Mass, weight, volume, density, specific gravity

Solution

Option (b) is correct.

Explanation:

The average density (\(\rho_{\text{avg}}\)) of the object can be calculated using the formula:

Given that the object is made of two equal parts by volume: - Volume of first part: \(V\), Density \(=\rho_0\) - Volume of second part: \(V\), Density \(=2\rho_0\)

Total Mass (\(M\)) is computed as:

Total Volume (\(V_{\text{total}}\)) is:

Substituting \(M\) and \(V_{\text{total}}\) into the average density formula:

Solution

Option (d) is correct.

Explanation:

Resistance \( R \) of a cylindrical wire is given by the formula: where \( \rho \) is the resistivity, \( l \) is the length, and \( A \) is the cross-sectional area of the wire. For the initial wire with radius \( r \) and length \( l \): \[ A_1 = \pi r^2 \] Thus, \[ R_1 = \frac{\rho l}{\pi r^2} \] For the second wire with radius \( 2r \) and length \( \frac{l}{2} \): \[ A_2 = \pi (2r)^2 = 4\pi r^2 \] So, \[ R_2 = \frac{\rho \left(\frac{l}{2}\right)}{A_2} = \frac{\rho \left(\frac{l}{2}\right)}{4\pi r^2} = \frac{\rho l}{8\pi r^2} = \frac{R_1}{4} \] Now substituting for \( R_1 \): \[ R_2 = \frac{R_1}{4} = \frac{\frac{\rho l}{\pi r^2}}{4} = \frac{\rho l}{4\pi r^2} = \frac{R}{8} \] Thus, the resistance of the second wire is: \[ R_2 = \frac{R}{8} \]

Correct Answer

📖 Governing Law / Core Concept

The principle governing the sinking of a body in fluids is Archimedes' Principle:

The buoyant force \( F_b \) acting on a submerged body is given by:

Where:

• \( \rho_{fluid} \) = Density of the fluid (water) in kg/m³
• \( V_{displaced} \) = Volume of fluid displaced in m³
• \( g \) = Acceleration due to gravity (9.81 m/s²)

🧪 Step-by-Step Breakdown

A body will sink if its overall density is greater than that of the fluid. This is determined as follows:

• Density of a body is defined as \( \rho_{body} = \frac{m_{body}}{V_{body}} \), where \( m_{body} \) is mass and \( V_{body} \) is volume.
• A body sinks when \( \rho_{body} > \rho_{water} \).
• Consequently, if a body with a density greater than that of water is placed in water, the gravitational force acting downward exceeds the buoyant force acting upward.

🔍 Option Analysis

• Option (a): Incorrect. This describes a body that will float.
• Option (c): Incorrect. Equal densities will result in the body neither sinking nor floating.
• Option (d): Incorrect. Weight alone doesn’t determine sinking; density relative to the fluid matters.

⚡ Mnemonic / Speed-Run

Remember: "Sink or Swim? Check the Density!" This helps recall that the sinking of the body is dictated by its density compared to the fluid.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The density (\( \rho \)) of an object is defined by the formula:

where:

• m = mass of the object (kg)
• V = volume of the object (m³)

🧪 Step-by-Step Breakdown

Given data:

• Weight of pumpkin \( W = 7.5 \, \text{N} \)
• Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)
• Volume of water displaced \( V_{\text{water}} = \frac{3}{4} \, \text{L} = 0.00075 \, \text{m}^3 \)

Extracting mass from weight:

Calculating density of pumpkin:

🔍 Option Analysis

An analysis of other options shows:

• A (10 kg/m³): Inaccurate, too low for a solid like pumpkin.
• B (100 kg/m³): Unreasonable, less than water's density.
• D (10,000 kg/m³): Improbable, denser than most solids.

⚡ Mnemonic / Speed-Run

To remember density calculation: "Mass over volume is the key, density is what you'll see!"

Correct Answer

📖 Governing Law / Core Concept

The governing principle is Archimedes' Principle, which states that a floating body displaces a volume of fluid equal to its weight.

🧪 Step-by-Step Breakdown

Both cubes C1 and C2 float in water, implying equilibrium such that the weight of the liquids plus the buoyant force equals the weight of the water displaced.

For each cube, if \( V_1 \) and \( V_2 \) are the volumes of liquid in C1 and C2, the buoyant forces are:

In equilibrium, \( \rho_1 \) (in C1) rises higher, indicating:

Both densities must be less than the density of water:

🔍 Option Analysis

• A: ρ2 < ρw < ρ1 - Incorrect, contradicts buoyant force principles.
• B: ρ2 < ρ1 < ρw - Incorrect, does not maintain the relationship for floating.
• D: ρ1 < ρw < ρ2 - Incorrect, contradicts Archimedes' principle.

⚡ Mnemonic / Speed-Run

Remember "lighter floats higher" to grasp density relationships: lower densities rise in denser fluids.