Problem 80 - Master Resource Book in JEE Main Physics

Question

The internal and external diameters of a hollow cylinder are measured with the help of a vernier callipers. Their values are \(4.23 \pm 0.01 \mathrm{~cm}\) and \(3.87 \pm 0.01 \mathrm{~cm}\) respectively. The thickness of the wall of the cylinder is (a) \(0.36 \pm 0.02 \mathrm{~cm}\) (b) \(0.18 \pm 0.02 \mathrm{~cm}\) (c) \(0.36 \pm 0.01 \mathrm{~cm}\) (d) \(0.18 \pm 0.01 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The wall thickness is (d) \(0.18 \pm 0.01 \text{ cm}\).

Step 1: Understand the Given Values

We are given the internal diameter \(d_i = 3.87 \pm 0.01 \text{ cm}\) and the external diameter \(d_e = 4.23 \pm 0.01 \text{ cm}\). We need to find the thickness of the wall of the cylinder.

Step 2: Calculate the Thickness of the Wall

To find the thickness \(t\) of the wall of the cylinder, we use the formula: \[ t = \frac{d_e - d_i}{2} \].Substituting the given values:\[ t = \frac{4.23 - 3.87}{2} = \frac{0.36}{2} = 0.18 \text{ cm} \].

Step 3: Determine the Uncertainty in Thickness

The uncertainty in thickness is calculated by adding the uncertainties of the internal and external diameters since the thickness relies on the difference between them. Thus, the uncertainty is:\[ \Delta t = \frac{\Delta d_e + \Delta d_i}{2} = \frac{0.01 + 0.01}{2} = 0.01 \text{ cm} \].

Step 4: Find the Correct Option

From the calculated result, the thickness with its uncertainty is \(0.18 \pm 0.01 \text{ cm}\). Therefore, the correct option is (d) \(0.18 \pm 0.01 \text{ cm}\).

Key Concepts

Vernier Calipers

Vernier calipers are precision instruments used to measure dimensions with high accuracy. They are especially useful for measuring internal and external diameters, lengths, and depths.
A vernier caliper consists of two scales. The main scale, which is fixed, and the vernier scale, which can slide alongside the main scale for more precise readings.
To use the vernier calipers effectively:

• Align the object to be measured with the jaws of the calipers.
• The main scale provides the measurement in centimeters (or inches) and the vernier scale provides additional precision.
• The difference between the main scale and vernier scale readings gives the most precise dimension.

In our exercise, the measurements of the internal and external diameters of a hollow cylinder are taken with vernier calipers, allowing for precise readings accurate to two decimal places. This precision facilitates the calculation of the thickness of the cylinder's wall.

Hollow Cylinder

A hollow cylinder is a 3D shape with a cylindrical shell. Its structure is defined by two concentric circles forming the outer and inner surfaces.
To understand the concept, imagine the cylinder like a paper towel roll. The outside is the external diameter, and the cardboard tube inside is the internal diameter.
The important measurements for a hollow cylinder include:

• External diameter - the diameter of the outer circle.
• Internal diameter - the diameter of the inner circle.
• Wall thickness - the distance between the outer and inner circles.

The exercise discussed measures these diameters to find the wall thickness, calculated by the difference between the external and internal diameters divided by two (since thickness is half the difference). By understanding these simple geometry concepts, it becomes clearer how to handle such problems in practical scenarios.

Measurement Uncertainty

In any measurement, accuracy is critical, and vernier calipers help increase this accuracy. However, no measurement is perfectly precise due to unavoidable errors or estimate differences, leading to "measurement uncertainty."
Measurement uncertainty quantifies the doubt in the result and is often represented by the symbol \( \pm \). For example, in the problem, the internal diameter is \(3.87 \ \pm 0.01 \) cm, meaning it could reasonably be as low as 3.86 cm or as high as 3.88 cm.
When calculating values like the thickness of a cylinder wall, it's crucial to combine uncertainties from all measurements involved to determine the overall uncertainty accurately.

• To find the uncertainty of the wall thickness, add the uncertainties of both the internal and external diameters.
• Subsequently, divide by two since the thickness formula divides the diameter difference by two.

This systematic approach ensures that the resulting measurement and its uncertainty realistically reflect the true values, providing confidence in the reliability of the measurement process.