\mathrm{A}}_{u}}$

For the same breaking stress, the maximum load that can be put is 140 N or 14 kg. The lower wire will break first if the load is increased.

(b) :
Tension in lower wire ${T}_{l}={m}_{1}g+w$
Here: g is the acceleration due to gravity
∴ Stress in lower wire:

Now, tension in upper wire ${T}_{2}={m}_{1}g+{m}_{2}g+w$

∴ Stress in upper wire:

For the same breaking stress, the maximum load that can be put is 20 N or 2 kg. The upper wire will break first if the load is increased.

#### Page No 300:

Given:
Force (F) applied by two persons on the rope = 100 N

Hence, the required Young's modulus  for the rope is .

#### Page No 300:

Given:
Cross-sectional area of steel rod A = 4 cm2 = 4 × 10−4 m2
Length of steel rod L = 2 m
Compression during night hours ΔL = 0.1 cm = 10−3 m
Young modulus of steel Y = 1.9 × 1011 N m−2

Let the tension developed at night be F.

∴ Required tension developed in steel rod during night hours = 3.8 × 104 N.

#### Page No 300:

Given:
Force (F) = m2g/2
Area of cross-section of the string = A
Young's modulus = Y
Let a be the acceleration produced in block m2 in the downward direction and be the tension in the string.
From the free body diagram: From equations (i) and (ii), we get:

Again, T = F + m1a

On applying the values of F and a, we get:

$⇒T=\frac{{m}_{2}g}{2}+{m}_{1}\frac{{m}_{2}g}{2\left({m}_{1}+{m}_{2}\right)}$

We know that:

∴ Required strain developed in the string .

#### Page No 300:

Given:
Mass of sphere (m) = 20 kg
Length of metal wire (L) = 4 m
Diameter of wire (d = 2r) = 1 mm
⇒ r = 5 × 10−4 m
Young's modulus of the metal wire = 2.0 × 1011 N m−2
Tension in the wire in equilibrium = T
T= mg
When it is moved at an angle θ and released, let the tension at the lowest point be T'​.
$⇒T\text{'}=mg+\frac{m{v}^{2}}{r}$
The change in tension is due to the centrifugal force.
∴​ $∆T=T\text{'}-T$

Now, using work energy principle:

Applying the value of v2 in (i):

Hence, the required maximum value of θ is 35.4˚.

#### Page No 301:

Given:
Original length of steel wire (L) = 1 m
Area of cross-section (A) = 4.00 mm2 = 4 × 10−2 cm2
Young's modulus of steel (Y) = 2 × 1011 N/m2
Acceleration due to gravity (g) = 10 m s−2

Let T be the tension in the string after the load is suspended and θ be the angle made by the string with the vertical, as shown in the figure: Expanding the above equation using the binomial theorem:

Increase in length:
ΔL = (AC + CB) − AB
AC = (l2 + x2)1/2
ΔL = 2 (l2 + x2)1/2 − 2l

We know that:

From the free body diagram:

On solving the above equation, we get x = 1.5 cm.
Hence, the required vertical depression is 1.5 cm.

#### Page No 301:

Given:
Cross-sectional area of copper wire A = 0.01 cm2 = 10−6 m2
Applied tension T = 20 N
Young modulus of copper Y = 1.1 × 1011 N/m2
Poisson ratio σ = 0.32

We know that:
$Y=\frac{FL}{A∆L}$

Hence, the required decrease in the cross -sectional area is .

#### Page No 301:

Given:
Bulk modulus of water (B) =
In order to decrease the volume (V) of a water sample by 0.01%, let the increase in pressure be P.

Hence, the required increase in pressure is .

#### Page No 301:

Given:

Depth (d) = 400 m
Density of water at the surface (ρ0) = 1030 kg/m3
We know that:

Here: ρ= density of water at a depth
m =  mass
V0 = volume at the surface
Vd = volume at a depth

Using equations (i) and (ii), we get:

Hence, the required density at a depth of 400 m below the surface is 2 kg/m3.​

#### Page No 301:

Given:
Face area of steel plate A = 4 cm2 = 4 × 10−4 m2
Thickness of steel plate d = 0.5 cm = 0.5 × 10−2 m
Applied force on the upper surface F   = 10 N
Rigidity modulus of steel = 8.4 × 1010 N m−2
Let θ be the angular displacement.
Rigidity modulus $m=\frac{F}{A\theta }$

∴ Lateral displacement of the upper surface with respect to the lower surface = θ × d
⇒ (0.297) × 10−6 × (0.5) × 10−2
⇒ 1.5 × 10−9 m

Hence, the required lateral displacement of the steel plate is 1.5 × 10−9 m.

#### Page No 301:

Given:
Length of thread l = 5 cm = 5 × 10−2 m
Surface tension of water T = 0.76 N/m
We know that:
F = T × = 0.76 × 5 × 10−2
= 3.8 × 10−3 N
Therefore, the water surface on one side of the thread pulls it with a force of 3.8 × 10−3 N.

#### Page No 301:

Given:

(a) Excess pressure inside mercury drop:

(b) Excess pressure inside the soap bubble:

(c) Excess pressure inside the air bubble:

#### Page No 301:

Given:
Surface area of mercury drop, A = 1 mm2 = 10−6 m2
Radius of mercury drop, r = 4 mm = 4 × 10−3 m
Atmospheric pressure, P0 = 1.0 × 105 Pa
Surface tension of mercury, T = 0.465 N/m

(a) Force exerted by air on the surface area:
F = P0A
⇒ F= 1.0 × 105 × 10−6 = 0.1 N

(b) Force exerted by mercury below the surface area:

(c) Force exerted by mercury surface in contact with it:

#### Page No 301:

Given:
Surface tension of water T = 7.5 × 10−2 N/m
Taking cos θ = 1:
Radius of capillary A (rA) = 0.5 mm = 0.5 × 10−3 m

Height of water level in capillary A:

Radius of capillary B (rB) = 1 mm = 1 × 10−3 m

Height of water level in capillary B:

Radius of capillary C (rC) = 1.5 mm = 1.5 × 10−3 m
Height of water level in capillary C:

#### Page No 301:

Let T be the surface tension, r be the inner radius of the capillary tube and ρ be the density of the liquid.

For cos θ = 1, height (h) of the liquid level is given as:

$h=\frac{2T\mathrm{cos}\theta }{r\mathrm{\rho }g}$

Now, for mercury:

...(i)

For water:

${h}_{w}=\frac{2{T}_{w}}{r{\mathrm{\rho }}_{w}g}$    ...(ii)

Dividing (ii) by (i), we get:

Height of the water level:

hw = 2 × 2.19 = 4.38 cm

Hence, the required rise in the water level in the capillary tube is 4.38 cm.

#### Page No 301:

Given:
Radius of tube r = 1.0 mm
Atmospheric pressure = 76 cm of Hg
Contact angle of mercury with glass θ = 135°
Surface tension of mercury T = 0.465 N/m
Density of mercury = 13600 kg m−3

Let h be the rise in level in the barometer.

∴ Net rise in level in the barometer tube = H − h
= 76 − 0.48
= 75.52 cm

#### Page No 301:

Given:
Radius of capillary tube r = 0.5 mm = 5 × 10−4 m
Depth (where pressure is to be found) h = 5.0 cm = 5 × 10−2 m
Surface tension of water T = 0.075 N/m
Excess pressure at 5 cm before the surface:
P = ρhg = 1000 × (5 × 10−2) × 9.8 = 490 N/m2
Excess pressure at the surface is given by:

Difference in pressure: P0 P

Hence, the required difference in pressure is 190 N/m2.

#### Page No 301:

Given:
Radius of cylindrical vessel, = 6.0 cm = 0.06 m
Surface tension of water, T = 0.075 J/m2
Area, A = πr2 = π × (0.06)2
Surface energy = T × A
= (0.075) × (3.14) × (0.06)2
= 8.5 × 10−4 J
Therefore, the surface energy of water kept in a cylindrical vessel is 8.5 × 10−4 J.

#### Page No 301:

Given:
Initial radius of mercury drop R = 2 mm = 2 × 10−3 m
Surface tension of mercury T = 0.465 J/m2
Let the radius of a small drop of mercury be r.
As one big drop is split into 8 identical droplets:
volume of initial drop = 8 × (volume of a small drop)
$\left(\frac{4}{3}\right)\pi {R}^{3}=\left(\frac{4}{3}\right)\mathrm{\pi }{r}^{3}×8\phantom{\rule{0ex}{0ex}}$

Taking cube root on both sides of the above equation:

$r=\frac{R}{2}=10$-3 m
Surface energy = T × surface area

∴ Increase in surface energy = TA' − TA
= (8 × 4 πr2 − 4πR2) T
$=4\pi T\left[8×\left(\frac{{R}^{2}}{4}\right)-{R}^{2}\right]\phantom{\rule{0ex}{0ex}}=4\pi T{R}^{2}$
= 4 × (3.14) × (0.465) × (4 × 10−6)
= 23.36 × 10−6
= 23.4 μJ

Hence, the required increase in the surface energy of the mercury droplets is 23.4 μJ.

#### Page No 301:

Given:
Radius of capillary tube r = 1 mm = 10−3 m

(a) Let T be the surface tension and ρ be the density of the liquid.

Then, for cos θ = 1, height (h) of liquid level:

$h=\frac{2T}{r\rho g}$    ...(i),
where g is the acceleration due to gravity

(b) Let the new length of the tube be h'.

The water surface in the capillary makes an angle of 60with the wall.

#### Page No 301:

Given:
Radius of tube r = 1 mm = 10−3 m
Contact angle of mercury with glass θ = 135°
Surface tension of mercury T = 0.465 N/m
Let ρ be the density of mercury.

(a) Depression (h) of mercury level is expressed as follows:

$h=\frac{2T\mathrm{cos}\theta }{r\rho g}$    ...(i)

(b) If the length dipped inside is half the result obtained above:
New depression h'= $\frac{h}{2}$
Let the new contact angle of mercury with glass be θ'.

∴ $h\text{'}=\frac{2T\mathrm{cos}\theta \text{'}}{r\mathrm{\rho }g}$    ...(ii)

Dividing equation (ii) by (i), we get:

$\frac{h\text{'}}{h}=\frac{\mathrm{cos}\theta \text{'}}{\mathrm{cos}\theta }\phantom{\rule{0ex}{0ex}}⇒\mathrm{cos}\theta \text{'}=\frac{\mathrm{cos}\theta }{2}$
$⇒\theta =112°$

#### Page No 301:

Given:
Surface tension of water T = 0.075 N/m
Separation between the glass plates d = 1 mm = 10−3 m
Density of water ρ = 103 kg/m3

Applying law of conservation of energy:

T (2L) = [1 × (10−3) × h] ρg

Therefore, the rise of water in the space between the plates is 1.5 cm.

#### Page No 301:

Given:
Edge of the ice cube (a) = 1.0 cm
The water that is formed due to the melting of ice acquires a spherical surface.
In the absence of gravity, let the radius of the spherical surface be r.
Volume of ice cube = volume of spherical surface of water

Surface area of spherical water surface = 4πr2

#### Page No 301:

Given:
Surface tension of soap solution T = 0.030 N m−1

The excess pressure inside the loop is expressed as follows:
$∆P=\frac{4T}{r}$

#### Page No 301:

Given:
Radius of metallic sphere r = 1 mm = 10−3 m
Speed of the sphere v = 10−2 m/s
Coefficient of viscosity η = 8 poise = 0.8 decapoise
Mass m = 50 mg = 50 × 10−3 kg
Density of glycerin σ = 1260 kg/m3

(a)
Viscous force exerted by glycerine on the sphere F = 6πηrv
⇒ F= 6 × (3.14) × (0.8) × 10−3 × (10−2)
= 1.50 × 10−4 N

(b)
Let V be the volume of the sphere.
Hydrostatic force exerted by glycerin on the sphere $F\text{'}=V\sigma g$
$⇒F\text{'}=\frac{4}{3}\pi {r}^{2}\sigma g$

(c)
Let the terminal velocity of the sphere be v'.
The forces acting on the drops are
(i) The weight mg acting downwards
(ii) The force of buoyance, i.e., $\frac{4}{3}\pi {r}^{3}\sigma g$ acting upwards
(iii) The force of viscosity, i.e., 6πηrv' acting upwards
From the free body diagram: #### Page No 301:

Given:
Radius of the drops r = 0.02 cm = 2 × 10−4 m
Viscosity of air η = 1.8 × 10−4 poise = 1.8 × 10−5 decapoise
Acceleration due to gravity g = 9.9 m/s2
Density of water ρ = 1000 kg/m3
Let v be the terminal velocity of a drop.
The forces acting on the drops are
(i) The weight $mg$ acting downwards
(ii) The force of buoyance, i.e.,  acting upwards
(iii) The force of viscosity, i.e., 6πηrv acting upwards

Because the density of air is very small, the force of buoyance can be neglected.
From the free body diagram: $6\pi \eta rv=mg\phantom{\rule{0ex}{0ex}}6\pi \eta rv=\frac{4}{3}\pi {r}^{3}\rho g$
$v=2{r}^{2}\frac{\mathrm{\rho }g}{9\eta }$

Hence, the required vertical speed of the falling raindrops is 5 m/s.

#### Page No 302:

Given:
Speed of water, v = 6 cm/s = 6 × 10−2 m/s
Radius of tube, r = 1 cm = 10−2 m
Diameter of tube, D = 2 × 10−2 m
Coefficient of viscosity, η = 0.01 poise

Let the Reynolds number be R and the density of water be ρ.

Here, the Reynolds number is less than 2000. Therefore, it is a steady flow.

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