High School Physics Notes

Chapter 7 Structural Properties of Matter

Intermolecular Forces

The forces that act between molecules are, obviously enough, called intermolecular forces. They arise from

Potential energy from the iteraction with surrounding molecules.
Thermal energy from the movement of the system of molecules. This is essentially kinetic energy.

Intermolecular forces between two molecules primarily depend on the distance \(r\) between them.

Types of Bonds

Each of the points in the uniform lattice structure which certain substances assume when forming a bond is called a lattice point.

Ionic Bonds

Each of the lattice points is occupied by an ion.
Attraction arises from the electrostatic force; which being strong, also makes the bonds comparatively strong.
The melting point is high due to the strength of the bonds.
The electro-conductivity is low.
Ionic bonds are only established between a metal and a non-metal.

Covalent Bonds

It's established only between non-metals by the sharing of electrons between two adjacent atoms.
Highly rigid.
High melting point.
Low electro-conductivity.
Compounds formed by covalent bonds generally act as electric dipoles.

Metallic Bonds

Established due to the presence of delocalized electrons, essentially like sharing the electrons of a metal atom with the rest of the atoms in a solid body.
The lattice points have positive ions and free electrons move around the whole structure.
High electro-conductivity.

Vandar Waals Bond

Arises between nonpolar molecules due to a temporary change in dipole moment.
Solids formed by this bond are called molecular solids.
Solids formed by this bond are generally soft.
Low melting point and bad electro-conductivity.

Definitions Regarding Elasticity

def Elasticity. The property of a deformed material to regain its original length, volume, and shape, once the deforming force has been removed, is called elasticity.

def Perfectly Elastic Body. A deformed body that fully returns to its initial length, volume, and shape, once the deforming force has been removed, is called a perfectly elastic body.

def Elastic Limit. The maximum applied force up to which a body remains perfectly elastic is called the elastic limit.

def Perfectly Rigid Body. A body that cannot be deformed by applying any magnitude of external force is called a perfectly rigid body. Such substances are not observed in nature.

def Perfectly Plastic Body. A body that retains its deformed shape and size once the deforming force has been removed, is called a perfectly plastic body.

def Breaking Force. The minimum applied force for which a body breaks or snaps is called the breaking force.

def Breaking Stress. The minimum perpendicular force applied per unit area for which a body breaks or snaps is called the breaking stress.

Elastic Properties

Strain is the change in the dimensions per unit dimension of a body due to the application of an external force. It's the rate of deformation of a body. \[ \text{strain} = \frac{B \text{~} A}{A} \] where \(A\) is the original dimensions of a body and \(B\) is the dimensions of the body after deformation.
Strain is a dimensionless scalar quantity.

Stress is the magnitude of the restoring force per unit area, perpendicular to the surface, aroused due to the application of an external force of equal magnitude. \[ \text{stress} = \frac{F}A \] where \(F\) is the magnitude of the applied or restoring force and \(A\) is the surface area perpendicular to the force.
The dimensions of stress is [ML\({}^{-1}\)T\({}^{-2}\)] and its units are Nm\({}^{-2}\) or Pa (Pascals).

The molecules in a crystal generally remain at a minimum potential energy state - the equilibrium, where the net intermolecular force is 0.

Longitudinal/tensile strain is the change in the length of a body per unit length. \[ \text{tensile strain} = \frac{l}{l_{0}} \] The perpendicular force that arises per unit surface area to resist the tensile strain, equal to the force applied to cause the tensile strain, is called the tensile stress. \[ \text{tensile stress} = \frac{F}{A} \] For a wire of radius \(r\) and cross-sectional area \(A = \pi r^2\) \[ \text{tensile stress} = \frac{mg}{\pi r^2} \] where \(m\) is the mass connected at the bottom of the wire and \(g\) is the acceleration due to gravity. Following the same reasoning as the tensile stress, the change in the volume per unit volume of a body is \[ \text{volumetric/bulk strain} = \frac{V}{V_{0}} \] Consequently, \[ \text{volumetric/bulk stress} = \frac{F}{A} \] Shearing strain is the relative displacement of two parallel planes situated at a unit distance. \[ \text{shearing strain} = \frac{\Delta x}{x_{0}}, \quad \Delta x \equiv x - x_0 \] And the strain angle \[ \theta = \arctan \left( \frac{\Delta x}{x_0} \right) \] recall that the body is of unit dimensions (length and width), therefore \(x_0\) is also the distance between the parallel planes.

Hook's Law states: within elastic limit, the stress of a body is directly proportional to the strain. \[ \text{stress} \propto \text{strain} \iff \frac{\text{stress}}{\text{strain}} = \text{constant} \] Continuously applying and removing stress to a body causes it elastic fatigue; as a result of which its elasticity decreases and it takes logner to return to its original state once the deforming force is removed. An elastically fatigued body can snap at a loewr breaking load even within its elastic limit.

Elastic Moduli

def Elastic moduli are the ratios of stress and strain of a material, which remains constant within the elastic limit.

Young's modulus or longitudinal modulus is given by \begin{align} \gamma &= \frac{\text{tensile stress}}{\text{tensile strain}} \\ &= \frac{ F/A }{ l/l_0 } \\ &= \frac{Fl_0}{Al} \end{align} where \(F\) is the magnitude of the applied force, \(A\) is the cross-sectional area perpendicular to the force, \(l\) is the stretched or compressed length of the material, and \(l_0\) is the original length of the material.

For a mass \(m\) attached at the bottom end of a wire of radius \(r\) and cross-sectional area \(A = \pi r^2\) \[ \gamma = \frac{mgl_0}{\pi r^2 l} \] Bulk modulus is given by \begin{align} \beta &= \frac{\text{bulk stress}}{\text{bulk strain}} \\ &= \frac{ F/A }{ V/V_0 } \\ &= \frac{FV_0}{AV} \\ &= \frac{PV_0}{V} \end{align} where \(V_0\) is the original volume of the material, \(V\) is the volume of the material after deformation, and \(P\) is the pressure applied to the material, which is the bulk stress.

Compressibility is the ratio of the bulk strain to the bulk stress on a material within the elastic limit. It's the reciprocal of the bulk modulus. \[ \text{compressibility} = \frac{\text{bulk strain}}{\text{bulk stress}} = \frac{1}\beta \] The rigidity modulus or shear modulus is given by \begin{align} G &= \frac{\text{shearing stress}}{\text{shearing strain}} \\ &= \frac{F/A}{x/x_0} \\ &= \frac{F/A}{\tan\theta} \end{align} where \(\theta\) is the strain angle.
For \(\theta \lt\lt 1\), \(\tan\theta \approx \theta\) \[ \therefore G = \frac{F}{A\theta} \] [Shear/rigidity modulus is denoted by an \(n\) in Dr. Tapan's text.]

Within elastic limit, the ratio of the lateral strain to the longitudinal strain is constant, and is termed as the poisson ratio. \[ \sigma = \frac{\text{lateral strain}}{\text{longitudinal strain}} \] For a wire of diameter \(d_0\) and and length \(l_0\) \[ \text{lateral strain} = \frac{d}{d_0} \] \[ \text{longitudinal strain} = \frac{l}{l_0} \] Therefore the poisson's ratio is given by \begin{align} \sigma &= \frac{d/d_0}{l/l_0} \\ &= \frac{dl_0}{d_0l} \\ \end{align} An equivalent but slightly different expression for the poisson's ratio is \[ \sigma = -\frac{rl_0}{lr_0} \] where \(r\) is the change in the radius of the wire due to deformation, and \(r_0\) is the original radius of the wire. The negative sign signifies that \begin{align} l \gt 0 &\implies r \lt 0, \quad \text{and} \\ l \lt 0 &\implies r \gt 0 \end{align} The poisson's ratio, being a ratio of two strains, is a dimensionless quantity.

In theory, sigma (poisson's ratio) remains within the range \(-1 \leq \sigma \leq 0.5\); but the negative value of \(\sigma\) is interpreted as the expansion of the diameter due to the expansion of the length (or vice versa), which is not observed in nature.

Generally, for metals \[ 0 \leq \sigma \leq 0.5 \] The work done to deform a body longitudinally is given by \[ W = \int_0^l \frac{\gamma A}L \, dl = \frac{1}2 \frac{\gamma A l^2}L \] where \(\gamma\) is the young's modulus, \(A\) is the cross-sectional area, and \(L\) is the initial length of the material.

Fluids

The type of fluid flow in which all particles follow the path of the adjacent particles moving forward with roughly the same velocity is called streamline flow.
In a streamline flow, the velocity of the flow is lower than the critical velocity of the fluid.

The type of fluid flow where the particles are flowing at a greater velocity than the critical velocity of the fluid is called turbulent flow.

Osborne Reynolds proved that the critical velocity of a fluid depends on the coefficient of viscocity (\(\eta\)) of the fluid, the density (\(\rho\)) of the fluid, and the radius (\(r\)) of the tube which the fluid is flowing through. \[ v_c \propto \frac{\eta}{\rho r} \] The proportionality constant is the Reynold's number \(R_e\). \[ v_c = R_e \frac{\eta}{\rho r} \] The type of fluid flow relates to the Reynold's number in the following ways \begin{align} R_e &\lt 2000 \implies \text{streamline flow} \\ R_e &\leq 3000 \implies \text{transitioning from streamline to turbulent flow} \\ R_e &> 3000 \implies \text{turbulent flow} \end{align} The property of a fluid to resist the relative motion of the different layers of it is termed as viscocity or internal friction.

Taking two layers of a fluid \(dy\) away from each other with velocities \(v\) and \(dv\), the velocity gradient is \[ \frac{dv}{dy} \] Newton's law of viscocity states - if there is relative velocity between two layers of a fluid at constant temperature, then the tangential viscous force is directly proportional to the surface area and the velocity gradient of the layers. \begin{align} F &\propto A, \quad \frac{dv}{dy} \text{ constant} \\ F &\propto \frac{dv}{dy}, \quad A \text{ constant} \\ F &\propto A \frac{dv}{dy}, \quad \text{ none are constant} \end{align} And the proportionality constant is the coefficient of viscocity. \[ F = \eta A \frac{dv}{dy} \] The dimensions of the viscous force is [ML\({}^{-1}\)T\({}^{-1}\)], and its units are Nsm\({}^{-2}\) or Pa\(\cdot\)s.

Another unit for the viscous force is poise. \[ 1 \text{ Nsm}^{-2} = 10 \text{ poise} \] The relation between the viscocity of a liquid with temperature is given by \[ \log \eta = A + \frac{B}T \] where \(A\) and \(B\) are constants.

The viscocity of gases is directly proportional to the square root of the temperature. \[ \eta \propto \sqrt{T} \] The relation between temperature and viscocity in liquids \[ T \uparrow \implies V \downarrow, \quad T \downarrow \implies V \uparrow \] And in gases \[ T \uparrow \implies V \uparrow, \quad T \downarrow \implies V \downarrow \] Pressure increases the viscocity in liquids, but almost no change in viscocity is observed in gases due to a change in pressure. However, some exceptions do occur at low pressures.

Stokes' Law and Drag Forces in Liquids

Stokes' Law states: the resistive force of a fluid on a sphere traversing through it is directly proportional to the radius of the sphere, the velocity of the sphere, and the viscocity of the fluid. \[ F = 6 \pi \eta r v \] where \(F\) is the resistive force, \(\eta\) is the coefficient of viscocity of the fluid, \(r\) is the radius of the sphere, and \(v\) is the velocity of the sphere.

The maximum constant velocity at which a body can travel in a fluid is called the terminal velocity. It's obtained when the applied force equals te resistive viscous force.

For a sphere of radius \(r\) and density \(\rho_s\) falling straight downwards in a fluid of density \(\rho_f\) and coefficient of viscocity \(\eta\) \begin{align} &\text{Gravitational force } &F_G &= mg = V\rho_s g \\ &\text{Buoyant force } &F_B &= V\rho_f g \\ &\text{Drag force } &F_{\text{drag}} &= 6 \pi r \eta v \end{align} Terminal velocity is reached when \begin{align} F_G &= F_B + F_{\text{drag}} \\ \implies V\rho_s g &= V\rho_f g + 6 \pi r \eta v \\ \implies v_{\text{terminal}} &= \frac{2r^2(\rho_s - \rho_f)g}{9\eta}, \quad V \equiv \frac{4}3 \pi r^3 \end{align} For air bubbles coming up in a fluid \[ v_{\text{terminal}} = \frac{2r^2(\rho_f - \rho_s)g}{9\eta} \] but since \(\rho_s \lt\lt \rho_f \implies \rho_f - \rho_s \approx \rho_f\), \[ v_{\text{terminal}} = \frac{2r^2\rho_f g}{9\eta} \]

Surface Tension and Related Phenomena

def Surface Tension. The force per unit area acting on both sides of an imaginary line drawn on the surface of a liquid at rest, tangent to the surface of the liquid and perpendicular to the immaginary line, is called surface tension. \[ T = \frac{F}l \] where \(l\) is the length of the imaginary line.

Molecular Theory of Surface Tension

def Cohesive force. The intermolecular attraction force between molecules of the same substance is called cohesive force.

def Adhesive force. The intermolecular attraction force between molecules of different substances is called adhesive force.

The maximum distance up to which two molecules are attracted with the cohesive force is called the range of intermolecular attraction. It's approximately \(10^{-10}\) m.
An imaginary sphere around a molecule whose radius is equal to the range of intermolecular attraction is called the influence sphere or range sphere.

The free surface of a liquid acts like an elastic membrane and tends to minimize the surface area by contracting itself. the work done per unit area of a surface to increase the potential energy (and therefore the surface area) per unit area of the surface is called the surface energy. \[ E_{\text{surface}} = \frac{W}{\Delta A} \] where \(\Delta A\) is the increase in surface area.

The expansion of the surface area for a larger drop breaking into smaller ones or the contraction of the safe after smaller drops gather into a larger one is given by \[ \Delta A = 4\pi (Nr^2 - R^2) \] where \(N\) is the number of smaller drops, \(R\) is the radius of the larger drop, and \(r\) is the radius of the smaller drops.

Then the energy dissipated due to the smaller drops combining into a larger one, or the work done to gather the smaller drops into a larger one is given by \[ W = \Delta A \times T = 4\pi(Nr^2 - R^2)T \] For hollow spheres like soap bubbles, the expansion or contraction in the surface area is twice that of a solid sphere such as a water drop because of surface tension acting on two surfaces (the inner and outer surfaces); hence the energy dissipated and/or the work done is also doubled.

Angle of Contact and Capilarity

The angle between the tangent to the curved liquid surface near the point of contact of a solid and a liquid, and the solid surface inside the liquid is called the angle of contact \(\theta\) for a given solid and a liquid.

If the liquid wets the solid, it bends upwards and \(\theta \lt \frac{\pi}2\).
If the liquid doesn't wet the solid, it bends downwards and \(\theta \gt \frac{\pi}2\)

The angle of contact depends on the following:

Properties of the solid and the liquid.
The medium above the free surface of the liquid.
Purity of the solid and the liquid.

A tube with a small and uniform cross-sectional area is called a capillary tube. The phenomenon of the rise or fall of liquid in a capillary tube is called capillarity.

Capillarity occurs due to surface tension.
Liquid moves up the capillary tube if the angle of contact \(\theta \lt \frac{\pi}2\) and moves down if \(\theta \gt \frac{\pi}2\).

The angle of contact and capillarity depend on the adhesive and cohesive forces between the liquid and the solid in the following ways:

cohesive force \(= \sqrt{2}\, \times\) adhesive force \(\implies\ \theta = 0\)
cohesive force \(\lt \sqrt{2}\, \times\) adhesive force \(\implies\ \theta \lt 0\)
cohesive force \(\gt \sqrt{2}\, \times\) adhesive force \(\implies\ \theta \gt 0\)

The height up to which a liquid rises due to capillarity is given by \[ h = \frac{2T\cos\theta}{r\rho g} \] where \(T\) is the surface tension, \(r\) is the radius of the capillary tube, \(rho\) is the density of the liquid, and \(\theta\) is the angle of contact.

For pure water, \(\theta \lt\lt 1\ \implies \cos\theta \approx 1\), then \[ h = \frac{2T}{r\rho g} \] At constant temperature, the surface tension \(T\) and the angle of contact \(\theta\) are constant. \begin{align} \therefore h &= \frac{K}r, \quad K = \frac{2T\cos\theta}{\rho g} \\ \implies h &\propto \frac{1}r \end{align} That is, at constant temperature, the height of the risen or fallen liquid in a capillary tube is inversely proportional to the radius of the tube - Jurin's Law.

Chapter 8 Periodic Motion

Periodicity is of two types:

Spatial periodicity - repeating in space
Temporal periodicity - repeating in time

Definitions Regarding Periodic Motion

def Periodic Motion. If the motion of a particle is such that it passes a certain point in its path from the same direction after a definite interval of time, the motion is called periodic motion.

def Time period. The time interval after which a particle in periodic motion passes a certain point of its path from the same direction is called the time period.

def Oscillatory/ Vibratory Motion. If the periodic motion of a particle is such that it moves in one direction for half the duration of the time period and in the opposite direction for the other half, it's called vibratory or oscillatory motion.

def Simple Harmonic Motion (SHM). If the acceleration of a body is directly proportional to its displacement with respect to a certain equilibrium point, and is always directed towards that point, then the motion is called simple harmonic motion.

Since force is proportional to acceleration, it is also proportional to the displacement. \begin{align} F & \propto a \propto -x \\ \implies F & \propto -x = -kx \end{align} where \(k\) is called the force constant.

The force in SHM is

Periodic
Oscillatory
Proportional to the magnitude of the displacement at any given instance

def Complete oscillation. In SHM, a complete back and forth movement of a body is called a complete oscillation.

def Time period. The time taken to complete one complete oscillation is called the time period.

def Frequency. The number of complete oscillations per second is called the frequency.

def Amplitude. The maximum displacement of a body in SHM from the equilibrium is called the amplitude.

def Phase. The phase of a particle in SHM is the state of its motion at a particular instant in time. It determines the displacement, velocity, acceleration, and so on, of the particle.

The Differential Equation of SHM

\begin{align} F = ma &= -kx \\ \implies m\ddot{x} &= -kx \\ \implies \ddot{x} &= -\frac{k}m x \\ \implies \ddot{x} &= -\omega^2x, \quad \omega \equiv \sqrt{\frac{k}m} \end{align} this differential equation is called the differential equation of periodic motion. Here's how to solve it. \begin{align} \ddot{x} &= -\omega^2x \\ \implies \frac{dv}{dt} &= -\omega^2x \\ \implies \frac{dx}{dt}\frac{dv}{dx} &= -\omega^2x \\ \implies \int v\, dv &= -\omega^2x\, dx \\ \implies \frac{v^2}2 &= -\omega^2 \frac{x^2}2 + C \\ \end{align} Now using the initial values \(v(0) = 0\) and \(x(0) = A\), we get \begin{align} C = \frac{\omega^2A^2}2 \end{align} Thus we obtain the general solution \begin{align} \frac{v^2}2 &= -\omega^2\frac{x^2}2 + \frac{\omega^2A^2}2 \\ \implies v &= \omega\sqrt{A^2 - x^2} \\ \implies \frac{dx}{A\sqrt{1 - \frac{x^2}{A^2}}} &= \omega\, dt \\ \end{align} Substituting \(x = A\sin\theta \implies dx = \cos\theta\, d\theta\) \begin{align} \frac{A\cos\theta}{A\sqrt{1 - \sin^2\theta}} d\theta &= \omega\, dt \\ \implies \int d\theta &= \int \omega \, dt \\ \implies \theta &= \omega t + \phi \\ \implies \arcsin\left(\frac{x}A\right) = \omega t = \phi \\ \implies x = A\sin(\omega t + \phi) \end{align} By instead substituting \(x = A\cos\theta\), we would've arrived at the following alternate solution \[ x = A\cos(\omega t + \delta) \] This is left as an exercise for the reader.

Observe in the obtained equation now that \begin{align} x &= A\sin[\omega\left(t + \frac{2\pi}\omega\right) + \phi] \\ &= A\sin(\omega t + 2\pi + \phi) \\ &= A\sin(\omega t + \phi) \\ &= x \end{align} meaning that the particle returns to its starting position after every \(\frac{2\pi}{\omega}\) seconds. Hence it's the time period \[ T = \frac{2\pi}\omega \] Now recall from earlier \[ \omega^2 = \frac{k}m \] Which then follows \[ T = 2\pi\sqrt{\frac{m}k} \] And because we have that \(f = \frac{1}T\), we also have that \[ f = \frac{1}{2\pi}\sqrt{\frac{k}m} \] We've been using \(\omega\) for a while but haven't mentioned what it is. It's the angular frequency of the periodic motion. It shouldn't be confused with angular velocity from rotational motion, albeit the two are very closely related. \[ \omega = \frac{2\pi}{T} = 2\pi f = \sqrt{\frac{k}m} \] The \(A\) in the position function is the amplitude, while the quantity \((\omega t + \phi)\) is the phase.

We can obtain an expression for the velocity by differentiating the position function \[ v = \frac{dv}{dx} = \omega A\cos(\omega t + \phi) \] Alternatively, we also have that \begin{align} v &= \omega \sqrt{A^2 - x^2} \end{align} From the former expression for the velocity, we can infer that the expression for the maximum velocity is given by \[ v_{\text{max}} = \omega A, \quad [\because \text{the maximum value for the cosine function is 1}] \] Differentiating once more, we find the expression for the acceleration \begin{align} a = \frac{dv}{dt} = -\omega^2 A\sin(\omega t + \phi) \end{align} And the expression that gives the maximum acceleration is therefore \[ a_{\text{max}} = \omega^2 A \]

Energy in Simple Harmonic Motion

Potential Energy

The potential energy is given by \[ U = \int_0^x F(x) \, dx \] And in SHM \[ F = -kx \] Therefore work is done by applying an equal and opposite force of \(F'=kx\) to it. Therefore the potential energy is given by \begin{align} U &= \int_0^x F'\, dx \\ &= \int_0^x kx\, dx \\ &= \frac{kx^2}2 \end{align} Which can be rewritten as the following \[ U = \frac{1}2kA^2\sin^2(\omega t + \phi) \] From which we can infer the maximum potential energy of a system in SHM to be \[ U_{\text{max}} = \frac{kA^2}2 \] which is also the total mechanical energy of the system.

Kinetic Energy

The kinetic energy of a system is given by \[ K = \frac{1}2 mv^2 \] And for SHM, we have that \[ v = \omega A \cos(\omega t + \phi) \] Therefore the kinetic energy of a system in SHM is given by \begin{align} K &= \frac{1}2 m \omega^2 A^2 \cos^2(\omega t + \phi) \\ &= \frac{1}2 k A^2 \cos^2(\omega t + \phi), \quad \left[\because \omega^2 \equiv \frac{k}m\right] \end{align} Then the maximum kinetic energy is \[ K_{\text{max}} = \frac{kA^2}2 \] which, again, is the total mechanical energy of the system.

And finally, we verify that our claim about the expressions for the maxima of the potential and the kinetic energies is true \[ E_{\text{mech}} = U + K = \frac{1}2 kA^2 \sin(\omega t + \phi) + \frac{1}2 k A^2 \cos(\omega t + \phi) = \frac{1}2 k A^2 \] and indeed we were not in the wrong to have been making that claim.

Applications of the Theory of Simple Harmonic Motion

Vertically Oscillating Spring

Let us take a vertical spring of neglegible mass with a body of mass \(m\) attached to the bottom end of it. As a result of the hanging mass, let the new equilibrium of it be at \(x_0\). Taking downwards as the positive direction, when the spring is at the equilibrium we have that \begin{align} \sum F_y &= 0 \\ \implies -kx_0 + mg &= 0 \\ \implies mg &= kx_0 \end{align} Now when the spring is given a downward pull or an upward push by \(x\) units, the hanging mass gets an acceleration \(a\) \begin{align} -k(x+x_0) + mg &= ma \\ \implies -kx + kx_0 &= ma \quad [\,\, \because mg = kx_0\,\,] \\ \implies -kx &= ma \\ \implies a &= -\frac{k}m x \\ \implies a &= -\omega^2 x, \quad \omega \equiv \sqrt{\frac{k}m} \end{align} Thus having obtained the acceleration of the hanging mass on the spring, we also get the rest of the properties of SHM holding true for the system as well, given three conditions are fulfilled:

The spring must not be stretched beyond its elastic limit so as to invalidate the applicability of Hook's Law.
The amplitude of the oscillation must be smaller than the stretched length due to the handing mass: \(\mid x \mid \lt \mid x_0 \mid \).
The mass of the spring is neglegible compared to the mass of the object.

The time period for the spring is given by \[ T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}k} = 2\pi\sqrt{\frac{x_0}g} \impliedby mg = kx_0 \] where \(m\) can be either the mass of the hanging mass alone or the comnbined mass of the spring and the hanging mass \(m \equiv m_{spring} + m_{object}\).

The square of the time period of the oscillating mass is directly proportional to the mass \begin{align} T &= 2\pi\sqrt{\frac{m}k} \\ \implies T^2 &= 4\pi^2\frac{m}k \\ \implies T^2 &\propto m, \quad [\,\, \because \left(\frac{4\pi^2}k\right) \text{ is a constant.} \,\,] \end{align}

Simple Pendulum

def Simple Pendulum. A system consisting of a point mass suspended by a massless, inextensible, but perfectly flexible thread which can oscillate in a vertical plane without friction is called a simple pendulum.

def Bob. The body of mass which is suspended by the thread in a simple pendulum is called the bob.

def Point of Suspension. The point from which the bob is suspended by the thread in a simple pendulum is called the point of suspension.

def Effective Length. The distance from the point of suspension to the center of mass of the bob is called the effective length of a simple pendulum. For a thread of length \(l\) and a bob of radius \(r\), the effective length \(L\) is \[ L = l + r \] The acceleration of a simple pendulum is given by \[ a = -g\sin\theta \] For \(\theta \lt\lt 1\), \(\sin\theta \approx \theta\). Then we have \[ a = -g\theta = -\frac{g}L x = -\omega^2 x, \quad \omega \equiv \sqrt{\frac{g}L} \] Then the time period for the simple pendulum becomes \[ T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{L}g} \]

Laws of the Simple Pendulum

1) Isochronism. For angular amplitude \(\theta \lt 4^{\circ}\) and constant effective length \(L\), each oscillation of a simple pendulum takes an eqaul amount of time at the same geographical location - the time period is independent of the angular amplitude.

2) Law of Length. For angular amplitude \(\theta \lt 4^{\circ}\) the time period of a simple pendulum is directly proportional to the square root of its effective length at the same geographical location. \[ T \propto \sqrt{L} \] 3) Law of Accerlation. For angular amplitude \(\theta \lt 4^{\circ}\) and constant effective length \(L\), the time period of a simple pendulum is inversely proportional to the square root of the acceleration due to gravity at the same geographical location. \[ T \propto \frac{1}{\sqrt{g}} \] 4) Law of Mass. For angular amplitude \(\theta \lt 4^{\circ}\) and constant effective length \(L\), the time period is independent of the bob's mass, volume, or composition at the same geographical location.

Uses of the Simple Pendulum

Determination of the acceleration due to gravity. \[ g = 4\pi^2\frac{L}{T^2} \]
Determination of the height of a geographical location. \[ h = \left[\frac{T'}T - 1\right]R = \left[\left(\frac{g}{g'}\right)^{0.5} - 1\right]R \] where \(T\) is the time period of the simple pendulum at the bottom of the mountain, \(T'\) is the time period of the same at the top of the mountain, and \(R\) is the radius of the earth.
Determination of time. If a clock runs \(n\) seconds fast/slow in one day, then its time period is given by \[ T = \frac{2\times 86400}{86400 \pm n} \]

The proofs of the above is left as an exercise for the reader.

def Second Pendulum. A pendulum that performs half an oscillation per second is called a second pendulum.

The length of a second pendulum is found thus \begin{align} T &= 2 = 2\pi\sqrt{\frac{L}g} \\ \implies L &= \frac{g}{\pi^2} \end{align} The observation to be made here is that the length of the second pendulum is directly proportional to the acceleration due to gravity.

Chapter 9 Waves

def Wave The periodic disturbance that carries energy by advancing through a material medium without permanently displacing the particles of the medium is called a wave.

Material media propagate mechanical waves. Electromagnetic waves propagate through the electromagnetic field which propagates all of spacetime and is therefore essentially everywhere; which is why electromagnetic waves have no difficulty traveling throught he vacuum of outer space.
A third type of waves is the matter wave which has to do with the fundamental particles of nature; it's discussion is beyond the scope of this chapter.

Definitions regarding Waves

def Complete Oscillation. The returning of an oscillating particle to its intitial position and moving in the same direction as its original motion is called a complete oscillation.

def Time Period. The time required for an oscillating particle to complete one complete oscillation is called the time period \(T\).

def Frequency. The number of complete oscillations of an oscillating particle in 1 second is called the frequency \(f\).

The time period and frequency are reciprocals \[ fT = 1 \]
def Amplitude. The maximum displacement of an oscillating particle from the equilibrium is called the amplitude \(A\).

def Phase. The state of the motion of an oscillating particle at a particular instant in time is called the phase.

def Wavelength. The distance traveled by a wave during the completion of one complete oscillation is called the wavelength \(\lambda\).
Alt: The distance between two consecutive peaks or troughs of a wave is called the wavelength.
Alt 2: The distance between two consecutive particles in a wave oscillating at the same phase is called the wavelength.

def Wave Velocity. The distance travelled by a wave in a particular direction in unit time is called the wave velocity \(v\).

def Angular Frequency. The rate of change of the phase of a particle in a wave is called the angular frequency \(\omega\). \[ \omega = \frac{2\pi}{T} = 2\pi f \] it has the units of \(\text{rad} \cdot \text{s}^{-1}\).

The following relations hold true \[ v = \frac{\lambda}T = \lambda f = \frac{\lambda \omega}{2\pi} \]
def Simple Harmonic Waves. The waves created due to SHM are called simple harmonic waves.
SHW are of two types:

Transverse waves are waves where the particles oscillate perpendicular to the direction of propagation of the wave.
Longitudinal waves are waves where the particles oscillate parallel to the direction of propagation of the wave.

Travelling or Progressive Waves

Travelling or Progressive waves are periodic disturbances propagating from one layer to another in a wide medium, continuously advancing in a particular direction.

A traveling wave is described by \[ y = A\sin(\omega t) = A\sin 2\pi f t \] With a non-zero phase-shift \(\phi\), \begin{align} y &= A\sin(\omega t - \phi) \\ &= A\sin(\omega t - \frac{2\pi}\lambda x) \\ &= A\sin(\omega t - kx), \quad k \equiv \frac{2\pi}\lambda \end{align} The \(k\) must not be confused with the spring constant. It's the wavenumber or spatial frequency, a spatial counterpart to the temporal frequency \(f = \frac{1}T\). It is sometimes also denoted with \(\frac{1}\lambda\) instead of \(\frac{2\pi}\lambda\).

\begin{align} y &= A\sin(\omega t - \phi) \\ &= A\sin\left( 2\pi f t - \frac{2\pi}\lambda x \right) \\ &= A\sin\left[\frac{2\pi}\lambda(vt - x)\right] \end{align} which is a wave travelling to the positive \(x\)-direction, which is generally to the right. On the other hand, a wave travelling to the negative \(x\)-direction (left) is described by \[ y = A\sin\left[\frac{2\pi}\lambda(vt + x)\right] \] a mere change in sign.

The Principal of Superposition. The resultant displacement from the equilibrium of a particle caused by two waves being incident on it simulteneously is equal to the vector sum of the displacements produced by the two waves individually.

The phase difference of two waves of the same amplitude and frequency, but different paths, is given by \[ \delta = \frac{2\pi}\lambda(x_2 - x_1) = \frac{2\pi}\lambda \Delta x \]

Standing or Stationary Waves

Standing or Stationary waves are resultant waves produced in a finite part of a medium due to the superposition of two waves of the same wavelength and amplitude travelling in opposite directions.

\begin{align} y &= y_1 + y_2 \\ &= a\sin \frac{2\pi}\lambda (vt - x) + a\sin \frac{2\pi}\lambda (vt + x) \\ &= 2a \, \sin\frac{2\pi}\lambda vt \, \cos\frac{2\pi}\lambda x \end{align} Here we have the amplitude \[ A = 2a\cos\frac{2\pi}\lambda x = 2a\cos kx \] then the wave is described by \[ y = A\sin\frac{2\pi}\lambda vt, \quad A \equiv 2a\cos kx \] The nodes of a standing wave are formed at the minima of the of the wave, where the amplitude is at its minimum.

The anti-nodes of a standing wave, on the other hand, are formed at the maxima of the wave, where the amplitude is at its maximum.

The expression we found earlier for the amplitude of a standing wave \[ A = 2a\cos\frac{2\pi}\lambda x \] Therefore the minimum is given by \begin{align} \cos\frac{2\pi}\lambda x &= 0 \\ \implies \frac{2\pi}\lambda x &= \frac{\pi}2,\, \frac{3\pi}2,\, \cdots,\, \frac{(2n+1)\pi}2 \\ \implies x &= (2n+1)\frac{\lambda}4, \quad n \in \mathbb{N} \end{align} And the maximum is given by \begin{align} \cos\frac{2\pi}\lambda x &= 1 \\ \implies \frac{2\pi}\lambda x &= 0,\, \pi,\, 2\pi,\, \cdots,\, n\pi \\ \implies x &= \frac{n}2 \lambda, \quad n \in \mathbb{N} \end{align} So we observe that the nodes are at \[ x = (2n+1)\frac{\lambda}4 \] and the anti-nodes are at \[ x = \frac{n}2\lambda \]

A few more Definitions

def Free Vibration. The vibration of a body due to a natural frequency of its own is called free vibration.

def Forced Vibration. When a body is made to oscillate by periodically applying a force to it at a frequency other than its natural frequency, it's termed as forced vibration.

The phenomenon of a body vibrating with the maximum amplitude because of applying a frequency to it equal to its natural frequency is called resonance.

Intensity

def. The amount of energy propagating per second per unit area perpendicular to the direction of propagation of a wave is called its intensity. \[ I = \frac{P}A = \frac{P}{4\pi r^2} \] it has the units of \(\text{Js}^{-1}m^{-2}\) or \(\text{Wm}^{-2}\). \begin{align} I &= \frac{P}A \\~\\ &= \frac{E}{tA} \\~\\ &= \frac{EL}{LtA}, \quad L \equiv \text{length of a portion of the medium} \\~\\ &= \frac{EL}{Vt}, \quad V \equiv \text{volume of a portion of the medium} \\~\\ &= \frac{Ev}V, \quad v \equiv \text{wave velocity} \end{align} The mechanical energy of a system in periodic motion is given by \[ E_{\text{mech}} = \frac{1}2 m v_{max}^2 = \frac{1}2 m (\omega A)^2, \quad A \equiv \text{amplitude of the wave} \] Then we have the intensity to be \begin{align} I &= E\frac{v}V \\ &= \frac{1}2 m (\omega A)^2 \frac{v}V \\ &= \frac{1}2 \frac{m}V v (2\pi f A)^2 \\ &= 2\rho v \pi^2 f^2 A^2 \end{align} The following hold true for intensity of sound waves

\(I \propto A^2\)
\(I \propto \frac{1}{r^2}\), where \(r\) is the distance between the source and the listener.
\(I \propto f^2\)
\(I \propto \rho\), where \(\rho\) is the density of the medium.
The intensity of sound is increased by the presence of a resonating body in vicinity.
The intensity increases if the sound is propagating along the direction of motion of the medium; the intensity decreases if it's propagating against the direction of motion of the medium.

The intensity level of a sound wave is given by \begin{align} \beta &= L - L_0 \\ &= a\log\frac{I}{I_0} \end{align} where \(L\) is some arbitrary loudness, \(L_0\) is some standard loudness, \(I\) is some arbitrary intensity, and \(I_0\) is the standard intensiity; \(a\) is a proportionality constant.

The unit of intensity level is Bel (B), but the usually used unit is a tenth of it - decibel, which is why the usual expression for the intensity level is \[ \beta = 10\log\frac{I}{I_0} \] def. The bel is defined as - the intensity level of the sound whose intensity is 10 times greater than that of the standard intensity \(I_0\) is called 1 bel.

def. The logarithm of the ratio of the intensity of a sound and the standard intensity is called the intensity level.

The following relation exists between intensity and power \[ \frac{I}{I_0} = \frac{P}{P_0} \implies \beta = 10\log\frac{P}{P_0} \] in which case, it's termed as the power level.

The change in the intensity or power level is given by \[ \Delta \beta = 10\log\frac{I_1}{I_2} = 10\log\frac{P_1}{P_2} \]

A few more Definitions 2

def Threshold of Audibility. The minimum intensity for audibility is called the threshold of audibility. It's \(10^{-12} \text{ Wm}^{-2}\) for humans.

def Threshold of Pain. The minimum intensity at which pain begins in the ear is called the threshold of pain. It's \(1 \text{ Wm}^{-2}\) for humans.

Beats

def. A beat is an interference pattern between two sound waves of the same amplitude but slightly different frequencies, perceived as periodic variation in volume whose rate is the difference of the two frequencies.

\begin{align} & y_1 = a\sin 2\pi f_1 t \\ & y_2 = a\sin 2\pi f_2 t \\ \therefore \, & Y = y_1 + y_2 = 2a\cos\left[2\pi\frac{f_1-f_2}2t\right]\sin\left[2\pi\frac{f_1+f_2}2t\right] \end{align} So we get that the resultant amplitude \[ A = 2a\cos\left[2\pi\frac{f_1-f_2}2t\right] \] is that of the superposition of the two waves.

And the resultant frequency \[ f = \frac{f_1 + f_2}2 \] is the mean of the two frequencies.

A beat is produced when the amplitude is at its maximum, and a period of silence is occurs when the amplitude is at its minimum.

Therefore a beat is produced when \begin{align} \cos\left[2\pi\frac{f_1-f_2}2t\right] &= \pm 1 \\ \implies 2\pi \frac{f_1-f_2}2t &= 0,\, \pi,\, 2\pi,\, \cdots,\, n\pi \\ \implies t &= 0,\, \frac{1}{f_1-f_2},\, \frac{2}{f_1-f_2},\, \cdots,\, \frac{n}{f_1-f_2}; \quad n \in \mathbb{N} \\ \end{align} And the silences are produced when \begin{align} \cos\left[2\pi\frac{f_1-f_2}2t\right] &= \pm 0 \\ \implies 2\pi\frac{f_1-f_2}2t &= \frac{\pi}2,\, \frac{3\pi}2,\, \frac{5\pi}2,\, \cdots,\, (2n+1)\frac{\pi}2 \\ \implies t &= \frac{1}{2(f_1-f_2)},\, \frac{3}{2(f_1-f_2)},\, \frac{5}{2(f_1-f_2)},\, \cdots,\, \frac{2n+1}{2(f_1-f_2)}; \quad n \in \mathbb{N} \end{align} We now observe that the duration between two consecutive beats as well as two consecutive silences is precisely \[ \frac{1}{f_1-f_2} \, \text{s} \] Therefore if we have 1 beat every \(\frac{1}{f_1-f_2}\) seconds, the number of beats per second is given by \[ N = f_1-f_2 \] This is what the whose rate is the difference of the two frequencies meant in our definition for the beat.

Music Theory and Organ Pipes

Definitions

def. The pitch of a sound is the characteristic that allows us to differentiate between a shrill and a dull sound. It is proportional to the frequency; however, frequency is a physical measurable phenomena and pitch is a sensation in the ear.

def. The characteristic of sound that allows us to differentiate between two sounds of the same intensity and pitch is called timbre.

If there is only one frequency present in a sound wave, it's called a tone. If there's more than one, it's called a note. The tone of the minimum frequency present in a note is called the fundamental tone; the rest are called overtones. Every overtone that is an integer multiple of the fundamental tone/frequency is called a harmonic.

The ratio of the frequencies of two tones is called the musical interval. The musical interval of two tones is the product of the intermediate musical intervals.

Two notes played simulteneously is called a chord. A pleasant combination of notes is called a conchord or consonance, while an unpleasant combination is called a discord or dissonance.
It is noticed that if the frequencies of the tones played in a note are in a ratio of integers, then the sound produced is pleasant.

def. The octave of a tone is a tone which has double the frequency of the original tone.

def. An assembly of tones having consonance is called a musical scale. And the lowest tone in the a musical scale is called the tonic or the key tone. There are 8 frequencies having cosonance in an ascending order in the musical scale, and the scale is called a diatonic scale. The three musical intervals in the diatonic scale are \(\frac{9}8,\, \frac{10}9,\, \frac{16}{15}\), which are called major, minor, and semi- tones respectively.

Depending on the voice of the vocalist, the diatonic scale is not always sufficient to produce melodious sound, which is when five other tones are included within one octave - then the scale is called a tempered scale.

A combination of sounds where the tones are in the ratio of \(4:5:6\) is called a triad. The sound produced by combining the octave of the lowest tone in a triad with the triad (at a ratio of frequencies of \(4:5:6:8])) is called a chord.

A harmony is when a number of sounds sound pleasant by being produced simulteneously. A melody is when a number of sounds sound pleasant by being produced successively. A number of musical instruments being used to produce either harmony or melody or both, is called an orchestra.

def. A flexible part of any material with uniform cross-sectional area and a length that is much greater than the cross-sectional area is called a string.

Marsenne's Laws of Transverse Vibrations in a Stretched String

Law of Length: with constant tension \(T\) and mass per unit length \(\mu\) \[ f \propto \frac{1}{L} \]
Law of tension: with constant length \(L\) and mass per unit length \(\mu\) \[ f \propto \sqrt{T} \]
Law of mass: with constant length \(L\) and tension \(T\) \[ f \propto \sqrt{1}{\mu} \]

And in general \[ f = K\frac{1}L \sqrt{\frac{T}\mu} \] where \(K\) is a proportionality constant.

For the fundamental frequency of a string, the value of \(K\) is experimentally determined to be \(\frac{1}2\). Thus we get \[ f = \frac{1}{2l}\sqrt{\frac{T}\mu} = \frac{1}\lambda \sqrt{\frac{T}\mu} \] And so the wave velocity is given by \[ v = f\lambda = \sqrt{\frac{T}\mu} \]

Vibration of the Air Collumn in an Organ Pipe

Simply put, an open organ pipe can only produce the odd harmonics of the fundamental tone of the pipe, while an open pipe can produce all harmonics.

In in open pipe, the \(n\)-th harmonic is given by \[ f_n = nf_1 = \frac{nv}{2L} \] And in a closed pipe, the \(n\)-th harmonic is given by \[ f_n = nf_1 = \frac{nv}{4L} \]