Finding Significant Figures in a Measurement

What is significant Figure?

The total number of digits (i.e. doubtful digit and confirm digits) in any measurement is known as significant figure.

Or

The digits that reflect the precision of the measurement are called significant figures.

Counting of Significant Figures in any Measurement:

Knowing the significant figures in a measure is based on the following rules-

(1) All non-zero numbers are significant figures.

For example, $46.3598$ has $6$ significant figures.

(2) All zero numbers between two non-zero numbers are significant figures.

For example, $600.8049$ has $7$ significant figures.

(3) If there is no non-zero number before the decimal point, then all the digits in the number except the zero numbers immediately after the decimal point are significant figures.

For example, $0.002809$ has $4$ significant figures.

(4) The zero after any non-zero digit to the right of the decimal point is a significant figure.

For example, $0.46920$ has $0$ significant figures.

(5) If a number is multiplied by a power of $10$, it does not affect the number of significant figures.

For example, $6.75\times 10^{3}$ has $3$ significant figures.

(6) Choosing different units does not change the number of significant figures.

For example, if a measurement is $2346 km$, then in different units, this measurement will be $23460 m$, $2346000 cm$ or $23460000 mm$. Here, the number of significant figures in each measurement is $4$.

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Limitations of Dimensional Analysis

Limitations of Dimensional Analysis:

(1) It is not possible to find the numerical value of constants $k$ (dimensionless) present in the formulas by this method. It can be obtained by experiment or other method.

(2) If any physical quantity depends on more than three quantities, then the mutual relation between these quantities cannot be established by this method. However, the dimensional correctness of any given equation of this type can be checked.

(3) If any physical quantity depends on only three physical quantities, but the dimensions of two of the three quantities are the same, then also the mutual relation between these quantities cannot be established by the dimensional method, but the dimensional correctness can be checked.

(4) If an equation has more than one term on one side, like $v=u+at$ (Here two terms on the right side), then this equation cannot be derived by dimensional method. That is, such relations cannot be derived in which there is a positive $(+)$ or negative $(-)$ sign anywhere. But whether the equation is dimensionally correct or not, can be checked.

(5) Deduction of equations containing trigonometric ratios ($ sin \theta$, $cos \theta$, $tan \theta$, etc.), variable exponential ($e^{x}$) and logarithmic ($log\:x$) terms is not possible by dimensional analysis method, but their dimensional truth can be checked.

(6) Whether a physical quantity is vector or scalar cannot be determined by the dimensional analysis method.

(7) If the constant in an equation is not dimensionless, then the dimensional analysis method cannot be used for the deduction of that equation.

(8) For a physical relation represented by an equation to be true, it is a necessary condition for this equation to be in dimensional balance, but only dimensional balance is not sufficient for the physical relation to be true. That is,

"Even if the equation is true physically and mathematically, it may not be true dimensionally."

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Difference between Forced Vibrations and Resonant Vibrations

Forced Vibrations:

1. The vibrations of a body under an external periodic force of frequency different from the natural frequency of the body, are called forced vibrations.

2. The amplitude of vibration is small.

3. The vibrations of the body are not in phase with the external periodic force.

4. These vibrations last for a very short time after the periodic force has ceased to act.

Resonant Vibrations:

1. The vibrations of a body under an external periodic force of frequency exactly equal to the natural frequency of the body, are called resonant vibrations.

2. The amplitude of vibration is very large.

3. The vibrations of the body are in phase with the external periodic force.

4. These vibrations last for a long time after the periodic force has ceased to act.

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Difference between Natural Vibrations and Damped Vibrations

Natural Vibrations:

1. The amplitude of natural vibrations or free vibrations remains constant and the vibrations continue forever.

2. Natural vibrations never lose energy during vibrations.

3. There are no external forces acting on the vibrating body. The vibrations are only under the restoring force.

4. The frequency of vibrations depends on the size and shape of the body and it remains constant.

Damped Vibrations:

1. The amplitude of damped vibrations gradually decreases or reduces with time and ultimately the vibrations cease.

2. In each vibration, there is some energy loss in the form of heat.

3. Besides the restoring force, a frictional or damping force acts on the body to oppose its motion.

4. The frequency of damped vibrations is less than the natural frequency. The decrease in frequency of vibrations depends on the damping force.

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Difference between Natural (Free) Vibrations and Forced Vibrations

Natural (Free) Vibrations:

1.) The vibrations of a body in absence of any resistive or external force are called natural vibrations.

2.) The frequency of vibration depends on the shape and size of the body.

3.) The frequency of vibration remains Constant.

4.) The amplitude of vibration remains constant with time (in absence of surrounding medium).

Forced Vibrations:

1.) The vibrations of a body in a medium in presence of an external periodic force are called forced vibrations.

2.) The frequency of vibration is equal to the frequency of the applied force.

3.) The frequency of vibration changes with change in the frequency of the applied force.

4.) The amplitude of vibration depends on the frequency of the applied force.

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Renewable Energy Sources

Renewable Energy Sources:

Various renewable energy sources are as follows:

Solar Energy:

The solar energy is a non conventional energy source. It is very cost effective clean and non polluted renewable energy source and reduces the greenhouse gas effect.

The Solar Energy is derived from the Sun radiation and can be utilised by photosynthesis, photo voltaic cell, photo thermoelectric system.

The sun release the enormous amount of energy and it's rate of radiation is $3.7 \times 10^{20}$ megawatt while the earth receive the radiation at rate by $1.85 \times 10^{11}$ megawatt. So energy radiation by the sun is several times more than the consumption of radiation in the earth.

Advantages:

1.) It is very clean energy.

2.) It is non polluted energy source.

3.) It is zero cost energy source and low maintenance cost.

4.) It has zero noise in operation.

Disadvantages :

1.) Energy source cannot utilise properly at night, cloudy atmosphere and rainy day.

2.) It has required large surface area to collect the energy source from Sun.

Hydro energy:

The hydro energy is derived from moving and falling water which convert into mechanical energy and utilise for production of electrical energy through turbine.

The water is stored in reservoir or dam has high potential energy and when water flow or fall under the gravity then it rotate the turbine and produces the electricity.

Advantages:

1.) It is very clean energy source.

2.) It produce zero pollution.

3.) It has zero fuel cost.

4.) It requires low maintenance cost.

5.) It is reliable energy source.

Disadvantages:

1.) It can cause climate change due to high amount of storage of water in mountains.

2.) It can cause flood and disrupt ecosystem.

Wind energy:

Wind energy is also non polluting energy source and it has tremendous potential to fulfill the demand of energy of the country.

It is estimated that only $2 \%$ of solar energy fall on the earth and converted into kinetic energy of the atmospheric molecules or atoms. The highest kinetic energy of atmosphere is found in the lower to mid troposphere layer which is lowest layer of atmosphere because of that this kinetic energy can be easily converted to the mechanical energy which can futher utilise for the production of electrical energy and other energy production.

Advantage:

1.) It is useful for remote places for the production of electricity.

2.) The availability of the source is zero cost.

Disadvantages:

1.) This energy source cannot be properly utilised where wind is available at very higher location.

2.) It is unreliable because flow of wind cannot be continuous all the time.

Wave Energy:

The wave energy is available on the surface of sea. The floating propeller is placed on the surface of the shallow water near to shores and due to motion of wave propeller get start to rotate and this rotational energy is used to derive the turbines.

Advantages:

1.) This is clean or cheap energy source.

2.) The size of the machine for the collection of wave is comparatively smaller than solar device.

Disadvantages:

Corrrosion of material used in plant.

Geothermal energy:

This is the energy is produced due to hot rocks present inside the earth. The temperature of the earth increases with increase in depth below the surface of the earth. The hot molted rock is present at center or core of the earth this causes volcano action. The hot rock is pull out from volcano and used to produce the steam by heating water. This steam is further utilised for the operation of turbine to produce the electricity.

Advantages:

It is cheap source which requires small area for the operation or production of the electricity.

Disadvantages:

1.) It causes the air pollution due to production of gases like $H_{2}S$ and $NH_{3}$in steam waste.

2.) It is also causes noise pollution due to drilling operation.

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