Physics Question Paper
SECTIOn – A
Q.01 What is the Young’s modulus of a perfectly rigid body?
Young’s modulus of a perfectly rigid body is infinite. A rigid body shows no deformation when stress is applied, meaning the strain is zero. Since Young’s modulus is defined as stress/strain, a zero strain value results in an infinite modulus.
Q.02 Water is more elastic than air. Why?
Elasticity is related to how much a substance resists deformation under stress. Liquids, like water, are much more difficult to compress than gases, like air. This resistance to compression is what makes them more elastic. In simpler terms, it takes a lot more pressure to change the volume of water compared to air.
OR (Alternative for Q.02):
Why bridges are declared unsafe after a long use?
Bridges are declared unsafe after long use due to fatigue and wear. Repeated stress and strain over time can weaken the materials, leading to cracks, corrosion, and other forms of damage. This can reduce the bridge’s load-bearing capacity and make it susceptible to failure.
OR (Alternative for Q.02):
Why does an air bubble in a liquid rise up?
An air bubble in a liquid rises due to buoyancy. The liquid exerts an upward force on the bubble equal to the weight of the liquid displaced by the bubble. Since air is much less dense than the liquid, this buoyant force is greater than the weight of the bubble, causing it to rise.
OR (Alternative for Q.02):
Why doesn’t Mercury wet the glass?
Mercury doesn’t wet glass because the cohesive forces (attraction between mercury atoms) are much stronger than the adhesive forces (attraction between mercury and glass). This means mercury atoms prefer to stick together rather than to the glass surface, resulting in a high contact angle and a non-wetting behavior.
Q.04 What is Stoke’s law?
Stoke’s law describes the viscous drag force on a spherical object moving through a fluid. The force is given by:
F = 6πηrv
Where:
- F is the viscous drag force
- η is the dynamic viscosity of the fluid
- r is the radius of the sphere
- v is the velocity of the sphere relative to the fluid
Q.05 Water pipes may burst in cold countries when the temperature goes down below zero degrees Celsius. Why?
Water expands when it freezes. When the temperature drops below zero, the water in the pipes freezes and expands, exerting tremendous pressure on the pipe walls. This pressure can exceed the pipe’s strength, causing it to burst.
Q.06 What is the change in the internal energy of a gas which is compressed adiabatically?
In an adiabatic process, there is no heat exchange with the surroundings (Q=0). The first law of thermodynamics states:
ΔU = Q – W
Where:
- ΔU is the change in internal energy
- Q is the heat added to the system
- W is the work done by the system
Since Q=0 in an adiabatic process:
ΔU = -W
When a gas is compressed, work is done on the gas, so W is negative. Therefore, ΔU is positive, meaning the internal energy of the gas increases.
Q.07 Name the thermodynamic variable defined by the zeroth law of thermodynamics.
The thermodynamic variable defined by the zeroth law of thermodynamics is temperature.
OR (Alternative for Q.07):
What is the value of specific heat of an ideal gas in an isothermal process?
In an isothermal process, the temperature remains constant. Specific heat is defined as the heat required to raise the temperature of a unit mass of a substance by one degree. Since the temperature doesn’t change in an isothermal process, the specific heat is infinite.
Q.08 State the law of equi-partition of energy.
The law of equipartition of energy states that for a system in thermal equilibrium, the total energy is distributed equally among all the degrees of freedom. Each degree of freedom contributes (1/2)kT to the average energy, where k is the Boltzmann constant and T is the absolute temperature.
Q.09 What is the basic condition for the motion of a particle to be SHM?
The basic condition for a motion to be simple harmonic motion (SHM) is that the restoring force must be directly proportional to the displacement from the equilibrium position and act in the opposite direction. Mathematically:
F = -kx
Where:
- F is the restoring force
- k is the spring constant (a positive constant)
- x is the displacement from equilibrium
Q.10 What is the total energy of a simple harmonic oscillator?
The total energy of a simple harmonic oscillator is constant and is proportional to the square of the amplitude. It can be expressed as:
E = (1/2)kA²
Where:
- E is the total energy
- k is the spring constant
- A is the amplitude of oscillation
Q.11 Assertion(A): A brass disc is just fitted in a hole in a steel plate. The system must be cooled to loosen the disc from the hole.
Reason(R):
Answer: (a) Both A and R are true and R is the correct explanation of A
Explanation: Brass expands more than steel for the same temperature change. Cooling the system will cause both the disc and the hole to shrink, but the brass disc will shrink more. This will create a gap, loosening the disc from the hole.
Q.12 Assertion(A): Reversible systems are difficult to find in the real world.
Reason (R) Most processes are dissipative in nature.
Answer: (a) Both A and R are true and R is the correct explanation of A
Explanation: Reversible processes are idealized concepts where the system can be returned to its initial state without any change in the surroundings. Real-world processes involve friction, heat loss, and other dissipative forces that make them irreversible.
Q.13 Assertion (A): Air pressure in a car tyre increases during driving.
Reason (R): Absolute zero-degree temperature is not zero energy temperature.
Answer: (b) Both A and R are true and R is NOT the correct explanation of A
Explanation: The air pressure in a car tire increases during driving because the tire temperature increases due to friction and flexing. This increases the kinetic energy of the air molecules inside, leading to more frequent and forceful collisions with the tire walls, hence increased pressure. While it is true that absolute zero temperature is not zero energy temperature (due to zero-point energy), this is not the reason for the pressure increase in the tire.
Q.14 Assertion(A): All oscillatory motions are necessarily periodic motion but all periodic motions are not oscillatory.
Reason (R) Simple pendulum is an example of oscillatory motion.
Answer: (a) Both A and R are true and R is the correct explanation of A
Explanation: Oscillatory motion means the motion is back and forth about a fixed point. Periodic motion means the motion repeats at regular intervals. All oscillatory motions repeat, making them periodic. However, a periodic motion may not be oscillatory. For example, a body moving in a uniform circular motion is periodic but not oscillatory. A simple pendulum repeats its motion (periodic) and moves back and forth (oscillatory), fitting both definitions.
SECTION – B
Q.15 Case study: Comparison of elasticity of wires (Answer any four)
Three wires A, B, and C of the same length and cross-sectional area but different materials are available. They are each stretched by applying the same deformative force to their ends. Wire A stretches the least and returns back to its original length when the deformative force is removed. Wire B stretches more than wire A and also returns back to its original length when the deformative force is removed. Wire C stretches the most and remains deformed, even after the deformative force is removed.
(i) Name the modulus of elasticity involved.
(a) Modulus of rigidity (b) Bulk modulus (c) Young’s modulus (d) none
Answer: (c) Young’s modulus
Explanation: Young’s modulus describes the elasticity of a solid in response to tensile or compressive stress, which is the scenario described (stretching).
(ii) Which of them has the greater modulus of elasticity?
(a) A (b) B (c) C (d) none
Answer: (a) A
Explanation: A stretches the least, meaning it requires more force to deform. A higher resistance to deformation indicates a greater modulus of elasticity.
(iii) Which of them has the least modulus of elasticity?
(a) A (b) B (c) C (d) none
Answer: (c) C
Explanation:C stretches the most, meaning it deforms the most easily under the same force. This indicates the lowest modulus of elasticity.
(iv) What is the SI unit of modulus of elasticity?
(a) Nm⁻² (b) Nm⁻³ (c) Nm (d) Nm⁻¹
Answer: (a) Nm⁻²
Explanation:Modulus of elasticity is stress/strain. Stress is force per unit area (Nm⁻²) and strain is dimensionless.
(v) Which of them is suitable for making thin wires?
(a) A (b) B (c) A & B (d) C
Answer: (c) A & B
Explanation:A and B both return to their original length, demonstrating elastic behavior. C remains deformed, indicating plastic behavior. Elastic materials are suitable for wires because they can withstand stress without permanent deformation. A is likely the best choice due to its higher elasticity.
Q.16 Case study: Normal mode of vibrations in an open organ pipe (Answer any four)
An organ pipe is the simplest musical instrument in which sound is produced by setting an air column into vibration just like any wind instrument like flute, shehnai, etc. A sound wave travels down the pipe and gets reflected at its open or closed end, producing stationary waves. If the frequency of these waves is equal to the frequency of the edge tone, resonance occurs and a loud sound is produced. If both
(i) Name the type of waves produced in an organ pipe.
(a) Transverse wave (b) Longitudinal wave (c) Electromagnetic wave (d) Matter wave
Answer: (b) Longitudinal wave
Explanation:Sound waves are longitudinal waves, meaning the particles of the medium vibrate parallel to the direction of wave propagation.
(ii) The velocity of sound in an air column depends upon:
(a) density (b) 1/density (c) 1/√density (d) √density
Answer: (a) density
Explanation:
The speed of sound in a gas is related to the square root of the bulk modulus and inversely related to the square root of the density.
(iii) What is the frequency in the second mode of vibration?
(a) v/2 (b) v (c) 2v (d) 3v
Answer: (c) 2v
Explanation:
For an open organ pipe, the frequencies of the harmonics are integer multiples of the fundamental frequency (v, 2v, 3v, etc.).
(iv) In the case of a stationary wave, the separation between successive nodes and antinodes is:
(a) λ (b) λ/2 (c) 3λ/4 (d) λ/4
Answer: (d) λ/4
Explanation:
A node is a point of zero displacement, and an antinode is a point of maximum displacement. They are separated by one-quarter of a wavelength.
(v) From the given figure, find the frequency of vibrations of the air column in the open organ pipe.
[You need to provide the figure to answer this question. It would likely show a standing wave pattern in the pipe. The frequency will depend on the number of loops (or nodes) in the standing wave pattern.]
However, based on the general information provided, the possible frequencies are integer multiples of the fundamental frequency (v/2L), such as v/2L, 2v/2L (or v/L), 3v/2L, and so on. Without the figure, I cannot give a specific answer.
SECTION – C
Q.17 Draw a stress-strain curve for a metallic wire. Depict elastic limit and permanent set.
Stress (σ)
^
|
| A (Proportionality Limit)
| /
| / B (Elastic Limit)
| /
|/
| C (Yield Point)
| /
| /
| /
| /
| / D (Ultimate Tensile Strength)
| /
| /
| /
|/_________________________ Strain (ε)
O E (Breaking Point)
- OA: Region where stress is proportional to strain (Hooke’s Law).
- A (Proportionality Limit): Point up to which stress is directly proportional to strain.
- B (Elastic Limit): Point up to which the material returns to its original length when stress is removed.
- BC: Region where the material undergoes plastic deformation.
- C (Yield Point): Point at which the material starts to deform plastically.
- CD: Region of strain hardening.
- D (Ultimate Tensile Strength): Maximum stress the material can withstand.
- DE: Necking region where the material begins to thin out.
- E (Breaking Point): Point at which the material fractures.
Permanent Set: If the wire is stressed beyond the elastic limit (B) and the stress is removed, it will not return to its original length. The residual strain is called the permanent set. This would be represented by the strain at point where line comes back to X-axis if we unload the wire after point C.
Q.18 To lift an automobile of 2000 kg, a hydraulic pump with a large piston of 900 cm² area is employed. Calculate the force that must be applied to the pump at a small piston of area 10 cm².
Using Pascal’s principle, the pressure applied to an enclosed fluid is transmitted equally and undiminished to all parts of the fluid and the walls of its container.
Pressure on large piston = Pressure on small piston
F₁/A₁ = F₂/A₂
Where:
- F₁ = Force on large piston = Weight of automobile = 2000 kg * 9.8 m/s² = 19600 N
- A₁ = Area of large piston = 900 cm²
- F₂ = Force on small piston (what we need to find)
- A₂ = Area of small piston = 10 cm²
F₂ = (F₁ * A₂) / A₁ = (19600 N * 10 cm²) / 900 cm² ≈ 217.78 N
Q.19 27 identical drops of water are falling down vertically in air, each with a terminal velocity of 0.2 m/s. If they are combined to form a single drop, what will be its terminal velocity?
The volume of the large drop is equal to the sum of the volumes of the 27 small drops.
Volume of 1 large drop = 27 * Volume of 1 small drop
(4/3)πR³ = 27 * (4/3)πr³
R³ = 27r³
R = 3r
Terminal velocity (v) is proportional to the square of the radius (r²) of the drop.
v₁/v₂ = r₁²/r₂²
v₁ = 0.2 m/s (terminal velocity of small drops) v₂ = terminal velocity of large drop
0.2/v₂ = r²/(3r)² = 1/9
v₂ = 9 * 0.2 = 1.8 m/s
Q.20 State Wien’s displacement law. What is the importance of Wien’s displacement law?
Wien’s displacement law states that the wavelength at which the black body radiation spectrum is most intense (λ_max) is inversely proportional to the absolute temperature (T) of the black body.
λ_max * T = b
Where b is Wien’s displacement constant (approximately 2.898 × 10⁻³ m·K).
Importance:
- Determining Temperature of Stars: By measuring the wavelength of maximum intensity of light from a star, we can estimate its temperature.
- Understanding Black Body Radiation: It helps explain the shift in the black body radiation spectrum towards shorter wavelengths (higher frequencies) as temperature increases.
- Thermal Imaging: It is the principle behind thermal imaging, which detects infrared radiation emitted by objects to determine their temperature.
OR (Alternative for Q.20):
Define the coefficient of thermal conductivity; derive its SI unit.
The coefficient of thermal conductivity (k) of a material is a measure of its ability to conduct heat. It is defined as the amount of heat that flows per unit time through a unit area of the material with a unit temperature gradient.
k = (Q * d) / (A * ΔT)
Where:
- Q = Heat transferred
- d = Thickness of the material
- A = Area of the material
- ΔT = Temperature difference across the material
SI Unit:
The SI unit of thermal conductivity is watt per meter-kelvin (W/m·K).
Q.21 In the given figure, an ideal gas changes its state from A to C by path ABC. Calculate the total work done on the gas.
(You need to provide the figure to calculate the work done. The work done is given by the area under the P-V curve. If the figure is a P-V diagram with A and C at the same volume, then work done is zero. If not, you’ll need to calculate the area under the curve ABC.)
Q.22 (a) State the first law of thermodynamics; express it mathematically also.
The first law of thermodynamics states that energy can neither be created nor destroyed, but can only be transformed from one form to another.
Mathematically:
ΔU = Q – W
Where:
- ΔU = Change in internal energy of the system
- Q = Heat added to the system
- W = Work done by the system
(b) A gaseous system absorbs 110 J of heat; its internal energy increases by 40 J. What is the amount of work done by the system?
Q = 110 J ΔU = 40 J
Using the first law of thermodynamics:
ΔU = Q – W
W = Q – ΔU = 110 J – 40 J = 70 J
Q.23 A simple harmonic motion is represented by x = 10 sin(20t + 0.5) metre. Find the maximum velocity and maximum acceleration.
The general equation for SHM is:
x = A sin(ωt + φ)
Where:
- A = Amplitude = 10 m
- ω = Angular frequency = 20 rad/s
- φ = Phase constant = 0.5 rad
Maximum velocity (v_max) = Aω = 10 m * 20 rad/s = 200 m/s
Maximum acceleration (a_max) = Aω² = 10 m * (20 rad/s)² = 4000 m/s²
Q.24 What are resonant oscillations? Give an example.
Resonant oscillations occur when a system is driven by an external force at its natural frequency. At resonance, the amplitude of oscillations becomes maximum.
Example: A swing pushed at its natural frequency will swing higher and higher.
OR (Alternative for Q.24):
With the help of an example, explain forced oscillations.
Forced oscillations occur when a system is driven by an external force at a frequency different from its natural frequency. The system will oscillate at the frequency of the driving force, but the amplitude of oscillations may not be very large unless the driving frequency is close to the natural frequency.
Example: A bridge vibrating due to the rhythmic footsteps of soldiers marching across it. If the marching frequency matches the bridge’s natural frequency, it could lead to resonance and potentially catastrophic results.
Q.25 Explain the phenomenon of beats. What is the essential condition for the formation of beats?
Beats occur when two sound waves of slightly different frequencies interfere with each other. The resulting sound wave has a frequency that is the average of the two original frequencies, and its amplitude varies periodically, creating a series of loud and soft sounds called beats.
Essential condition for the formation of beats:
SECTION – D
Q.26 (a) Define surface tension. Write its formula and dimensions.
Surface tension is the tendency of liquid surfaces to shrink into the minimum surface area possible. It is defined as the force acting per unit length on the surface of a liquid, tending to pull the surface together.
Formula:
Surface Tension (T) = Force (F) / Length (L)
Dimensions:
[ML⁻²] (Mass × Length⁻² × Time⁻²)
(b) What happens to the surface tension of water when a detergent is added to it?
The surface tension of water decreases when a detergent is added to it. Detergents are surfactants, which reduce the cohesive forces between water molecules, thus lowering the surface tension.
OR (Alternative for Q.26):
Explain the following:
(a) Coefficient of viscosity:
The coefficient of viscosity (η) is a measure of a fluid’s resistance to flow. It describes the internal friction within a fluid. A higher viscosity means a greater resistance to flow.
(b) Streamline flow:
Streamline (or laminar) flow is a type of fluid flow in which the fluid moves in smooth, parallel layers, with no disruption between the layers. The velocity at any given point in the fluid remains constant with time.
(c) Turbulent flow:
Turbulent flow is a chaotic and irregular flow regime characterized by eddies, swirls, and unpredictable variations in velocity and pressure. It occurs at high Reynolds numbers, where inertial forces dominate over viscous forces.
Q.27 What do you understand by the following? Write their SI units:
(a) Specific heat:
Specific heat (c) is the amount of heat required to raise the temperature of 1 kg of a substance by 1 Kelvin (or 1 degree Celsius).
SI unit: J kg⁻¹ K⁻¹ (joules per kilogram per kelvin)
(b) Latent heat of fusion:
Latent heat of fusion (Lf) is the amount of heat required to change 1 kg of a substance from solid to liquid at its melting point, without any change in temperature.
SI unit: J kg⁻¹ (joules per kilogram)
OR (Alternative for Q.27):
(a) What is a black body?
A black body is an idealized object that absorbs all electromagnetic radiation incident on it. It also emits radiation at all wavelengths, depending on its temperature. It is a perfect absorber and a perfect emitter of radiation.
(b) State Stefan-Boltzmann law of black body radiation:
The Stefan-Boltzmann law states that the total energy radiated per unit area per unit time by a black body is directly proportional to the fourth power of its absolute temperature.
E = σT⁴
Where:
- E is the energy radiated per unit area per unit time
- σ is the Stefan-Boltzmann constant (approximately 5.67 × 10⁻⁸ W m⁻² K⁻⁴)
- T is the absolute temperature
Q.28 On the basis of the kinetic theory of gases, derive an expression for the pressure exerted by an ideal gas.
Derivation:
Consider an ideal gas enclosed in a cubic container of side length ‘l’. The gas molecules are in random motion, colliding with the walls of the container.
-
Momentum change: The change in momentum of a molecule when it collides elastically with the wall is 2mvₓ (assuming the molecule is moving along the x-axis with velocity vₓ).
-
Force: The force exerted by the molecule on the wall is the rate of change of momentum, which is 2mvₓ/Δt (where Δt is the time between collisions).
-
Number of collisions: The number of collisions per unit time is proportional to the velocity vₓ and the number density of molecules.
-
Pressure: Pressure is force per unit area. Combining the above factors, we get:
P = (1/3)ρv²
Where:
- P is the pressure
- ρ is the density of the gas
- v² is the mean square velocity of the gas molecules
Q.29 Derive Newton’s formula for the speed of sound in air. What is Laplace’s correction to it?
Newton’s formula:
Newton assumed that sound propagation in air is an isothermal process. Based on this assumption, he derived the formula:
v = √(B/ρ)
Where:
- v is the speed of sound
- B is the bulk modulus of air
- ρ is the density of air
Laplace’s correction:
Laplace argued that sound propagation is actually an adiabatic process, not isothermal. He corrected Newton’s formula by introducing the adiabatic index (γ):
v = √(γB/ρ)
Where γ is the ratio of specific heat at constant pressure to specific heat at constant volume.
Q.30 How are stationary waves produced? Discuss the formation of stationary waves in a string fixed at both ends and show that the first four harmonic frequencies are in the ratio 1:2:3:4.
Production of stationary waves:
Stationary waves are produced when two identical waves traveling in opposite directions interfere with each other. This interference results in points of maximum displacement (antinodes) and points of zero displacement (nodes).
Formation in a string fixed at both ends:
When a string fixed at both ends is plucked or vibrated, waves travel along the string and reflect back from the fixed ends. The superposition of the incident and reflected waves creates stationary waves. The fixed ends must be nodes.
Harmonics:
The possible frequencies of vibration are called harmonics. For a string fixed at both ends, the wavelengths (λ) and frequencies (f) of the first four harmonics are:
- Fundamental (n=1): λ₁ = 2L, f₁ = v/2L
- First overtone (n=2): λ₂ = L, f₂ = v/L = 2f₁
- Second overtone (n=3): λ₃ = (2/3)L, f₃ = 3v/2L = 3f₁
- Third overtone (n=4): λ₄ = (1/2)L, f₄ = 2v/L = 4f₁
Where:
- L is the length of the string
- v is the speed of the wave
Therefore, the frequencies of the first four harmonics are in the ratio 1:2:3:4.
SECTION – E
Q.31 (a) State and prove Bernoulli’s theorem. Draw necessary diagram.
Bernoulli’s Theorem:
Bernoulli’s theorem states that for an ideal, incompressible, non-viscous fluid in steady flow, the sum of the pressure energy per unit volume, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.
Mathematical Expression:
P + (1/2)ρv² + ρgh = constant
Where:
- P is the pressure
- ρ is the density of the fluid
- v is the velocity of the fluid
- g is the acceleration due
to gravity
Proof:
The proof is based on the principle of conservation of energy. Consider a fluid flowing through a tube of varying cross-section. The work done by the pressure forces is equal to the change in kinetic and potential energies of the fluid. By equating the work done to the energy change, we arrive at Bernoulli’s equation.
(Diagram: A diagram should show a tube with varying cross-sectional area, with labels for pressure (P), velocity (v), and height (h) at different points along the tube. Streamlines should be drawn to indicate the direction of flow.)
(b) Why are the roofs of huts blown off in a storm?
During a storm, high-speed winds create a region of low pressure above the roof of the hut, according to Bernoulli’s principle. The pressure inside the hut is higher. This pressure difference creates an upward force on the roof, which can be strong enough to lift it off.
OR (Alternative for Q.31):
(a) What do you understand by capillarity? Give an example.
Capillarity is the phenomenon of a liquid rising or falling in a narrow tube (capillary tube) due to the combined effects of surface tension, adhesion, and cohesion.
Example: Water rising in a thin glass tube, or the absorption of water by blotting paper.
(b) Derive a formula for the rise (h) of a liquid in a capillary tube of uniform diameter. How does it vary with the radius of the tube?
Derivation:
Consider a capillary tube of radius ‘r’ dipped in a liquid of density ‘ρ’ and surface tension ‘T’. The liquid rises to a height ‘h’ in the tube.
The upward force due to surface tension is balanced by the weight of the liquid column.
Upward force = 2πrTcosθ (where θ is the contact angle)
Weight of liquid column = πr²hρg
Equating the two forces:
2πrTcosθ = πr²hρg
h = (2Tcosθ)/(ρgr)
Variation with radius:
The height ‘h’ is inversely proportional to the radius ‘r’ of the tube. This means that the narrower the tube, the higher the liquid will rise.
Q.32 What is an isothermal process? State the conditions under which such a process takes place, and hence derive an expression for work done by one mole of an ideal gas during isothermal expansion.
Isothermal Process:
An isothermal process is a thermodynamic process in which the temperature of the system remains constant.
Conditions:
- The system must be in contact with a heat reservoir to exchange heat and maintain constant temperature.
- The process must be carried out slowly to allow for heat exchange and maintain thermal equilibrium.
Derivation for work done:
For an isothermal process, PV = nRT (ideal gas law).
Work done (W) = ∫PdV (integral from V₁ to V₂)
W = nRT ∫(dV/V) (integral from V₁ to V₂)
W = nRT ln(V₂/V₁)
For one mole of gas (n=1):
W = RT ln(V₂/V₁)
OR (Alternative for Q.32):
What is an adiabatic process? State the conditions under which such a process takes place, and hence derive an expression for work done by one mole of an ideal gas during adiabatic expansion.
Adiabatic Process:
An adiabatic process is a thermodynamic process in which no heat is exchanged between the system and its surroundings (Q=0).
Conditions:
- The process must be carried out quickly so that there is no time for heat exchange.
- The system must be insulated from its surroundings.
Derivation for work done:
For an adiabatic process, PV^γ = constant.
Work done (W) = ∫PdV (integral from V₁ to V₂)
Using the adiabatic relation, we can express P in terms of V and integrate:
W = (P₂V₂ – P₁V₁)/(1-γ)
For one mole of an ideal gas:
W = R(T₂ – T₁)/(1-γ)
Q.33 (a) What is a simple pendulum? Derive an expression for its time period.
Simple Pendulum:
A simple pendulum is an idealized model consisting of a point mass (bob) suspended from a massless, inextensible string.
Derivation for time period:
The time period (T) of a simple pendulum is given by:
T = 2π√(L/g)
Where:
- L is the length of the pendulum
- g is the acceleration due to gravity
The derivation involves considering the restoring force due to gravity and using Newton’s second law to set up a differential equation for the angular displacement. For small angles of oscillation, the equation simplifies to simple harmonic motion, leading to the above expression for the time period.
(b) What is the effect on the frequency of oscillation of a simple pendulum mounted in a cabin that is falling freely under gravity? Explain.
When a simple pendulum is mounted in a cabin falling freely under gravity, the effective acceleration due to gravity becomes zero (g_eff = g – a = 0, where a is the acceleration of the cabin, which is equal to g).
Since T = 2π√(L/g_eff), as g_eff approaches zero, the time period (T) approaches infinity, and the frequency (f = 1/T) approaches zero. This means the pendulum will not oscillate.
OR (Alternative for Q.33):
(a) Show that simple harmonic motion may be regarded as the projection of uniform circular motion along the diameter of the circle. Hence, derive an expression for displacement and velocity of a particle executing SHM.
Explanation:
Imagine a particle moving in a uniform circular motion with constant angular velocity (ω). If we observe this motion along a diameter of the circle, the projection of the particle’s position on this diameter will execute simple harmonic motion.
Derivation:
Let the radius of the circle be ‘A’. The angular displacement at time ‘t’ is θ = ωt.
The projection of the particle’s position on the x-axis (along the diameter) is given by:
x = Acos(ωt) (displacement)
The velocity (v) along the x-axis is the derivative of displacement with respect to time:
v = -Aωsin(ωt)
(b) What is the phase relation between displacement and velocity? Show graphically.
The velocity leads the displacement by π/2 (90 degrees) in SHM.
Graphical Representation:
(A graph should be drawn with displacement (x) and velocity (v) plotted against time (t). The displacement curve is a cosine wave, and the velocity curve is a negative sine wave, shifted by π/2 to the left. The velocity reaches its maximum when the displacement is zero, and vice versa.)