Gravitation

Correct Answer

📖 Governing Law / Core Concept

The gravitational constant \( G \) is used in Newton's law of universal gravitation, which states: where:

• F = gravitational force between two masses (N)
• G = gravitational constant (N m²/kg²)
• m₁, m₂ = masses (kg)
• r = distance between masses (m)

🧪 Step-by-Step Breakdown

From the formula: \[ F = G \frac{m_1 m_2}{r^2} \] we rearrange for \( G \): \[ G = \frac{F r^2}{m_1 m_2} \] Identifying SI units:

• Force \( [F] \) = kg m/s²
• Distance \( [r] \) = m
• Mass \( [m] \) = kg

Substituting units into the equation: Simplifying gives: \[ G = \frac{\text{kg} \cdot \text{m}^3/\text{s}^2}{\text{kg}^2} = \text{m}^3\text{kg}^{-1}\text{s}^{-2} \] Thus, the dimensional formula of \( G \) is:

🔍 Option Analysis

• Option A: [ML³T⁻²] - Incorrect, does not match dimensional derivation.
• Option C: [M²L⁻²T⁻²] - Incorrect, does not match.
• Option D: [M²L⁻¹T⁻²] - Incorrect, does not match.

⚡ Mnemonic / Speed-Run

Remember \( G = \frac{F r^2}{m_1 m_2} \) for deriving dimensions quickly. The negative exponents indicate inverse relationships.

6. Visual Suggestion

📖 Factual Verification & Reference

Solution

Option (a) is correct.

Explanation:

For the first system: \[ F = G \frac{M \cdot M}{R^2} = G \frac{M^2}{R^2} \] For the second system with masses \(2M\) at distance \(\frac{R}{2}\): \[ F' = G \frac{(2M) \cdot (2M)}{\left(\frac{R}{2}\right)^2} = G \frac{4M^2}{\frac{R^2}{4}} = G \frac{4M^2 \cdot 4}{R^2} = G \frac{16M^2}{R^2} \] Now, relate \(F'\) to \(F\): \[ F' = 16 \times G \frac{M^2}{R^2} = 16F \] Thus, the gravitational force between the bodies in the second system is \(16F\).

Correct Answer

📖 Governing Law / Core Concept

The acceleration due to gravity (g) on the surface of a planet is given by the formula:

Where G is the gravitational constant, M is the mass of the planet, and R is its radius. For a planet of uniform density (ρ), mass can be expressed as:

🧪 Step-by-Step Breakdown

Substituting the expression for mass into the equation for g:

This indicates that g is directly proportional to the radius R when density is constant.

Let g₁ and g₂ be the gravitational accelerations on planets 1 and 2, respectively.

Where k = \frac{4 \pi G \rho}{3}. Since R₁ > R₂, it follows that:

📌 Hints / Properties Used:

• Gravitational formula: \( g = \frac{GM}{R^2} \)
• Relationship between mass, density, and volume
• Proportionality of gravitational acceleration to radius in uniform density

📊 Visual Diagram Suggestion:

A diagram showing two planets with labeled radii R₁ and R₂, highlighting that R₁ > R₂. Arrows indicating greater gravitational pull g₁ compared to g₂ could aid in understanding.

Correct Answer

📖 Governing Law / Core Concept

The falling apple demonstrates the Law of Conservation of Energy, which states that energy can neither be created nor destroyed; it merely transforms from one form to another.

🧪 Step-by-Step Breakdown

The energy transfer when the apple falls is as follows:

• Gravitational Potential Energy (GPE): Initially, the apple possesses gravitational potential energy due to its height.
• Kinetic Energy (KE): As the apple falls, GPE converts to kinetic energy, increasing its velocity.
• Heat Energy to Air (Q_air): As the apple accelerates, it generates heat due to air friction.
• Heat Energy to Ground and Apple (Q_ground): Upon hitting the ground, some kinetic energy is converted to heat, raising the temperature of both the apple and the ground.
• Sound Energy: A portion of energy transforms into sound energy, which we hear upon the apple's impact.

🔍 Option Analysis

• Option (a) incorrectly lists sound energy before heat energy transfers.
• Option (b) fails to sequence kinetic energy after gravitational potential energy effectively.
• Option (d) places sound energy before heat energy to the ground, which is inaccurate.

⚡ Mnemonic / Speed-Run

To remember the sequence: Gravity → Kinetic → Heat (air) → Heat (ground) → Sound can be abbreviated as GKHHs.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

A conservative force is defined as a force for which the work done is independent of the path taken and depends only on the initial and final positions. Mathematically, for a conservative force \( F \), the work done \( W \) can be expressed as:

Where \( A \) and \( B \) are the initial and final points, \( dr \) is the displacement vector.

🧪 Step-by-Step Breakdown

In contrast, the frictional force is path-dependent, primarily arising from the interaction between surfaces. In any movement where friction is present, the work done against it will vary with the distance traveled:

Where \( f \) is the magnitude of the frictional force and \( d \) is the distance over which it acts.

🔍 Option Analysis

• Electric Force: A conservative force where potential energy exists due to charge distribution.
• Gravitational Force: A conservative force characterized by energy conservation in gravitational fields.
• Spring Force: Another conservative force, described by Hooke's Law: \( F = -kx \), where \( k \) is the spring constant and \( x \) is the displacement.

⚡ Mnemonic / Speed-Run

Remember: "Conservative Forces Keep Their Path" - implies they depend only on endpoints.

6. Visual Suggestion

📖 Factual Verification & Reference