Category: Physics Notes PPT

Impedance meaning – It is the measure of overall opposition of an AC circuit to current denoted by Z. In simple words, it gives the amount of circuit that impedes the flow of change. Impedance is like resistance, which also takes into account the effects of inductance and capacitance. The measurement unit for Impedance is ohms.

As impedance considers the effects of inductance and capacitance and varies with the frequency of current passing through the circuit, it is more complex than resistance. As compared to resistance, which is constant regardless of frequency, the impedance varies with frequency.

When it comes to defining reactance, it is the measure of the opposition of inductance and capacitance to current. Let’s learn more about these two terms in brief.

Impedance Formula

The mathematical symbol of impedance is Z, and the unit of measurement is the ohm. It is the superset of both resistance and reactance combined.

In phasor terms, impedance Z is characterized as a summation of resistance R and reactance X as:

X = R + j X

Where reactance X is the summation of iInductive reactance X[_{L}] and capacitive X[_{C}].

X = X[_{L}] – X[_{C}]

Impedance, Z = [frac{V}{I}]

V = voltage in volts (V)

I = current in amps (A)

Z= impedance in ohms (Ω)

R= resistance in ohms (Ω)

Impedance can be split into two parts:

The alternating current lags or leads the voltage depending upon the nature of the reactance component of impedance (whether predominantly inductive or capacitance).

The inductance and capacitance cause phase shifts between current and voltage, which means the resistance and reactance cannot be simply summed up to give impedance. Instead, they must be summed up as vectors with reactance at right angles to resistance, as shown in the figure below.

Impedance Z  = [sqrt{R^{2}+X^{2}}]

There are four electrical quantities that determine the impedance (Z) of a circuit: These are resistance (R), capacitance (C), inductance (L), and frequency (f).

What is Reactance?

The measure of the opposition of inductance and capacitance to current in an AC circuit is known as reactance, denoted by X. It varies with the frequency of electrical signals and is measured in ohms.

Reactance is of two types:

• Capacitive reactance ( X[_{C}] ), and

• Inductive reactance ( X[_{L}] ).

Reactance Formula

The total reactance (X) is equal to the difference between the two:

Total reactance, X = X[_{L}]– X[_{C}]

a. Capacitive Reactance X[_{C}]

The reactance, which is large at low frequencies and small at high frequencies is known as capacitive reactance ( X[_{C}] ). X[_{C}] is infinite for steady DC, at zero frequency (f=0Hz). This means that the capacitor passes AC but blocks DC.

Capacitive Reactance, X[_{C}] = 1/2fC

Where,

X[_{C}] = reactance in ohms (Ω)

f = frequency in hertz (Hz)

C = capacitance in farads (F)

For example, 1µF capacitor has a reactance of 3.2k for a 50Hz signal, but when a frequency is higher at 10kHz, the reactance is only 16.

b. Inductive Reactance, X[_{L}]

The reactance which is small at low frequencies and large at high frequencies is known as inductive reactance. X[_{L}] is zero for steady DC, at zero frequency (f=0Hz). This means that the inductor passes DC but blocks AC.

The formula for calculating the inductive reactance of a coil is:

Inductive reactance, or X[_{L}] is a product of 2 times (pi), or 6.28, frequency of the ac current in hertz, and the inductance of the coil, in henries.

X[_{L}] = 2x f x L

L = the inductance value of coil in henries.

Inductive reactance, X[_{L}] = 2fL

Where,

X[_{L}] = reactance in ohms (Ω)

f    = frequency in hertz (Hz)

L   =inductance in henry (H)

For example, a 1mH inductor has a reactance of only 0.3 for a 50Hz signal, but when the frequency is higher at 10 kHz, its reactance is 63 .

Reactance and Impedance Formula

An element in the DC circuit can be easily described by using only its resistance. The resistance of a capacitor in DC circuits is regarded to be an open connection whereas the resistance of an inductor in a DC circuit will be regarded as a short connection or zero resistance. As opposed to DC circuits, it is seen that in AC circuits, the impedance of an element is a value of how much the element tends to oppose the flow of current when an AC voltage is being applied across it. Impedance can be represented as  a complex number that consists of both real and imaginary parts and can be represented as follows:

Z = R + jX

Where Z represents the impedance

R represents the value of resistance

X which is the imaginary part will represent reactance

It is seen that the resistance seen in the circuit will always be positive while the reactance that is seen will be either positive or negative.

Do you know?

When current and voltage are out of step with each other, it means there is a phase shift. For example, when you charge a capacitor, the voltage across it is zero. However, the current is maximum. When the capacitor is charged, the voltage will be maximum, and the current will be at a minimum. The charging and discharging occur continually with AC, where the current reaches maximum shortly before the voltage reaches the maximum, so it is called current leading voltage.

Regelation is a phenomenon in which the freezing point of water is lowered by the application of pressure. Regelation is defined as the phenomenon in which at below 0°C temperature, on the application of pressure the ice melts to water, and on the removal of pressure refreezes back to ice. It is demonstrated by the process of compressing the ice under pressure and turning it into the water, and when the pressure is removed it solidifies again.

That’s why the refreezing of water is derived from the melting of ice under pressure when the pressure is relieved. Meaning that as a liquid changes into a solid when heat and pressure are released, freezing is almost always an exothermic process. Glaciers act as a source of a river due to regelation.

Regelation of Ice

The ability of water molecules to coalesce together after being separated by solid matter is called the regelation of ice. This process is what enables ice cubes in a drink to form and eventually melt; the water molecules on the outside of the cube will freeze onto the sides of the cube, while those on the inside will continue to flow until they reach the bottom. The longer an ice cube is left in a drink, the more Regelation that will occur and the smaller it will become.

Examples of Regelation

Here are a few examples:

1. Glaciers act as a source of the river due to regelation. The mass of the glacier exerts pressure on the lower surface, which lowers the melting point of the ice at its base. This results in the melting of ice and propels the glacier to slide over the liquid. Under appropriate conditions, liquid water flows from the base of the glacier to lower altitudes when the temperature of the air is above the freezing point of water.

2. Preparation of an ice ball: The ice slab is shredded into pieces and the shredded pieces are pressurized around the tip of a stick to prepare the ice ball. If two small pieces of ice are taken and pressed against each other, they stick to each other.

3. Glaciers act as a source of a river due to regelation.

4. Skating is possible on snow due to the formation of water only for the regelation. Water is formed due to the increase of pressure and it serves as a lubricant.

5. This process is often observed in the Arctic and Antarctic regions, where ice sheets are exposed to warmer air temperatures. It’s also sometimes seen with thin pieces of sea ice that have broken off into chunks; these can melt away if they’re left out on land, but regelation may occur when they come back into contact with one another or other objects (e.g., rocks) at lower elevations. 

If you’ve ever put an object onto a block of ice and had it stick there for a while before sinking down below its surface, this was likely due to regelation happening right above your hand.

6. Regelation is observed in plants when the water droplets on their leaves freeze. The tiny ice crystals that form grow and spread, until eventually, they take over the entire leaf. This happens during cold weather conditions and is one-way plants can protect themselves from losing too much water.

7. Regelation can be seen when the temperature of a material is below the freezing point but there is still liquid water present. The water droplets on the surface will freeze and create an ice layer. This happens when frost forms on objects like grass, metal wires, or leaves.

8. Regelation moreover occurs in animals like frogs and toads, who use this process as a way of surviving the winter. It’s not an active type of behaviour; rather, it occurs automatically whenever these amphibians are exposed to cold air temperatures for extended periods (e.g., during hibernation).

9. Regelation is also similar to another process called nucleation, which can cause ice crystals to form in water. Nucleation happens when the temperature of a liquid or gas phase decreases below its normal freezing point but doesn’t quite reach 0 degrees Celsius yet; this triggers an increase in molecular activity and forces some molecules to become locked together through ionic bonds (i.e., they start forming solid material).

Experiment on Regelation of Ice

Using a metallic wire by performing an experiment on an ice slab, the regelation of ice can be better understood. The procedure of the experiment is as follows:

1. Take a slab of ice.

2. Take a wire and fix two blocks, say 5 kg each, at its ends.

3. Put the wire over the slab.

4. If you observe, you can see that the wire passes through the ice slab. 

5. Due to the fact that just below the wire, ice melts at a lower temperature, this happens due to an increase in pressure.

6. Water above the wire freezes again when the wire has passed.

7. Thus the wire passes through the slab and the slab doesn’t split. This process of refreezing is called regelation.

Here, the melting point of the ice becomes lower than 0°C due to the applied pressure. This implies that at 0°C ice is converted into water and as soon as the pressure is removed the melting point is restored back to 0°C and the water is converted back to ice again.

Skating is possible on snow due to the formation of water only for the regelation. Water is formed due to the increase of pressure and it serves as a lubricant.

Fun Facts

1. Microscopy, Astrid Döppenschmitt, and Hans-Jürgen Butt, (These all are research works) measured the thickness of the liquid-like layer on ice using atomic force which is roughly 32 nm at −1°C, and 11 nm at −10°C, in 1998. 

2. For normal crystalline ice, its melting point is far below,  there will be some relaxation of the atoms near the surface. Near its melting point, the simulations of ice show that there is significant melting of the surface layers rather than asymmetric relaxation of atom positions. Evidence for a liquid layer on the surface of ice is being provided by Nuclear magnetic resonance.

3. The surface melting can account for the low coefficient of friction of ice, as experienced by skaters, ease of compaction of ice, high adhesion of ice surfaces.

4. A glacier can exert a sufficient amount of pressure on its lower surface to lower the melting point of its ice. It allows it to m
   ove from a higher elevation to a lower elevation with the melting of the ice at the glacier’s base. at lower elevations when the temperature of the air is above the freezing point of water the Liquid water may flow from the base of a glacier.

5. Glaciers act as a source of a river due to regelation.

Conclusion

This is what the regelation of ice stands for. Understand the concept by following the simpler explanation. Learn how this natural phenomenon occurs and can be used in different ways.

We know that resistance is the opposition created to the current flowing through the circuit. The resistance is the prevention of major disasters like short-cut or high damage to the property.

However, the resistance has a good relationship with the length.

Let’s suppose that the resistance is a speed breaker and the speed of your vehicle is the current. Now, when the speed breaker is in the middle of the road, not on its ends. You will try to take out your speedy vehicle from the side of the road, hit by a vehicle, and meet an accident.

What is Resistance?

An electron traveling through the wires encounters resistance, which is basically a hindrance to the flow of charge. For a moving electron, the journey from one end to the other end is not a direct route, but a zig-zag path, because they collide with the ions in the conducting material. So, the electrons encounter a hindrance to their movement, making it difficult for the current to flow. This causes resistance.

While the electric potential difference between two terminals encourages the movement of charge, the resistance discourages it. The rate at which charge flows from one terminal to another is thus a result of the combined effect of these two factors.

Relationship Between Length and Resistance

In the above example, we discussed how length and resistance are related to each other. Now, let’s talk about it in detail.

Now, you encounter a road that has twice the speed breakers as that were earlier. Now, you will have to be very sure before you reach the edge of the speed breaker because at this time, your very high-speed vehicle will pass through many resistors (speed breakers) and your vehicle will slow down eventually.

So, mathematically, the equation can be expressed as:

R ∝ L ……(1)

You are driving your vehicle on the road and it is compulsory to cross the speed breakers because in front of you there is a big jam on the road. Now, if the length is less and instead of spreading these breakers by a distance, these are joined end-to-end, so what you observed here is, the area is halved but if you drive it fast, your vehicle will jump, again there is a risk.

So, here even if the length is lesser; however, the area is halved, still you have to be slow. It means the resistance is directly proportional even if the area is halved.

So, mathematically, we can write the equation as:

R ∝ 1/A ……(2)

Now, let’s understand the resistance length of wire in terms of Physics.

Relationship Between Resistance and Length of Wire

Let’s suppose that there are two conductors in the form of cuboidal slabs (they are identical in shape and size) joined end-to-end. Each of these has a length as ‘L’ and the area of cross-section as ‘A’.

When the potential difference ‘V’ is applied across either slab, the current ‘I’ starts flowing. So, by Ohm’s law, we have the relation as:

ROLD = V/I….(3)

Where R is the resistance across conductors, which is the same in each and it is measured in Ohms. As these two conductors are placed side-by-side, so the total length becomes ‘2L’, while the current them becomes ‘I/2’ because if ‘I’ is the total current flowing through both conductors and ‘V’ is the same potential difference across the conductors, so each of these conductors gets ‘I/2’ current.

So, the new resistance of the combination is Rc, and mathematically, we derive our expression in the following manner:

      [R_{c} = frac{V}{I/2} = frac{2V}{I}]

          

Looking at equation (3), we find a unique relationship between the old resistance and the resistance of combination, which is as follows:

Rc =  2 ROLD …..(4)

Equation (4) implies that on doubling the length, the resistance of the combined slabs, i.e., Rc becomes the double of the old resistance ‘R’.

Resistance and Length of Wire

Now, consider the same two slabs again. Here, instead of placing them side-by-side, we place them one above the other. We can see this arrangement below:

We can notice one thing here the length of each conductor remains ‘L’, however, the area of cross-section, i.e., ‘A/2’ instead of ‘A’ because the area of each conductor added to become ‘A’. One thing is common here and that the total current is ‘I’ across both the conductors, so across each conductor, again the current will be ‘I/2’.

Using Ohm’s law again, we get the equation as:

ROLD1 = V/I….(5)

Now, writing the equation for the resistance of the combination as:

        [R_{p} = frac{V}{I/2} = frac{2V}{I}] ……(6)

From equations (5) and (6), we get a new relationship as:

RP = 2 ROLD1 …..(7)

From equation (7), we can notice that on halving the area, the resistance doubles.

We came to the conclusion that on doubling the length and halving the area of cross-section, the resistance doubles in each case, which means we proved the relationships in equations (1) and (2). Now, we will find a new relationship, so let’s get started.

Relation Between Resistance and Length

Here, we will combine equations (1) and (3):

R ∝ L/A

Now, removing the sign of proportionality, we get the following resistance per unit length formula:

R = ⍴ L/A …..(8)

Or,

⍴ = RA/L

Here, ⍴ is called the proportionality constant or the resistivity or the specific resistance of the material conductor. It is measured in Ohm-m.

So, the resistance per unit length is also called the resistivity of the material (conductor).

⍴ = R/L (Where A is a constant value).

Other Factors that Affect Resistance

Resistance has a relationship with two other factors apart from the length. 

Cross-Sectional Area and Resistance: The cross-sectional area of the wires has a direct effect on the amount of resistance. Wider wires have a larger cross-sectional area, and the wider the wire, the less the resistance to the flow of electric charge. When all other variables are left unchanged, the charge will flow at higher rates through wider wires than through thinner wires. The resistance of a thinner wire is less than the resistance of a thick wire as the thin wire has fewer electrons to carry the current. Thus, the relationship between resistance and the area of the cross-section of a wire is inversely proportional.

Material and Resistance: Not all materials are equal in terms of their ability to conduct. Some materials offer less resistance to the flowing charge than others and that’s why they are better conductors. So, the conducting ability of a material depends on its resistivity. The resistivity depends on the material’s electronic structure and temperature. Resistivity increases with rising temperature for most materials. 

Silver is not used in wires even if it is the best conductor because of its high cost. Copper and aluminum have a high conducting ability and are also among the least expensive materials. 

Summary

In short, the resistance in a wire increases as the length of the wire increases. A long wire has higher resistance than a short wire. This is because of the fact that in a long wire, the electrons end up colliding with a lot of ions as they pass through. So, the relationship between resistance and wire length is proportional.

The density of a substance can be defined as the mass of a substance per unit volume i.e. Density = Mass of substance/Volume of substance, It is one of the basic physical properties of a substance that can be used along with its other unique properties to characterise it and every substance has a different density Its unit is kg/m³. Relative density on the other hand can be defined as the ratio of the density of a substance to the density of the standard substance. Relative Density is also called Specific Gravity. Given below are some of the substances whose relative densities are mentioned at room temperature. 

Usually water at 4 ^oC (used as a standard for a liquid or solid and the air is used for gas). It is a unitless quantity. Relative density finds its huge application in the petroleum industry where the products obtained are based on the measurements done according to the relative densities of the liquids. In this topic, we will discuss more on the relative density formula, the relative density of solid, etc.

Relative Density Formula

The density of the substances varies with pressure and temperature so it is necessary to specify the pressure and the temperatures at which the densities and the masses are to be determined.

It is said that measurements are done mostly at 1 atmosphere which is 101.325 kPa ignoring the variations caused by changing weather patterns and external affairs. But as we know relative density refers to highly incompressible aqueous solutions so the other incompressible substances like petroleum products that show variations in density caused by pressure are mostly neglected at least where apparent relative density is to be measured.

The formula of relative density or [R.D = frac{ text{Density of Substance}}{ text{Density of Water}}]

[R.D = frac{(rho_{substance})}{(rho_{reference})}]

Relative density is a dimensionless quantity. If a substance is said to have a relative density less than one then it is less dense compared to a reference substance. If it is greater than 1 then it is denser than the reference substance. If the density which is relative is exactly 1 then the densities are equal. Similarly, the relative density of solid can be calculated as:

[Relative Density= frac{ text{(Loss of weight of solid in the air)}}{( text{Loss of weight of solid in water)}}]

In this topic, we have understood what is relative density in physics. Let’s understand some of its uses and factors affecting the measurements.

SI Unit of Relative Density

Since relative density is the ratio of two same quantities, therefore there is no SI unit of relative density. Relative density is a dimensionless quantity. 

Applications of Relative Density

1. The major application of relative density is in the petroleum industry where the products obtained are mostly based on the measurements done according to the relative densities of the liquids under process.

2. Heavy molecular weight hydrocarbons can be converted to low molecular weight hydrocarbons such as gasoline, jet fuel, and diesel based on the chemical processes involving the measurements based on relative densities of compounds.

3. It is used for determining the density of an unknown substance from the known density of another substance.

4. It is also used by geologists to find out the mineral content of the rock.

5. Testing the purity of a substance (Eg: gold)

Solved Examples of Density and Relative Density

1. Find the density of a block of ice if its mass is 500 kg and volume is 5 metre cube. 

Answer: Mass of the block of ice = 500 kg

The volume of the block of ice= 5 metres cube

The density of the block of ice= Mass of the block of ice/ Volume of the block of ice

= 500 kg/ 5 m³

= 100 kg/ m³

Therefore, the density of a block of ice is 100 kg/ m³. 

2. Find the density of a 3500 kg cuboid whose length, breadth and height are 22 metres, 10 metres, and 12 metres respectively. 

Answer: Mass of the cuboid= 3500kg

Volume of the cuboid= l x b x h where l is length, b is breadth and h is the height. 

= 22 x 10 x 12

= 2640 m³ 

Density of the cuboid= Mass of the cuboid/ Volume of the cuboid

= 3500 kg / 2640 m³

= 1.326 kg/m³

Therefore, the density of the cuboid is 1.326 kg/ m³

3. Find the density of natural oil whose specific gravity is 0.65. Express the answer in kg/m³. 

Answer: As we know, specific gravity is relative density, therefore, 

Relative Density of the oil= 0.65

Density of water= 1000 kg/ m³ 

Relative Density= Density of natural oil/ density of water

Density of oil= Relative density x density of water

= 0.65 x 1000

= 650 kg/ m³ 

Therefore, the density of the natural oil is 650 kg/ m³. 

Factors Affecting Measurement of Relati
ve Density

• Air Bubbles: A small bubble with a diameter of 1 mm can yield a 0.5 mg increase and those with 2 mm can yield a 4 mg increase. So, make sure that the solid object or sinker immersed in the liquid is not adhered to by air bubbles.

• Solid Matter Sample: A sample with a very large volume immersed in the fluid will result in an increase in the level of fluid within the pitcher of the glass.

• Temperature: Solids are generally not affected by temperature changes so that the corresponding density changes are not relevant. However, according to the Archimedes Principle, while determining the density of a liquid or a solid, its temperature is taken into consideration. The temperature change affects liquids greater and causes changes in the density in the order of 0.1 to 1 per ^oC.

Measurement of Relative Density

1. Hydrometer: It is an instrument that is used to determine the relative density or specific gravity. It works on the model of the Archimedes Principle. Archimedes’ principle states that any body, which is partially or fully immersed in the water, is acted upon by an upward force called the buoyant force. This force is equal to the weight of the liquid displaced by the object which is immersed in the water. The hydrometer is kept in the hydrometer jar which has the liquid. When the level of the sample liquid in the jar aligns with a point on the hydrometer scale, that tells the relative density of the liquid. 

2. Pycnometer: This device is used to determine the specific gravity of various liquids. The steps to determine the relative density of a liquid requires measurement of the empty flask first. The flask is weighed with the reference liquid and then finally weighed with the testing liquid. These weights are used together to find the relative density. These weights are used to calculate the relative density of the liquid. Pycnometers can also be used for measuring the density of solids and gases also. 

The resolving power of an optic instrument, say a telescope or microscope, is its capability to produce separate images of two nearly zonked objects/ sources. The plane swells from each source after passing through an orifice from diffraction pattern characteristics of the orifice. It is the lapping of diffraction patterns formed by two sources that sets a theoretical upper limit to the resolving power. 

Resolving Power of a Microscope 

For microscopes, the resolving power is the antipode of the distance between two objects that can be just resolved. 

Where n is the refractive indicator of the medium separating object and orifice. Note that to achieve high- resolution n sin θ must be large. This is known as the Numerical aperture.

Thus, for good resolution :

• sin θ must be large. To achieve this, the objective lens is kept as close to the instance as possible. 

• An advanced refractive indicator (n) medium must be used. Canvas absorption microscopes use canvas to increase the refractive indicator. Generally, for use in biology studies, this is limited to1.6 to match the refractive indicator of glass slides used. (This limits reflection from slides). Therefore, the numerical orifice is limited to just 1.4-1.6. Therefore, optic microscopes (if you do the calculation) can only image to about0.1 microns. This means that generally organelles, contagions, and proteins can not be imaged. 

• Dwindling the wavelength by using X-rays and gamma shafts. While these ways are used to study inorganic chargers, natural samples are generally damaged by x-rays and hence aren’t used.

Resolving Power of a Telescope

Resolving power is another essential point of a telescope. This is the capability of the instrument to distinguish easily between two points whose angular separation is lower than the lowest angle that the bystander’s eye can resolve. The resolving power of a telescope can be calculated by the following formula resolving power = 11.25 seconds of bow/ d, where d is the periphery of the objective expressed in centimetres. Therefore, a 25-cm- periphery ideal has a theoretical resolution of 0.45 seconds of bow and a 250-cm (100- inch) telescope has one of0.045 seconds of a bow. 

An important operation of resolving power is in the observation of visual double stars. There, one star is routinely observed as it revolves around an alternate star. Numerous lookouts conduct expansive visual binary observing programs and publish registers of their experimental results. One of the major contributors in this field is the United States Naval Observatory in Washington, D.C. 

A rigid body possesses two kinds of energy: kinetic energy and potential energy. A rigid body’s potential energy is the energy stored up in the body due to its position and other stresses on the body. The kinetic energy of a rigid body is a form of energy possessed by a moving body by means of its motion. If work is done on an object by applying a net force, the object gains speed which in turn increases its kinetic energy. The kinetic energy of a body in motion is dependent on its mass and speed. This article will cover kinetic energy in rotational motion and learn about the formula for rotational energy.

Rotational Kinetic Energy

When an object spins about an axis, it possesses rotational kinetic energy. The kinetic energy of a rotating body is analogous to the linear kinetic energy and depends on the following factors:

• The speed at which the object is rotating, the faster the speed more is the energy.

• The angular kinetic energy is directly proportional to the mass of the rotating object.

• The position of the point mass from the axis of rotation also determines its energy. The particles that are further from the rotation axis possess more rotational kinetic energy than the ones closer to the rotational axis.

Moment of Inertia

The total of mr² for all the point masses that make up an object’s moment of inertia I, where m is the mass and r is the distance of the mass from the centre of mass, may be described as the object’s moment of inertia I. It may be mathematically stated as I = ∑mr2. Here, I is analogous to m in translational motion. In translational motion, I is comparable to m.

The axis around which you spin an object also affects the moment of inertia. Objects spin about their centre of mass by default, but they may be configured to rotate around any axis. The parallel axis theorem calculates the moment of inertia when rotating around an axis other than the centre of mass. The moment of inertia for an object rotated about a different axis parallel to the axis passing through the centre of mass isIcm+mr2, where r is the distance between the two axes and I cm is the moment of inertia when rotated about the centre of mass, which you learned how to calculate in the previous paragraph.

The Formula for Rotational Energy

To calculate the kinetic energy of rolling motion, let us first consider the linear kinetic energy of a body with mass m which is moving with a velocity of v.

Linear kinetic energy = ½ * m * v2

This straightforward formula holds for all objects moving in a straight line and applies to the center of mass of the object. This way the object is approximated to a point mass.

Now, to describe the formula for Rotational Energy of a rotating object we need to describe the mass distribution of the object along the axis of rotation, denoting it by the moment of inertia kinetic energy, I. Moment of inertia is the measure of difficulty in changing the rotational motion of a body around the axis of rotation. The kinetic energy moment of inertia depends on the mass of the body and the distribution of the mass around the axis of rotation.

 

I = m*r2, where I is the moment of inertia, m is the mass of the body and r is the distance of point mass from the rotational axis. The unit of moment of inertia is Kg.m2.

The Formula for Rotational Energy is Given by 

K.Erotational = ½ (I * ω2). Where I is the moment of inertia and ω is the angular velocity of the object expressed in radians per second. The unit of rotational kinetic energy is Joule.

The Analogy between Rotational and Translational Kinetic Energies

The formulas expressed for the rotational and translational kinetic energy are analogous to each other in the sense:

If an object is rotating as well as its center of mass is moving in a straight line then the total kinetic energy is given by the sum of rotational and translational kinetic energies.

K.E.total = K.E.rotational + K.E.linear = ½ (I * ω2) + (½ * m * v2)

Examples and Application of the Rotational Kinetic Energy Formula

The formula for Rotational Energy has many applications and can be used to:

• Calculate the simple kinetic energy of an object which is spinning.

• Calculate the kinetic energy of an object that is rolling, i.e. there are rotational and translational kinetic energies involved.

Let us look at the example of the movement of Earth about its axis. Earth is spinning on its axis at an approximate rate of once in 24 hours. How do you calculate its rotational kinetic energy, assuming it has uniform density?

Solution

Earth’s radius = 6.37 × 106 m

Mass of Earth = 5.97 × 1024 kg

We need to first find the moment of inertia to calculate rotational kinetic energy. Considering the shape of the Earth as a sphere we get:

Moment of inertia I of Earth = ⅖ * m * r2 = ⅖ * (5.97 × 1024 kg * (6.37 × 106 m)2) = 9.69 * 1037 Kg.m2

Angular velocity of Earth = 2π radians in a day, which is converted into rad/s as:

2π/86400 seconds = 7.27 * 10-5 rad/sec

The rotational Kinetic energy of Earth = ½ (I * ω2)  = ½ * (9.69 * 1037 Kg.m2) * (7.27 * 10-5 rad/sec) = 2.56 * 1029 Joules

Newton’s Second Law of Rotation

To explain Newton’s 2nd law of rotation, let us first understand a few terms related to the theorem:

• Torque- The twisting or rotational effect of a force on an object is called torque. It is expressed in a newton-meter (Nm). The symbol for torque is τ.

• Angular Acceleration- This is the change in the angular velocity of a rotating object per unit of time. The symbol for angular acceleration is 𝞪.

Newton’s 2nd law of rotation states that if several torques are acting on a rigid body about a fixed axis, then the total of the torques is equal to the moment of inertia times angular acceleration. In formula it is
expressed as:

iτi = I * 𝞪

Here I𝞪 is a scalar quantity that can either be positive or negative depending on whether the rotation is clockwise or counterclockwise. Counterclockwise angular acceleration is considered positive, and clockwise is considered negative.

Torque

The rotational equivalent of linear force is torque. Depending on the subject of study, it is also known as the moment, moment of force, rotating force, or turning effect. Archimedes’ study of the use of levers gave birth to the idea. Torque is a twist of an item around a given axis, similar to how a linear force is a push or a pull. The product of the magnitude of the force and the perpendicular distance of the line of action of a force from the axis of rotation is another definition of torque.

Torque is usually represented by the lowercase Greek letter tau or symbol 𝛕. When the term “moment of force” is used, it is usually abbreviated as M.

 

Flywheel 

The usage of a flywheel can smooth out energy fluctuations and make the energy flow of an intermittently working machine more consistent. In most combustion piston engines, flywheels are employed.

In a flywheel, energy is stored mechanically as kinetic energy. A flywheel that stores and provides mechanical energy in the form of rotating kinetic energy has been studied for decades, but its use in automotive systems has been limited due to its hefty weight and expensive cost. The development of flywheel energy storage systems has been expedited by recent advancements in frictionless magnetic bearings, carbon-fiber composite materials, manufacturing techniques, and advanced power electronic controllers.

Conclusion

Rotational Kinetic Energy is a form of energy possessed by a moving body by means of its motion. The kinetic energy of a body in motion is dependent on its mass and speed. This article will cover kinetic energy in rotational motion and learn about the formula for rotational energy.