Ideal Gas (chapter 3)

• Ideal Gas: A Gas that obeys PV=nRT at all values of temperature, pressure and volume.

The Assumptions of Ideal Gas

1. It contains the same type of molecules.
2. All collisions with the wall of container are elastic.
3. Kinetic energy of the atoms are directly proportional to the temperature.
4. There are no inter-molecular forces of attraction, subsequently, there is no potential energy among the atoms.
5. At 0 Kelvin the mean kinetic energy becomes 0.

Proofs

• Show that PV=NKT

We know that PV=nRT

we also know that n=N/N_a

so PV=NRT

now since K=R/N_a

PV=NKT

• Pressure of a single atom along the x-axis

Suppose there is a cube shaped container with length L that has a single atom of ideal gas and the atom has speed v:

ΔP = mv - (-mv)

ΔP=2mv

F= ΔP/Δ T =2mv/t

since the collision is elastic T=2L/V

Now putting equation (2) in (1) we get mv² /L³

Now since Pressure = Force/Area

Pressure = mv²/L³

• Total Pressure

Pressure of N atoms on the wall of the container along x-axis:

P_N=mv²_N/l³ 

Total Pressure:

P=P₁+P₂..... P_N

mv²₁/L³+mv²₂/L³+mv²_N/L³

m/L³ (v²₁+v²₂+v²_N)

Now since the mean square speed is denoted by

<v²> = v²₁+v²₂+v²_N / N

Plugging the above mentioned expression in the expression of total Pressure:

P=Nm<v²> /L³

• Total pressure for N atoms

P_x/y/z = Nm <v²_x/y/z> / L³

<c²> = <v²_x>+<v²_y>+ <v²_z>

since it is an ideal gas so <v²_x> = <v²_y> = <v²_z>

<c²> = 3 <v²>

1/3<c²> = <v²>

Plugging this in the expression of Total Pressure

P=1/3 Nm<c²>/L³

• Mean Kinetic Energy of an atom of ideal gas:

PV=NKT

PV=1/3Nm<c²>

3KT=m<c²>

multiplying both sides with 1/2

3/2KT = Kinetic Energy

Brownian Motion

• It is the random motion of microscopic particles in a fluid as a result of countinuous bombardment from molecules of the surrounding medium

The following is observed:

1. Bright tiny Smoke particles are jerking sideways in a random motion
2. Particles randomly moved into and out of focus of the microscope, It indicates random vertical motion of the particles

Explanation for the Brownian motion: The tiny smoke particles are bombarded by millions of small air molecules moving in random directions.

Oscillations (Chapter 3)

• Oscillation: To and fro motion of the object about mean position.
• Simple Harmonic Motion: To and fro motion in which the acceleration of the object is directly proportional to its displacement and the acceleration is always directed towards the mean position.

Difference between oscillation and simple harmonic motion

Consider the motion of an object on a ramp;

Here the driving force is mgsinθ (we assume no resistive forces are acting on the ball)

mgsinθ=ma

gsinθ=a

notice that both g and θ remain constant throughout the motion and hence acceleration is constant. This motion is thus known as oscillation.

Now consider the motion of a pendulum;

Here the driving force is again mgsinθ however the value of θ changes throughout the motion and hence the value of acceleration changes. This is a perfect example of simple harmonic motion.

Acceleration ∝ - displacement

a = -xω²

Motion Graphs of Simple Harmonic Motion

Displacement - Time Graph

Velocity-Time Graph

Acceleration-Time Graph

Velocity-Displacement Graph

Acceleration-Displacement Graph

Potential Energy - Displacement Graph

Kinetic Energy - Displacement Graph

Total Energy - Displacement Graph

Equations of Energy for Simple Harmonic Motion

Energy (total) = Energy (kinetic) + Energy (potential)

When Displacement = 0

Energy (total) = Energy (kinetic)

Energy = 1/2mv²

Now since v=ωx
Energy = 1/2mω²x²

Types of Oscillations 

1. Free Oscillation: Oscillation in which the net loss of energy is 0.
2. Forced Oscillation: Vibration of the particle at the frequency of the applied force.
3. Damped Oscillation: Decrease in the amplitude of vibrations due to external resistive forces.

Note: (To provide damping force to a system we can either change the medium of the oscillation of the system i.e if its in air put it in water or we can attach a light card perpendicular to the motion to provide resistive force)

Differences between Critical Damping and Over Damping

Critical Damping: In critical damping the system does not oscillate and the displacement of the system decreases to 0 in a short period of time

Over damping/Heavy damping: The system does not oscillate and the displacement decreases to 0 in a comparatively longer period of time.

• Resonance: Resonance occurs when the natural frequency of vibration of the object becomes equal to the driving frequency, giving maximum amplitude.

Note: Notice that as the damping increases, the curve shifts to the left and becomes flatter, in addition the amplitude of the vibration decreases

Applications of Resonance:

• Resonance is useful in nuclear magnetic resonance (we will learn about this later) and in RLC circuits.
• Resonance also has some undesirable effects including shattering of glass (at the right pitch of sound) and resonance in rigid structures.

Gravitational fields (chapter 2)

• Field of Force: Region of space where there is force acting on an object
• Line of force: An imaginary line which represents the direction and the strength of a field
• Newton's Law of Gravitation: It states that the gravitational force between two points is equal to the product of their masses and is inversely proportional to the square of their separation (remember that the separation is to be taken from the centers of the respective masses).

F ∝ m₁m₂/r²

F=Gm₁m₂/r²

(G is the universal Gravitational Constant)

note: objects are considered as point masses when the seperation between them is very large than their relative sizes.

• Gravitational field strength: Gravitational field strength is defined as force per unit mass.

GMm/r² ÷ M = Gm/r²

Graphs representing the variation of gravitational field strength relative to the separation from Earth's surface

Graph representing the gravitational field strength between 2 planets

• Gravitational potential: It is the work done per unit mass in bringing an object from infinity to that point. Its unit is J/kg and the symbol used to represent it is Ф. 

Ф= -Gm/r

Gravitational potential energy is negative because gravitational forces are all attractive.

Graph representing the variation in gravitational potential relative to the separation

Graph representing the gravitational potential between 2 planets

the graph shows that more energy is required to escape from the larger planet as it has greater gravitational field strength.

• Gravitational potential energy at a point: It is the work done by the whole body to move it from infinity to that point.

Relating the linear motion of a particle towards the center of the planet with law of conservation of energy

ΔK.E = ΔP.E

1/2mv²- 0 = 0 - (-GMm/r)
v² = 2Gm/r
v² = 2Ф

• Geostationary satellites: Geostationary satellites are satellites that appear to be above a certain point on earth all the time. It is because the geostationary satellites have the same period as the period of the Earth's rotation. The following are their properties:

1. They are launched from the equator. (they have equatorial orbits)
2. Their rotation is from West to east.
3. Their altitude above the surface of earth is 3600 km.

Finding the Linear speed of satellite

since the gravitational force provides the satellite with centripetal force:

Gravitational force = centripetal force

GMm/r² = mv²/r

v² = GM/r

Circular Motion (Chapter 1)

Definitions

• Angular Displacement: It is the angle through which an object moves on a circular path. Its unit is radian.

• Angular Velocity: It is the rate of change of Angular Displacement of An object. Its Unit is ω.

 ω=Δθ/T

ω=2π/T=2πf (Since f=1/T)

• Radian: It is the Angle at which the arc length of a circle becomes equal to its radius.

s=rθ

θ=Arc-length/Radius=s/r

Proof That 1 Radian=2π

We know that s=rθ and we also know that s=2πr. Now equating the two equations we get

rθ=2πr

θ=2π

• Linear Velocity: It is the rate of Change of displacement of an object that is moving in a straight path.

v=rω

• Centripetal Force: Resultant force acting on an object moving in a circle always directed towards the center of the circle.

F=mv²/r

putting in v=rω we get F=mrω²

• Centripetal Acceleration: It is the acceleration of an object moving in uniform circular motion.

a=v²/r

Using v=rω in the equation a = r²ω²/r = rω²

Situations for Vertical circular motion

Case 1

Centripetal force = Tension - Weight 

mrω² =Tension - mg

Case 2

 Centripetal force = Tension + Weight 

 mrω² =Tension + mg

Case 3

When object is at the top

mg-R= mrω² 

When object is at the bottom

R-mg= mrω² 

Situations for Horizontal circular motion

Case 1

friction provides centripetal force

F=mrω²

Case 2

Tcosθ = mg

Tsinθ = mrω²

Case 3

Rsinθ = mg

Rcosθ = mrω²