Problem 92 - Master Resource Book in JEE Main Physics

Question

Length is measured in metre and time in second as usual. But a new unit of mass is so chosen that \(G=1\). This new unit of mass is equal to (a) \(1.5 \times 10^{7} \mathrm{~kg}\) (b) \(1.5 \times 10^{10} \mathrm{~kg}\) (c) \(6.67 \times 10^{-11} \mathrm{~kg}\) (d) \(6.67 \times 10^{-8} \mathrm{~kg}\)

Step-by-Step Solution

Answer

The new unit of mass is \(6.67 \times 10^{-11} \mathrm{~kg}\), option (c).

Step 1: Understand the Exercise

The problem requires us to find the new unit of mass when the gravitational constant \(G\) is set to 1. \(G\) is usually \(6.67 \times 10^{-11} \mathrm{~m}^3 \, \mathrm{kg}^{-1} \, \mathrm{s}^{-2}\). We need to change the unit of mass so that \(G = 1\) in the new system.

Step 2: Analyze the Required Conversion

When \(G = 1\), the unit of mass should adjust to make the units consistent. The traditional equation for \(G\) sets \(G = 6.67 \times 10^{-11}\), where length is in meters, time in seconds, and mass in kilograms.

Step 3: Determine the New Unit Factor

We set the equation \(G_{new} = 1\) and compare it to the actual \(G\). Since length and time remain constant: \[ G_{new} \times \text{new mass unit} = G_{old} \] Thus, \(1 \times \text{new mass unit} = 6.67 \times 10^{-11}\). This means the new unit of mass is \(6.67 \times 10^{-11} \mathrm{~kg}\).

Step 4: Verify with Provided Options

Comparing our calculated new mass unit \(6.67 \times 10^{-11} \mathrm{~kg}\) with the given options, option (c) matches our result.

Key Concepts

Unit Conversion

Unit conversion is a crucial technique in physics and helps us switch between different measurement units without altering the actual physical quantity. In the context of this exercise, we need to modify the unit of mass such that the gravitational constant, denoted as \( G \), equals 1. The standard value of \( G \) is \( 6.67 \times 10^{-11} \, \mathrm{m}^{3} \, \mathrm{kg}^{-1} \, \mathrm{s}^{-2} \). By doing this conversion, we're essentially choosing a different basis for measurement that affects how we physically interpret mass with respect to gravity.

To achieve this, a simple comparison is used:

• Keep the units of length (meters) and time (seconds) unchanged.
• Only adjust the unit of mass so the numerical value of \( G \) becomes 1.

This conversion essentially redefines how we perceive mass in relation to gravitational forces, by effectively changing the unit to \( 6.67 \times 10^{-11} \, \mathrm{kg} \). We'll explore why this specific value makes all the units consistent in the following "Measurement Units" section.

Newton's Law of Gravitation

Newton's Law of Gravitation describes the gravitational force between two masses. It states that the force \( F \) between two objects is directly proportional to the product of their masses (\( m_1 \) and \( m_2 \)) and inversely proportional to the square of the distance \( r \) between their centers. The formula is represented as:\[ F = \frac{G \, m_1 \, m_2}{r^2} \]Here, \( G \) is the gravitational constant, which normally has the value \( 6.67 \times 10^{-11} \, \mathrm{m}^3 \, \mathrm{kg}^{-1} \, \mathrm{s}^{-2} \).

In this exercise, we're asked to find a new unit of mass such that \( G \) becomes 1. This change simplifies Newton’s Law of Gravitation to:\[ F = \frac{m_1 \, m_2}{r^2} \]meaning that in our new unit system, calculations involving gravity become more straightforward. The influence of the gravitational constant is absorbed into the unit of mass, allowing us to focus on the interaction of the masses and their separation only.

Measurement Units

Measurement units allow us to quantify and express physical quantities consistently in science. In this exercise, we focus on three fundamental units used in physics:

• Length: Measured in meters (m).
• Time: Measured in seconds (s).
• Mass: Initially measured in kilograms (kg), but altered in this exercise.

For this problem, while length and time remain unchanged, mass is redefined to simplify calculations involving gravity. By setting \( G = 1 \), a new mass unit \( 6.67 \times 10^{-11} \, \mathrm{kg} \) aligns with the adjusted gravitational constant. This redefinition keeps formulas consistent and ensures better clarity when applying Newton's Law of Gravitation.

Understanding these conversions in units allows students to navigate between various measurement systems confidently, thereby broadening their comprehension of physics.