Answer in brief, any six questions. a. The energy of a photon is given by E = hf . Find the dimension and unit of Plank's constant h , where f is the frequency of radiation. [2] b. From a flying…

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a. Dimension and Unit of Planck’s Constant $h$

Step 1: Recall the given equation The energy of a photon is given by: $$E=hf$$ where:

• $E$ = Energy of the photon (in joules, J)
• $h$ = Planck’s constant (unknown dimension and unit)
• $f$ = Frequency of radiation (in hertz, Hz or s⁻¹)

Step 2: Express dimensions of each quantity

• Energy ($E$): In SI units, energy is measured in joules (J), which is equivalent to $\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-2}$.
• Frequency ($f$): Frequency is the number of oscillations per second, so its dimension is ${\text{s}}^{-1}$.

Step 3: Rearrange the equation to solve for $h$ $$h=\frac{E}{f}$$

Step 4: Substitute dimensions $$[h]=\frac{[E]}{[f]}=\frac{\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-2}}{{\text{s}}^{-1}}=\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$$

Step 5: Determine the unit Since $E$ is in joules (J) and $f$ is in hertz (Hz or s⁻¹), the unit of $h$ is: $$\text{J}\cdot \text{s}(\text{Joule-second})$$

Final Answer: The dimension of Planck’s constant $h$ is $\boxed{{\text{ML}}^2{\text{T}}^{-1}}$ and its unit is $\boxed{\text{Joule-second (J⋅s)}}$.

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b. Why Should a Body Be Dropped in Advance from a Flying Aeroplane to Hit the Target?

Explanation: When an object is dropped from a moving aeroplane, it retains the horizontal velocity of the aeroplane due to inertia (Newton’s First Law of Motion). This means the object continues to move forward horizontally while simultaneously falling vertically under gravity.

• If the object is dropped directly above the target, it will lag behind due to its horizontal motion, missing the target.
• To ensure the object hits the target, it must be released ahead of the target so that by the time it falls vertically, it has also moved horizontally to reach the target’s position.

Key Concept: This phenomenon is due to the independence of horizontal and vertical motions in projectile motion.

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c. Which Has Greater Kinetic Energy: A Light Body or a Heavy Body with the Same Momentum?

Given:

• Two bodies have the same momentum ($p$).
• Let:
  • Mass of light body = $m_1$ (smaller)
  • Mass of heavy body = $m_2$ (larger)
  • Velocities = $v_1$ and $v_2$ respectively.

Step 1: Recall the formula for momentum and kinetic energy $$p=mv\text{and}KE=\frac{1}{2}m{v}^2$$

Step 2: Express velocities in terms of momentum Since $p$ is the same for both: $$v_1=\frac{p}{m_1}\text{and}{v}_2=\frac{p}{m_2}$$

Step 3: Write kinetic energy expressions $$K{E}_1=\frac{1}{2}{m}_1{v}_1^2=\frac{1}{2}{m}_1{\left(\frac{p}{m_1}\right)}^2=\frac{p^2}{2m_1}$$ $$K{E}_2=\frac{1}{2}{m}_2{v}_2^2=\frac{1}{2}{m}_2{\left(\frac{p}{m_2}\right)}^2=\frac{p^2}{2m_2}$$

Step 4: Compare $K{E}_1$ and $K{E}_2$ Since $m_1<m_2$: $$\frac{p^2}{2m_1}>\frac{p^2}{2m_2}⟹K{E}_1>K{E}_2$$

Conclusion: The light body has greater kinetic energy when both have the same momentum.

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d. Why Do Passengers Feel Weightlessness in an Artificial Satellite but Not on the Moon?

Comparison Table:

Key Concepts:

1. Free-fall condition: In a satellite, both the satellite and the person are accelerating toward Earth at $g$, so no contact force (normal reaction) exists.
2. Normal force on Moon: The Moon’s surface exerts an upward force on the person, balancing part of gravity, resulting in apparent weight.

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e. Why Are Passengers in a Small Boat Not Allowed to Stand?

Explanation: A small boat has a small base area, which means its center of gravity (COG) is higher and less stable compared to larger boats or ships. When passengers stand up:

1. The COG shifts upward and sideways, increasing the risk of toppling.
2. The boat’s buoyancy and stability depend on its metacentric height (distance between COG and center of buoyancy). Standing disrupts this balance.
3. The moment of inertia decreases, making the boat more prone to rolling or capsizing due to minor disturbances (e.g., wind or waves).

Safety Measure: Passengers must sit or lie down to keep the COG low and the boat stable.

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f. Effect of Melting Polar Ice on the Duration of the Day

Step 1: Understand Earth’s Rotation and Angular Momentum Earth rotates about its axis with a constant angular momentum ($L$) given by: $$L=I\omega$$ where:

• $I$ = Moment of inertia of Earth
• $\omega$ = Angular velocity (related to day length)

Step 2: Effect of Melting Ice

• Ice at the poles is close to the axis of rotation, contributing less to Earth’s moment of inertia ($I$).
• When ice melts, water spreads toward the equator, increasing Earth’s moment of inertia (since mass moves farther from the axis).

Step 3: Conservation of Angular Momentum Since $L$ is conserved ($L_{\text{initial}}=L_{\text{final}}$): $$I_{\text{initial}}{\omega }_{\text{initial}}=I_{\text{final}}{\omega }_{\text{final}}$$ If $I_{\text{final}}>I_{\text{initial}}$, then ${\omega }_{\text{final}}<{\omega }_{\text{initial}}$.

Step 4: Relate Angular Velocity to Day Length $$\omega =\frac{2\pi }{T}$$ where $T$ = Duration of the day. If $\omega$ decreases, $T$ increases.

Conclusion: Melting polar ice increases Earth’s moment of inertia, decreasing angular velocity, and thus lengthening the day (making it slightly longer).

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g. Why Does Hot Soup Taste Better Than Cold Soup?

Explanation: The perception of taste is influenced by thermal and chemical factors:

1. Volatile Compound Release:

   • Heat increases the kinetic energy of molecules in soup, causing more aromatic compounds (e.g., spices, herbs) to evaporate.
   • These volatile molecules reach the olfactory receptors in the nose more easily, enhancing flavor perception.

2. Saliva and Taste Buds:

   • Warm soup stimulates saliva production, which helps dissolve food particles and carry taste molecules to taste buds more efficiently.
   • Cold soup may numb taste buds slightly, reducing sensitivity.

3. Texture and Mouthfeel:

   • Hot soup has a softer, smoother texture, which is more pleasant and easier to swallow.
   • Cold soup may feel thicker or clingy, altering the eating experience.

4. Psychological Factor:

   • Culturally, hot food is often associated with comfort and richness, subconsciously making it taste better.

Scientific Basis: The Flavor-Temperature Interaction suggests that warmth enhances the perception of bitterness, sweetness, and umami, while coldness may mute these tastes.

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