Part B: Oscillations, Acoustics and Thermal Physics
1. Derive an expression for the time period of the torsional pendulum and arrive at the equation of torsional rigidity. Also mention its uses.
A torsional pendulum consists of a rigid body (like a disc) suspended by a thin wire fixed at its upper end. When the body is twisted by a small angle θ from its equilibrium position and released, it executes angular simple harmonic motion (SHM).
Theory:**
- When the wire is twisted by an angle θ, it exerts a restoring torque (τ) proportional to the angular displacement θ, acting in the opposite direction.
- This is given by τ = -Cθ, where C is the **torsional rigidity** or torsional constant of the wire (the restoring couple per unit angular twist). The negative sign indicates that the torque opposes the displacement.
- According to Newton's second law for rotation, this restoring torque is also equal to the product of the moment of inertia (I) of the suspended body about the axis of rotation (the wire) and the angular acceleration (α).
- Angular acceleration α = d²θ/dt².
- So, τ = Iα = I (d²θ/dt²).
Equating the two expressions for torque:
I (d²θ/dt²) = -Cθ
Rearranging the equation:
(d²θ/dt²) + (C/I)θ = 0
This equation is in the standard form for simple harmonic motion: (d²x/dt²) + ω²x = 0.
Comparing the two equations, the angular frequency (ω) of the torsional pendulum is:
ω² = C/I ⟹ ω = √(C/I)
The time period (T) of the oscillation is related to the angular frequency by T = 2π/ω.
T = 2π / √(C/I)
Therefore, the expression for the time period of a torsional pendulum is:
T = 2π √(I/C)
Equation for Torsional Rigidity (C):**
From the time period equation, we can rearrange to find the torsional rigidity C:
T² = (2π)² (I/C) = 4π² I / C
C = 4π² I / T²
The torsional rigidity C can also be expressed in terms of the material's properties (Shear Modulus G), the wire's radius (r), and its length (L): C = πGr⁴/(2L). Equating the two expressions for C allows for the determination of G if I and T are known, or vice versa.
Uses of Torsional Pendulum:**
- Determining Moment of Inertia (I): If the torsional constant (C) of the wire is known, the moment of inertia of an irregularly shaped object can be found by measuring its period of oscillation (T) when suspended from the wire.
- Determining Shear Modulus (G): If the moment of inertia (I) of the suspended body (usually a regular shape like a disc) is known, the shear modulus (modulus of rigidity) of the wire's material can be determined by measuring the period (T), radius (r), and length (L) of the wire.
- Torsional Clocks: Used in timekeeping devices like anniversary clocks, where the slow, long-period oscillation is utilized.
2. What is ultrasonics? Explain the magnetostriction method of producing ultrasonic waves.
Ultrasonics:**
Ultrasonics refers to sound waves having frequencies **above the upper limit of human hearing**, which is typically around 20 kilohertz (20 kHz). These high-frequency waves possess unique properties like high energy, short wavelength, and directionality, making them useful in various applications.
Magnetostriction Method for Producing Ultrasonic Waves:**
- Principle: This method is based on the **magnetostriction effect**. Ferromagnetic materials (like iron, nickel, cobalt, and their alloys) undergo a small change in length when placed in a magnetic field parallel to their length. When the magnetic field is alternating, the material vibrates longitudinally.
- Construction:
- A rod (XY) made of a ferromagnetic material (e.g., nickel) is clamped at its center.
- Two coils, L₁ and L₂, are wound around the ends of the rod without touching it.
- Coil L₁ (Grid coil) is part of the resonant tank circuit (LC circuit) of an oscillator (like a Hartley oscillator using a transistor or vacuum tube).
- Coil L₂ (Plate/Collector coil) is connected to the output (Plate/Collector) circuit of the oscillator and provides positive feedback to the L₁C circuit.
- A DC polarizing magnetic field might be applied initially to operate in an optimal region of the magnetostriction curve.
- Working:
- When the oscillator circuit is switched on, the tank circuit (L₁C) starts oscillating electrically at its natural frequency, $f = 1 / (2π√(L₁C))$.
- The alternating current flowing through L₁ produces an alternating magnetic field along the length of the rod XY.
- Due to the magnetostriction effect, the rod starts vibrating longitudinally, changing its length at the frequency of the alternating magnetic field.
- These vibrations create mechanical stress in the rod, which in turn causes a change in the magnetization of the rod (inverse magnetostriction effect or Villari effect).
- This change in magnetization induces an alternating EMF in coil L₂, which is fed back to the input coil L₁ via the oscillator circuit.
- If the frequency of the electrical oscillations matches the natural frequency of longitudinal vibration of the rod, resonance occurs. The amplitude of vibrations becomes maximum.
- The natural frequency of the rod is given by $f_{rod} = (1 / 2L) √(E/ρ)$, where L is the length, E is Young's modulus, and ρ is the density of the rod material.
- By adjusting the variable capacitor C, the frequency of the tank circuit is tuned to match the rod's natural frequency, achieving resonance.
- The high-amplitude longitudinal vibrations of the rod's ends produce high-frequency sound waves (ultrasonics) in the surrounding medium.
- Frequency Limit: This method is typically suitable for producing frequencies up to about 100 kHz. For higher frequencies, the efficiency decreases.
3. Explain with a neat diagram, principle, construction, working and application of piezoelectric method to produce ultrasonics.
Principle:**
This method utilizes the **inverse piezoelectric effect**. Certain anisotropic crystals (like quartz, tourmaline, Rochelle salt) develop an electric potential difference across their faces when subjected to mechanical stress (direct piezoelectric effect). Conversely, when an electric field is applied across specific faces of these crystals, they undergo mechanical deformation (strain) - they either expand or contract (inverse piezoelectric effect).
If a high-frequency alternating voltage is applied across the crystal, it vibrates mechanically at the same high frequency, generating ultrasonic waves.
Construction:**
- A thin slice (wafer) of a piezoelectric crystal (e.g., quartz) is cut with its faces perpendicular to specific crystallographic axes (e.g., X-cut for longitudinal vibrations).
- The crystal slice (Q) is placed between two metal plates (electrodes), A and B, forming a parallel plate capacitor arrangement.
- These plates are connected to the output of a high-frequency electronic oscillator (e.g., a Hartley or Colpitts oscillator). The crystal itself often forms part of the oscillator's resonant tank circuit, replacing the conventional LC circuit or coupled to it.
Working:**
- The electronic oscillator generates a high-frequency alternating voltage (typically radio frequencies).
- This alternating voltage is applied across the metal plates A and B, creating a high-frequency alternating electric field across the quartz crystal.
- Due to the inverse piezoelectric effect, the crystal alternately contracts and expands with each half-cycle of the applied voltage.
- These mechanical vibrations occur at the frequency of the applied voltage.
- The crystal has a natural frequency of mechanical vibration determined by its dimensions (especially thickness for longitudinal vibrations) and elastic properties. The fundamental frequency is given by $f = (P / 2t) √(E/ρ)$, where P is an integer (P=1 for fundamental), t is the thickness, E is Young's modulus, and ρ is the density.
- When the frequency of the applied alternating voltage from the oscillator is tuned (often by adjusting a variable capacitor in the oscillator circuit) to match the natural frequency of the crystal, **resonance** occurs.
- At resonance, the amplitude of the crystal's mechanical vibrations becomes very large.
- These high-amplitude, high-frequency vibrations of the crystal faces generate powerful ultrasonic waves in the surrounding medium (air, liquid, or solid).
Advantages:**
- Can produce very high ultrasonic frequencies (up to several MHz or even GHz).
- High efficiency in converting electrical energy to ultrasonic energy.
- Stable output frequency determined by the crystal dimensions.
Applications (of Ultrasonics Produced):**
- Medical diagnostics (imaging) and therapy (lithotripsy).
- Non-destructive testing (NDT) of materials.
- SONAR systems.
- Ultrasonic cleaning.
- Welding of plastics and metals.
- Chemical processing (sonochemistry).
4. Describe Forbe’s method to determine thermal conductivity of metals with relevant theory and experiment.
Forbe's method is a classical technique used to determine the thermal conductivity (k) of a good conductor, typically in the form of a long bar.
Theory:**
The method involves measuring temperatures along the bar under two conditions: a steady state and a transient (cooling) state.
- Steady State:
- One end of a long, uniform metal bar is heated (e.g., by steam), while the sides are exposed to the surroundings, allowing heat loss through radiation and convection.
- Heat flows along the bar from the hot end towards the cold end.
- After some time, a **steady state** is reached where the temperature at each point along the bar becomes constant.
- In the steady state, the heat flowing into any cross-section of the bar must equal the heat flowing out of that cross-section plus the heat lost from the surface area beyond that cross-section.
- Consider a cross-section at distance x from the hot end. The heat flowing across this section per second is given by Fourier's law: $Q₁ = -kA (dT/dx)$, where A is the cross-sectional area and dT/dx is the temperature gradient at x.
- The heat lost per second from the surface of the bar beyond x (from x to the end of the bar) is given by $Q₂ = ∫ₓᴸ eP(T-T₀) dx'$, where e is emissivity, P is perimeter, T is temperature at x', T₀ is ambient temperature, and L is the length where T ≈ T₀.
- In steady state, $Q₁ = Q₂$. Therefore: $-kA (dT/dx) = ∫ₓᴸ eP(T-T₀) dx'$.
- Transient (Cooling) State:
- The same bar is uniformly heated to the temperature of the steam and then allowed to cool in the same surroundings.
- The rate at which a small element dx at distance x cools depends on the heat lost from its surface.
- Rate of heat loss from element dx = $dQ/dt = mc (dT'/dt)$, where m is mass of element, c is specific heat, and dT'/dt is the rate of cooling at temperature T (which was the steady state temperature at x).
- Mass of element $m = (A dx) ρ$, where ρ is density.
- Rate of heat loss also = Surface heat loss = $eP(T-T₀) dx$.
- Equating these: $(A dx) ρ c (dT'/dt) = eP(T-T₀) dx$.
- $eP(T-T₀) = A ρ c (dT'/dt)$.
- Combining States:
- Substitute the expression for $eP(T-T₀)$ from the transient state into the steady state equation:
- $-kA (dT/dx) = ∫ₓᴸ A ρ c (dT'/dt) dx'$.
- Cancel A: $-k (dT/dx) = ∫ₓᴸ ρ c (dT'/dt) dx'$.
- Rearranging for k: $k = - \frac{ρ c ∫ₓᴸ (dT'/dt) dx'}{(dT/dx)}$.
- The integral represents the area under the (dT'/dt) vs. x' graph from x to L. The term (dT/dx) is the slope of the steady-state temperature vs. distance graph at x.
Experimental Procedure:**
- Setup: Take a long uniform metal bar with holes drilled along its length for thermometers. Coat the surface uniformly (e.g., with lampblack) for consistent emissivity.
- Steady State Measurement:
- Insert thermometers into the holes.
- Heat one end of the bar using a steam chamber. Shield the rest of the bar from direct steam.
- Allow the system to reach steady state (thermometer readings become constant). Record the steady-state temperatures (T) at different distances (x) from the hot end.
- Plot a graph of T vs. x. Determine the temperature gradient (dT/dx) at various points by finding the slope of the tangents to this curve.
- Transient (Cooling) Measurement:
- Take an identical bar (or the same bar after steady state) and heat it uniformly to the steam temperature.
- Place it in the same environment and allow it to cool.
- Record the temperature (T') as it falls over time (t) for different positions (x) along the bar (or use thermocouples placed at the same positions as the steady state thermometers).
- Plot cooling curves (T' vs. t) for each position x.
- Determine the rate of cooling (dT'/dt) at different temperatures T by finding the slopes of tangents to the cooling curves at those temperatures. (Note: The temperature T here corresponds to the steady-state temperature recorded earlier at position x).
- Plot a graph of the rate of cooling (dT'/dt) against the distance (x) for the temperatures observed in the steady state.
- Calculation:
- For a chosen point x, find the slope (dT/dx) from the steady-state graph.
- Calculate the area under the (dT'/dt) vs. x graph from the chosen point x to the end of the bar (where T ≈ T₀). This gives the integral $∫ₓᴸ (dT'/dt) dx'$.
- Measure the density (ρ) and determine the specific heat capacity (c) of the bar material.
- Calculate the thermal conductivity k using the formula: $k = - \frac{ρ c × (\text{Area under cooling rate graph from x to L})}{(\text{Slope of steady state graph at x})}$. (The negative sign handles the negative slope dT/dx).
This method accounts for heat loss along the bar but requires careful measurements and graphical analysis.
5. Describe the process of non destructive testing of material, using ultrasonic waves by pulse echo techniques( A-scan, B -scan and TM- Mode Scanning)
Non-Destructive Testing (NDT):** NDT techniques evaluate the properties or integrity of a material, component, or system without causing damage. Ultrasonic NDT uses high-frequency sound waves to detect internal flaws, measure thickness, or characterize materials.
Pulse-Echo Technique:**
This is the most common ultrasonic NDT method. It works on the principle of measuring the time taken for a short burst (pulse) of ultrasound to travel through the material, reflect off a boundary (like the back wall or a defect), and return to a receiver.
- A transducer (probe) generates a pulse of ultrasound and sends it into the test material.
- The transducer also acts as a receiver, detecting echoes reflected back.
- The time of flight (ToF) of the echo is measured. Knowing the speed of sound in the material (v), the distance (d) to the reflector can be calculated: $d = (v \times ToF) / 2$ (divided by 2 for the round trip).
- Internal flaws (cracks, voids, inclusions) act as reflectors because they represent a boundary with different acoustic impedance. They produce echoes that arrive earlier than the echo from the back wall.
- The amplitude of the echo provides information about the size and nature of the reflector.
Different ways of displaying the received echo information lead to different scan types:
1. A-Scan (Amplitude Scan):**
- Display: Presents the received echo information as a graph of **amplitude vs. time of flight** (or distance) for a single point on the material.
- How it works: The transducer is placed at one location. The display shows spikes (peaks) corresponding to echoes. The horizontal position of a spike indicates the depth of the reflector, and the height of the spike indicates the strength (amplitude) of the echo.
- Information: Provides precise depth information about defects at a specific location. Primarily used for flaw detection and thickness measurement at discrete points.
- Representation: One-dimensional view (depth).
2. B-Scan (Brightness Scan):**
- Display: Creates a **two-dimensional cross-sectional view** of the material. It plots the depth of reflectors (time of flight) along one axis versus the position of the transducer as it scans along a line on the surface.
- How it works: The transducer is moved along a line on the material's surface. A-scan data is collected at multiple points along this line. The display shows dots or lines where echoes occur. The brightness of the dot can represent the amplitude of the echo.
- Information: Provides a visual profile or cross-section, showing the depth and length of defects along the scan line. Useful for visualizing the shape and extent of flaws.
- Representation: Two-dimensional view (scan direction vs. depth).
3. TM-Scan / C-Scan (Time-Motion / Constant-Depth Scan):**
(Note: The question asks for TM-Mode, which is often related to Time-Motion studies in medical ultrasound. However, in NDT context, C-Scan is the more common planar view. Assuming C-Scan is intended based on common NDT scans.)
- Display (C-Scan): Presents a **two-dimensional planar view (top-down view)** of the material, similar to an X-ray image.
- How it works (C-Scan): The transducer scans across the surface in a raster pattern (like mowing a lawn). The display plots the echo amplitude or time-of-flight information corresponding to a specific depth range (gate) onto an x-y grid representing the surface area scanned. Color or grayscale intensity typically represents the echo amplitude or depth.
- Information (C-Scan): Shows the location and size of defects projected onto the surface plane. Excellent for mapping the distribution of flaws over an area, assessing corrosion, or checking bond integrity.
- Representation (C-Scan): Two-dimensional view (X-Y surface coordinates).
- TM-Scan (Time-Motion): More common in medical ultrasound, it displays the movement of structures over time along a single scan line. One axis represents depth, and the other represents time. Less common in standard materials NDT compared to C-scan.