Unit 6: Mechanical Properties of Matter — Long Questions

(1) Deforming force: An external force applied on an object can change its size or shape. Such a force is known as deforming force.

(2) Elasticity:
An object is said to be elastic, if after removal of the deforming force, it restores to its original size and shape. This property of the material is known as elasticity. Due to this property, we can determine the strength of a material and the deformation produced under the action of a force.

Example (i) An appropriate force applied to a spring can increase its length called extension or cause compression thus reducing its length. If this force is removed, the spring will restore its original size and shape.
(ii) Similarly, stretched rubber strip or band comes to its original shape and size on removing the applied force.
(iii) When a tennis ball is hit by a racket, the shapes of tennis ball and also racket strings are distorted or deformed (Fig. 6.1). They regain their original shape after bouncing of the ball by the racket.

(3) Elastic limit:
Most of the materials are elastic up to a certain limit known as elastic limit. Beyond the elastic limit, the change becomes permanent. The object or material does not regain its original shape or size even after the removal of the deforming force.

Statement of Hooke's law
"If force F is applied on a spring to stretch or compress it, the extension or compression x has been found directly proportional to the applied force within the elastic limit." Thus,

Mathematically,
F ∝ x
Or
F = kx or k = F/x ............(6.1)

Spring Constant Where k is the constant of proportionality and is known as spring constant. In fact, it is a measure of stiffness of the spring. The greater the value of spring constant, the greater will be the stiffness or strength of the spring. Its unit is Nm^-1

Graphical representation A graph of force against extension is a straight line passing through the origin. If the applied force or load exceeds the elastic limit of the spring, it is permanently deformed and its graph will no longer remain linear. The gradient or slope of force-extension graph is a measure of spring constant k.

Applications of Hooke's Law
Hooke's law serves as the basic principle in wide range of applications. In the field of technology and engineering, springs in many devices rely on Hooke's low for their functions such as spring scales, balance wheel of the mechanical clocks, galvanometer, suspensions system in vehicles and motorbikes, door hinges, mattresses, material testing machines. etc.

However, Hooke's law applies within a specific range of forces. Exceeding the range or limit results in permanent deformation and no longer follows Hooke's law. Some of the uses are elaborated below:

(i) Spring scales
Spring scales use the extension or compression of a spring to determine the weight of objects. In a common spring balance the extension or elongation produced is a measure of the weight. In compression balance, the spring is compressed by the load (force) and the compression produced is measured by means of a pointer moving over a scale. Weighing machine usually use this type of balance.

(ii) Balance wheel of mechanical clocks
The balance wheel in mechanical clocks use spring to control the back and forth motion that regulates the speed of the hands of a clock (Fig. 6.4).

(iii) Galvanometer
Galvanometer is a current detecting device. It makes use of a tiny spring called hair spring (Fig. 6.5) which provides electrical connections to the galvanometer coil - also restores the pointer back to zero position. The deflection of the pointer is proportional to the current flowing through it within the range.

Definition Density of a substance is defined as its mass per unit volume.

Mathematically,
Density = Mass/volume ............(6.2)

Unit The SI unit of density is kgm^-3. Other unit also in use is gcm^-3. Table shows the density of some substances.

Density of some substances Substance - Density (kgm^-3)
Air - 1.3
Patrol - 800
Water - 1000
Concrete - 2400
Aluminum - 2700
Steel - 7800
Lead - 11400
Gold - 19300
Osmium - 22600

Importance The architects and engineers take special care of the density of the building material to be used in designing and constructing roads, bridges and buildings. The density of building material is essential for estimating the strength required in foundations and supporting pillars.

Density measurement Density of a substance can be determined by measuring its mass and volume. The mass can be easily measured by a physical balance.
If the substance is solid and has a regular shape, its volume can be found by measuring its dimensions.

Example If the substance is in the form of a sphere, its diameter can be measured by a Vernier Calipers and volume is thereby calculated. Knowing mass and volume, the density can be found out.

Definition Pressure is defined as the force exerted normally on unit area of an object.

Mathematically,
If F is the force acting normally on a surface of area A, then pressure P on the surface is given by

P = F/A ............(6.3)

Unit In the system international, the unit of pressure is Nm^-2 and is called pascal (Pa). The area A on which the force acts is usually referred as contact area. Equation (6.3) shows that for a certain force, the pressure can be very large if the contact area A is small.

Explanation If a wooden rod has a flat end, it will be very difficult to push it into ground. On the other hand, if it has a pointed end, it can be easily pushed into the ground. In the first case, the applied force is spread over a large area, whereas in the second case, the force is concentrated on a small area. The force applied on the rod will exert greater pressure in the second case than in the first one.

Daily Life Examples

(i) Chopper: The edge of the blade of a chopper is made very sharp. When we apply force on the handle of the chopper to cut an object, the pressure on the object, at the contact surface, due to its small area becomes very high and the object is easily cut (Fig. 6.6).

(ii) Thumb Pin: The top of a thumb pin is flat but the end of the pin is very sharp. So, the contact area is very small. When we apply a force at the top, the pressure at the end of pin is so high that it pierces into the wooden board (Fig. 6.7).

(iii) Walking on ground: When we walk on ground, we exert a force on it due to which we experience a reaction force. When the ground is flat, this reaction force is spread over the whole area of the foot and the pressure due to reaction force is not painful. But when we walk on pebbles, the contact area is reduced. Then the pressure due to reaction force becomes so high that it becomes painful.

(iv) Elephant feet: Heavy animals like elephant have thick legs and large flat feet so that due to large contact area, pressure becomes less otherwise, their bones would not tolerate the pressure.

Explanation Let us determine the pressure at a certain depth of a liquid. Figure: 6.8 shows a container of liquid. Consider an area A in the liquid at depth h. The force acting on this area is equal to the weight of the liquid column over surface A. The volume of this liquid is V=Ah. If ρ is the density of liquid, then mass m of the liquid column will be:

m = ρV = ρAh

Therefore, force acting on area A will be
F= mg = ρAhg

The pressure P at area A will be,
P = F/A = ρAhg/A
Or
p = ρgh ............(6.4)

Equation 6.4 shows that pressure in a liquid increases with depth.

Dependence The value of pressure depends on the depth and density of the liquid. Pressure produces force at right angle to the surface. A force or its component that is parallel to the surface, does not contribute to pressure. The pressure, by definition, is only contributed by the normal component of the force. That is, the force in a liquid that push directly against the surface and add up to a net force is perpendicular to the surface. It there is a hole in the surface of the liquid container, the liquid spurts at right angle to the surface before curving downward due to gravity.

Definition The atmosphere exerts pressure on the surface of the Earth and on everything on the Earth. This pressure is called atmospheric pressure.

Explanation (i) The Earth is surrounded by a layer of air which we call atmosphere. We know that air is a mixture of gases. Their molecules are always in motion. They collide with one another and with all other objects coming in their way. Thus, they exert force on the objects. This force per unit area is the atmospheric pressure. Since the molecules of air have random motion, therefore, atmospheric pressure acts equally in all direction.

(ii) Atmospheric pressure extends up to a height of about 100 kilometers. The density of air is not the same in the atmosphere. It decreases continuously with altitude.

(iii) We live at the bottom of the Earth's atmosphere which is a fluid that exerts pressure on our bodies. At sea level, the value of atmospheric pressure is about 1.013 × 10⁵ Pa. This value is referred to as standard atmospheric pressure. It is an enormous pressure which can crush anything. We do not feel it because practically all the bodies have air inside them. As atmospheric pressure acts in all direction, so it balances the pressure inside.

Evidence of Atmospheric Pressure We can observe the force of the atmospheric pressure if we remove the inside air from a vessel.

Variation of Atmospheric Pressure with Height
We have studied that pressure in a liquid increases with depth. At depth h, the pressure of liquid is given by P = pgh

This formula is applicable to all the fluids. As the gases of the atmosphere are also fluid, therefore, the atmospheric pressure should be maximum on the ground at sea level. As we go up in the air, atmospheric pressure decreases. At a height of about 5km it falls to 55 kPa and at a height of 30 km it to 1 kPa. By measuring the atmospheric pressure at a point in air, altitude of that point can be determined. The lower the atmospheric pressure, the greater is the altitude.

Changes in Atmospheric Pressure as Weather Indicator
The atmospheric pressure does not always remain uniform but fluctuates. By observing the variation, the meteorologists can forecast the weather condition. Atmospheric pressure depends upon the density of air. At height altitudes, where the air is less dense, the atmospheric pressure falls down. Similarly, increase in the quantity of water vapours also decreases the density. Thus, atmospheric pressure becomes low in cloudy region. Weather casters use this knowledge to predict rains. A fall in pressure often means that rain clouds are on the way and the rain is to follow.

Measurement of Atmospheric Pressure by barometer:
Atmospheric pressure is usually measured by the height of mercury column which it can support. Instruments which measure the atmospheric pressure are called barometers.

Construction A simple mercury barometer consists of a glass tube about one metre long that is closed at one end. It is completely filled with mercury, then it is inverted vertically in a dish of mercury. A metre scale is placed by the side of the tube to measure the height of mercury column. The space in glass tube over the top of the mercury is completely empty. The pressure is almost zero.

Working The pressure P, at point A in the mercury column is the same as at point B at the surface of mercury column is the same as at point B at the surface of mercury in the dish because both the points are at the same level. This is equal to the atmospheric pressure P = pgh acting at the surface of mercury in the dish. If we put P = 1.013 × 10⁵ Pa at sea level, ρ = 13.6 × 10^-3 kg m^-3 for mercury, the height of mercury column comes out to be 760 mm. By using this instrument atmospheric pressure at any altitude in the air can be measured in terms of height of mercury column.

Measurement of Pressure by Manometer:

Construction A simple manometer consists of a U shaped glass tube which contain mercury. In the beginning, the atmospheric pressure at the two open ends of the tube is the same and hence, mercury level in the two arms remains same.

Working If on connecting a gas cylinder with short arm keeping the longer arm of the tube open, the mercury level in short arm is lower than that in the long arm, then the unknown pressure is more than the atmospheric pressure. If the mercury level in the short arm is more than the long arm, then the unknown pressure is less that atmospheric pressure.

Definition When pressure is applied at one point in an enclosed fluid, it is transmitted equally to all parts of liquid without loss.

Application The technology of hydraulic systems is bases on Pascal's law. Its main advantages are:
i. Liquids does not absorb any of the supplied energy.
ii. They are capable of moving much heavy loads and providing great forces due to incompressibility.

Some useful hydraulic systems are:
1. Hydraulic press
2. Car lift at service stations
3. Hydraulic brakes of vehicles

Evidence from daily life examples:
(i) Take water in a flask with piston and having a few side tubes fixed at different position. If such flask is not available, you can join a syringe at the mouth of a pet bottle. For side tubes, bendable transparent drinking straws can be glued on the holes punched on sides of the bottle.

You will observe that the level of water in all the side tubes is the same. This is because a liquid seeks its own level and rises to the same height at all points Now push the piston through some distance. The level of water in all the side tubes rises to the same height. This is because the pressure applied at one point of the liquid is transmitted equally to every point of the liquid. Since gases (air) and liquids are termed as fluid.

(ii) When we inflate a balloon, we blow air in it with a certain pressure but the balloon blows uniformly from all sides. It means that the pressure applied at its mouth has been transmitted uniformly in all direction.

(iii) When a motorbike tyre is inflated, air pressure is applied at one point but the tyre is uniformly inflated from all sides. This indicates that pressure is transmitted to each part of the tyre.

(i) Hydraulic press:

Construction Hydraulic press is a specially designed container. In this container there are two cylinders joined by means of a pipe. The cross-sectional area of the smaller cylinder is A1 and that of the larger one is A2. The cylinders are filled with some incompressible liquid.

Workings Suppose that the small piston is pressed down by applying a force F1. The pressure P1 = F1/A1 produced by small piston is transmitted equally to the large piston. Due to this pressure P1 a force F2, will act on A2, which is given by F2 = PA2

Putting the value of P1
F2 = F1/A1 × A2 .............(i)

Since A2> A1, therefore, F2 > F1. The result indicates that a small force applied on the smaller piston, results into a large force on the larger piston. Such a system is known as force multiplier.

Applications (i) Cotton bale or any other object to be compressed is placed over the larger piston. A force F1 is applied on the smaller piston. The pressure P produced by smaller piston is transmitted equally to the larger piston. A much greater force F2 acts on it. This force lifts the larger piston and compresses the cotton bale.

(ii) This principle is also used at service stations to lift cars for washing.

(ii) Hydraulic Brakes

Construction The brakes of some vehicles work on Pascal's law. In such type of brakes, cylinders with pistons are attached to the wheels. The brake pedal is attached to a master cylinder having smaller area of cross-section. Master cylinder is connected to all the larger cylinders attached to the wheel through pipes as shown in fig. Oil is filled in this system.

Working When petal is pushed down, the piston applies pressure on the liquid in the master cylinder. The liquid in the master cylinder. The liquid pressure is transmitted equally to all the larger pistons of other cylinders. This pressure causes these pistons to move outward pressing the brake pads and brake discs or brake drums. Force of friction between the pads and discs or drums slow down the vehicle. When pressure is released from the pedal, the springs pull back the brake pads and wheels again turn freely.