Problem 16 - Master Resource Book in JEE Main Physics

Question

One slug is equivalent to \(14.6 \mathrm{~kg}\). A force of 10 pound is applied on a body of \(1 \mathrm{~kg}\). The acceleration of the body is (a) \(44.5 \mathrm{~ms}^{-2}\) (b) \(4.448 \mathrm{~ms}^{-2}\) (c) \(44.4 \mathrm{~ms}^{-2}\) (d) None of these

Step-by-Step Solution

Answer

Option (c) \(44.4 \, \mathrm{m/s^2}\) is correct.

Step 1: Understanding the Question

We are given a force of 10 pounds applied to a 1 kg mass and need to find the acceleration in meters per second squared (\(\mathrm{m/s}^2\)). We will use Newton's Second Law of Motion, \(F = ma\), where \(F\) is force, \(m\) is mass, and \(a\) is acceleration.

Step 2: Convert Force from Pounds to Newtons

The conversion factor is 1 pound = 4.44822 newtons. Thus, a force of 10 pounds is equivalent to \(10 \times 4.44822 = 44.4822\) newtons.

Step 3: Apply Newton's Second Law

Using Newton's Second Law, \(F = ma\). We rearrange it to find acceleration: \(a = \frac{F}{m}\). Substitute \(F = 44.4822\, \mathrm{N}\) and \(m = 1\, \mathrm{kg}\) to get \(a = \frac{44.4822}{1} = 44.4822 \, \mathrm{m/s}^2\).

Step 4: Choose the Closest Answer

The calculated acceleration is approximately 44.4822 \(\mathrm{m/s}^2\), which is closest to option (c) \(44.4 \, \mathrm{m/s^2}\).

Key Concepts

Force Conversion

When working with forces, it's crucial to convert the units to a common system before performing any calculations.
In the given problem, the force is expressed in pounds, a unit often used in the United States. However, for scientific calculations, especially when using Newton's Second Law, it's standard to work with the International System of Units (SI), where the unit of force is the newton (N).

• **Conversion Factor:** There is a specific conversion factor to change pounds to newtons. This factor is 1 pound = 4.44822 newtons.
• **Calculation:** If you have a force of 10 pounds, you multiply it by 4.44822 to convert it to newtons. So, 10 pounds is equal to 44.4822 newtons.

Understanding and correctly converting these units is essential to ensure calculations are accurate and meaningful.

Acceleration Calculation

Determining acceleration is the main goal of this exercise, which requires the application of Newton's Second Law of Motion. This law ties together force, mass, and acceleration through the formula:
\[ F = ma \]

• **Rearranging the Formula:** To find acceleration, rearrange the formula to solve for \(a\): \(a = \frac{F}{m}\).
• **Known Variables:** In this problem, we have a force (now in newtons) and a mass. The force is 44.4822 N and the mass is 1 kg.
• **Plug in Values:** Substituting the values into the rearranged formula gives us \(a = \frac{44.4822}{1} = 44.4822 \mathrm{~m/s^2}\).

This calculation shows how force and mass determine acceleration.

Unit Conversion

Unit conversion is critical when dealing with physics problems, particularly when different measurement systems are involved.

These conversions ensure that all quantities are compatible and can be correctly integrated into formulas and calculations.

• **System of Units:** The mass is given in kilograms, which is the SI unit for mass, and the force needs to be in newtons, also an SI unit. When units are consistent, calculations become seamless.
• **Importance in Physics:** Without consistent units, it's impossible to directly compare or compute physical quantities.
• **Prevention of Errors:** Ensure all measurements adhere to the necessary units prior to inserting them into formulas. This step prevents computation mistakes that can lead to incorrect results.

Through precise unit conversions, solving physics problems like this one becomes manageable and intuitive.