Problem 19 - Master Resource Book in JEE Main Physics

Question

If \(1 \mathrm{~g} \mathrm{~cm} \mathrm{~s}^{-1}=x\) newton-sec, then the number \(x\) is equal to (a) \(1 \times 10^{-3}\) (b) \(3.6 \times 10^{-3}\) (c) \(1 \times \underline{10^{-5}}\) (d) \(6 \times \underline{10^{-4}}\)

Step-by-Step Solution

Answer

The value of \(x\) is \(1 \times 10^{-5}\). Option (c) is correct.

Step 1: Understanding the Conversion

The problem asks us to convert the units of \(1 \text{ g cm s}^{-1}\) into newton-seconds. This involves understanding the relationship between the given units of mass, length, and time with those of force (newton) and time (second).

Step 2: Relationship of Units

We know that \(1 \text{ g} = 1 \times 10^{-3} \text{ kg}\) and \(1 \text{ cm} = 1 \times 10^{-2} \text{ m}\). The equation \(1 \text{ g cm s}^{-1}\) can be rewritten as \[1 \times 10^{-3} \text{ kg} \cdot 1 \times 10^{-2} \text{ m} \cdot \text{s}^{-1}\]

Step 3: Rewriting in Base Units

Therefore, \[1 \text{ g cm s}^{-1} = (1 \times 10^{-3} \text{ kg}) \cdot (1 \times 10^{-2} \text{ m}) \cdot \text{s}^{-1}\]This is equivalent to \[1 \times 10^{-5} \text{ kg m s}^{-1}\]

Step 4: Comparing to Newton-Second

We know that 1 newton is defined as \(1 \text{ kg m s}^{-2}\). Therefore, 1 newton-second is \(1 \text{ kg m s}^{-1}\).Thus, converting to newton-second, \[1 \times 10^{-5} \text{ kg m s}^{-1} = 1 \times 10^{-5} \text{ newton-second}\]

Step 5: Conclusion

The conversion shows us that \(x = 1 \times 10^{-5}\). Therefore, the answer is option (c).

Key Concepts

newton-second

The newton-second is a unit used in physics to measure momentum, which is the product of mass and velocity. It combines the unit of force (newton) and time (second). Understand that:

• It represents how much momentum an object changes over a given second when subjected to a constant force.
• One newton-second is equal to a force of one newton applied for one second.
• Its formula is simply mass (in kg) multiplied by velocity (in m/s).

This concept is crucial for understanding how forces in motion interact over time. It helps predict how fast and in what direction an object will move after a force has been applied. The calculation of newton-seconds can provide insights into various physical phenomena, including car crashes and rocket launches.

mass conversion

Mass conversion involves changing the mass of an object from one unit to another, typically from grams to kilograms in the SI unit system.

• The SI unit for mass is the kilogram (kg).
• 1 gram (g) equals 0.001 kilograms (kg).
• This conversion is essential for calculations in physics where SI units are preferred.

Mass conversion appears often in problems dealing with force and motion, as forces are calculated using mass. Virtually all calculations involving newtons require mass to be in kilograms. It's important to securely grasp this conversion to avoid errors in solving physics problems, as even small mistakes in unit conversion can lead to incorrect results.

length conversion

Length conversion in physics is often necessary when dealing with various unit systems. Typically, this involves converting smaller units like centimeters (cm) to meters (m).

• The base unit of length in the SI system is the meter (m).
• 1 centimeter (cm) is equivalent to 0.01 meters (m).
• This conversion is crucial for working with equations needing length as a factor.

Understanding length conversion is fundamental in physics, especially in force and motion calculations. Dimensional consistency in formulas and calculations requires converting all units to a common system, often the metric system, to ensure accuracy and validity of results.

force and motion

In physics, force and motion are closely related concepts. Force is what changes the state of motion of objects and is measured in newtons (N).

• Newton’s laws of motion describe how forces affect motion.
• Force = Mass × Acceleration (F = ma), showing the direct relationship between mass, acceleration, and force.
• Force is a vector, meaning it has both magnitude and direction.

Understanding how force affects motion helps in analyzing everyday occurrences, like how a ball accelerates when kicked. Forces can initiate motion, slow an object down, or change its direction. Physics problems often require estimating the forces involved to predict how they affect motion.

SI units

The International System of Units (SI) is the fundamental metric system that ensures uniformity in measurement across science and technology.
Standardized units include:

• Meter (m) for length
• Kilogram (kg) for mass
• Second (s) for time
• Newton (N) for force

SI units enable professionals and students to perform calculations that are understandable globally. Their usage eliminates confusion caused by different measurement systems, facilitating clear communication and precise calculations in physics. When solving physics homework, it’s essential to always convert quantities to the SI unit system for all calculations, ensuring accuracy and consistency.