Maharashtra Board 2026 Class 12 Physics(S) 54 Question Paper with Solution - Available

Concept:
The shape of a liquid meniscus depends on the angle of contact (\(\theta\)) between the liquid and the container wall.



Step 1: Angle of contact definition
Angle of contact is the angle between the tangent to the liquid surface and the solid surface, measured inside the liquid.



Step 2: Cases based on angle


\(\theta < 90^\circ\) (acute angle):
Adhesive forces \(>\) cohesive forces → liquid wets the surface → concave meniscus.
Example: Water in glass.

\(\theta > 90^\circ\) (obtuse angle):
Cohesive forces \(>\) adhesive forces → liquid does not wet the surface → convex meniscus.
Example: Mercury in glass.




Step 3: Given condition
Since the angle of contact is obtuse (\(> 90^\circ\)):
\[ Meniscus shape = Convex \]



Final Answer:
\[ \boxed{Convex meniscus} \] Quick Tip: Angle of contact rule: Acute angle → Concave meniscus Obtuse angle → Convex meniscus Mercury in glass is the classic convex example.

Concept:
Magnetization describes how strongly a material becomes magnetized when placed in a magnetic field. It represents the alignment of magnetic dipoles within a substance.



Step 1: Definition
Magnetization (\(M\)) is defined as:
\[ M = \frac{Magnetic dipole moment}{Volume} \]

It indicates the net magnetic moment per unit volume of the material.



Step 2: Physical meaning
When a material is placed in an external magnetic field:
- Magnetic domains align
- Net magnetic dipole moment develops
This gives rise to magnetization.



Step 3: SI unit
Magnetic dipole moment has unit:
\[ A·m^2 \]

Dividing by volume (\(m^3\)):
\[ \frac{A·m^2}{m^3} = A/m \]



Final Answer:
\[ \boxed{Magnetization (M) = magnetic moment per unit volume, unit = A/m} \] Quick Tip: Magnetization = Magnetic moment density. Always remember unit: Ampere per metre (A/m).

Concept:
The First Law of Thermodynamics is based on the principle of conservation of energy applied to thermodynamic systems.



Step 1: Statement
It states that the total energy of an isolated system remains constant. Energy may change form (heat, work, internal energy) but cannot be created or destroyed.



Step 2: Mathematical expression
If heat \(Q\) is supplied to a system and work \(W\) is done by the system, then:
\[ Q = \Delta U + W \]

where: \(\Delta U\) = change in internal energy.



Step 3: Interpretation
- Part of heat increases internal energy.
- Remaining heat is used in doing external work.



Final Form:
\[ \boxed{Q = \Delta U + W} \] Quick Tip: First Law = Energy conservation in thermodynamics. Heat supplied = Increase in internal energy + Work done.

Concept:
Angular momentum is the rotational analogue of linear momentum. It depends on the moment of inertia and angular velocity of a body.



Step 1: Definition of angular momentum
For a particle:
\[ \vec{L} = \vec{r} \times \vec{p} \]

For a rigid body rotating about a fixed axis:
\[ L = I\omega \]

where \(I\) = moment of inertia, \(\omega\) = angular velocity.



Step 2: Relation between torque and angular momentum
Torque is defined as:
\[ \vec{\tau} = \frac{d\vec{L}}{dt} \]

This is the rotational form of Newton’s second law.



Step 3: Condition for conservation
If the net external torque on the system is zero:
\[ \vec{\tau}_{ext} = 0 \]

Then:
\[ \frac{d\vec{L}}{dt} = 0 \]



Step 4: Integrating
\[ \frac{d\vec{L}}{dt} = 0 \Rightarrow \vec{L} = constant \]

Thus angular momentum remains conserved.



Step 5: Alternative form
For a rotating rigid body:
\[ I_1 \omega_1 = I_2 \omega_2 \]

This shows that if moment of inertia changes, angular velocity adjusts to keep angular momentum constant.



Conclusion:
In absence of external torque, angular momentum remains conserved in both magnitude and direction. Quick Tip: No external torque → Angular momentum conserved. Example: Figure skater spins faster when pulling arms inward.

Concept:
Surface tension is explained by intermolecular forces acting between liquid molecules, particularly cohesive forces.



Step 1: Molecules inside the liquid
A molecule deep inside a liquid is surrounded by other molecules on all sides.
Cohesive forces act equally in all directions, so the net force is zero.



Step 2: Molecules at the surface
A surface molecule has:
- Neighbouring molecules below and sideways
- Very few molecules above (air)

Thus, cohesive forces are unbalanced and act inward.



Step 3: Result of inward force
Due to this inward pull:
- Surface molecules are drawn closer together
- Surface area tends to decrease

This gives rise to surface tension.



Step 4: Surface behaves like a stretched membrane
The liquid surface acts like an elastic film trying to contract, similar to a stretched rubber sheet.



Step 5: Molecular energy explanation
Surface molecules possess higher potential energy than interior molecules.
To minimize energy, the liquid reduces surface area, producing surface tension effects.



Conclusion:
Surface tension is caused by unbalanced cohesive molecular forces at the liquid surface, making the surface contract and behave like a stretched membrane. Quick Tip: Inside liquid → Balanced forces Surface molecules → Inward pull → Surface tension That’s why drops become spherical.

Concept:
Simple harmonic motion (SHM) is a type of oscillatory motion in which the restoring force is directly proportional to displacement and directed towards the mean position.



Step 1: Restoring force in SHM
In SHM, restoring force is proportional to displacement and opposite in direction:
\[ F \propto -x \]
\[ F = -kx \]

where \(k\) is a constant.



Step 2: Apply Newton’s second law
\[ F = ma \]
\[ m\frac{d^2x}{dt^2} = -kx \]



Step 3: Rearranging
\[ \frac{d^2x}{dt^2} + \frac{k}{m}x = 0 \]



Step 4: Define angular frequency
Let:
\[ \omega^2 = \frac{k}{m} \]

Substitute into the equation.



Final Differential Equation
\[ \boxed{\frac{d^2x}{dt^2} + \omega^2 x = 0} \]



Interpretation:
This equation shows that acceleration is always proportional to displacement and directed towards the equilibrium position, which is the defining property of SHM. Quick Tip: SHM key idea: Restoring force \(\propto\) displacement and opposite in direction. That always leads to: \[ \frac{d^2x}{dt^2} + \omega^2 x = 0 \]

Concept:
Lenz’s Law is a consequence of Faraday’s law of electromagnetic induction and determines the direction of induced current.



Step 1: Statement of Lenz’s Law
The induced emf or current is always in such a direction that the magnetic field produced by it opposes the change in magnetic flux causing it.

Mathematically included as the negative sign in Faraday’s law:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]



Step 2: Example of increasing flux
Consider a magnet moving towards a conducting loop.
- Magnetic flux through the loop increases.
- Induced current produces a magnetic field opposing the approaching magnet.

This creates a repulsive force.



Step 3: Energy consideration
To push the magnet towards the loop, external work must be done against this repulsive force.
This mechanical work is converted into electrical energy (induced current).



Step 4: If Lenz’s law were not true
If the induced current supported the change instead of opposing it:
- Magnet would accelerate without external work.
- Energy would be produced without input.
This would violate conservation of energy.



Step 5: Conclusion
Thus, Lenz’s Law ensures that:
- Induction always resists the cause producing it.
- Energy input is required for energy output.
Hence it is consistent with the law of conservation of energy. Quick Tip: Lenz’s Law = Nature resists change. Induced current always opposes flux change → Prevents free energy → Ensures energy conservation.

Concept:
A conical pendulum consists of a mass tied to a string moving in a horizontal circular path such that the string makes a constant angle \(\theta\) with the vertical.

The motion is uniform circular motion with centripetal force provided by components of tension.



Step 1: Forces acting on the bob
Let tension in the string be \(T\).
Resolve tension into components:


Vertical component: \(T\cos\theta\) balances weight
Horizontal component: \(T\sin\theta\) provides centripetal force




Step 2: Vertical equilibrium
\[ T\cos\theta = mg \quad \cdots (1) \]



Step 3: Centripetal force
Radius of circular path:
\[ r = l\sin\theta \]

Centripetal force:
\[ T\sin\theta = \frac{mv^2}{r} \]
\[ T\sin\theta = \frac{mv^2}{l\sin\theta} \quad \cdots (2) \]



Step 4: Divide equations (2) by (1)
\[ \frac{T\sin\theta}{T\cos\theta} = \frac{mv^2/(l\sin\theta)}{mg} \]
\[ \tan\theta = \frac{v^2}{gl\sin\theta} \]



Step 5: Solve for velocity
\[ v^2 = gl\sin\theta \tan\theta = gl\frac{\sin^2\theta}{\cos\theta} \]
\[ v = \sqrt{\frac{gl\sin^2\theta}{\cos\theta}} \]



Step 6: Time period of circular motion
\[ T = \frac{2\pi r}{v} \]
\[ T = \frac{2\pi l\sin\theta}{\sqrt{\frac{gl\sin^2\theta}{\cos\theta}}} \]



Step 7: Simplify
\[ T = 2\pi \sqrt{\frac{l\cos\theta}{g}} \]



Final Expression:
\[ \boxed{T = 2\pi \sqrt{\frac{l\cos\theta}{g}}} \] Quick Tip: Conical pendulum shortcut: Replace effective length by \(l\cos\theta\) in simple pendulum formula. So: \[ T = 2\pi \sqrt{\frac{l\cos\theta}{g}} \]

Concept:
According to the kinetic theory of gases, the average kinetic energy of gas molecules is directly proportional to the absolute temperature.



Step 1: Kinetic energy of one molecule
The translational kinetic energy of a molecule is:
\[ KE = \frac{1}{2}mv^2 \]

For RMS speed:
\[ \frac{1}{2}m v_{rms}^2 \]



Step 2: Relation from kinetic theory
Average kinetic energy of a gas molecule:
\[ \frac{1}{2}m v_{rms}^2 = \frac{3}{2}kT \]

where \(k\) = Boltzmann constant \(T\) = absolute temperature



Step 3: Solve for RMS speed
\[ m v_{rms}^2 = 3kT \]
\[ v_{rms}^2 = \frac{3kT}{m} \]
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]



Step 4: Proportionality
Since \(k\) and \(m\) are constants for a given gas:
\[ v_{rms} \propto \sqrt{T} \]



Conclusion:
The RMS speed of gas molecules is directly proportional to the square root of the absolute temperature. Quick Tip: Temperature increases → molecular speed increases. Doubling temperature does NOT double speed — it increases by \(\sqrt{2}\).