Units and Measurement Important Key Questions and Detailed Explanations

Units and Measurement - Solutions

1. Define the term "physical quantity."

A physical quantity is any quantity that can be measured and expressed in terms of units. Examples include length, mass, time, temperature, etc.

2. What is the SI unit of luminous intensity?

The SI unit of luminous intensity is the candela (cd).

3. Name the device used to measure the diameter of a small spherical object.

The device used to measure the diameter of a small spherical object is a screw gauge or micrometer.

4. How many significant figures are there in the number 0.00234?

The number 0.00234 has 3 significant figures (2, 3, and 4).

5. Write the dimensional formula of velocity.

The dimensional formula of velocity is \([M^0 L^1 T^{-1}]\).

6. Convert 1 angstrom to meters.

1 angstrom (\(\unicode{x212B}\)) = \(1 \times 10^{-10}\) meters.

7. State one advantage of using the SI system over other systems of units.

The SI system is universally accepted and used, making it easier to communicate and share scientific data globally.

2-Mark Questions Solutions

1. Differentiate between fundamental and derived units with one example each.

• Fundamental units: The basic units that are independent of any other units. For example, the meter (m) is the fundamental unit of length.
• Derived units: These units are derived from fundamental units. For example, the unit of velocity (m/s) is derived from the units of length (meter) and time (second).

2. What are systematic errors? Give one example.

Systematic errors are consistent, repeatable errors associated with faulty equipment or a flawed experimental design. For example, an error caused by a zero offset in a weighing scale.

3. Explain the term "dimensional analysis." How is it useful in verifying physical equations?

Dimensional analysis involves checking the consistency of units in physical equations. It is useful for verifying the correctness of physical equations, converting units, and deriving relationships between different physical quantities.

4. A physical quantity \(X\) is given by \(X = \frac{a^2 b}{\sqrt{c}}\). If the percentage errors in \(a\), \(b\), and \(c\) are 1%, 2%, and 4%, respectively, find the percentage error in \(X\).

Given: \(X = \frac{a^2 b}{\sqrt{c}}\)
Percentage error in \(X\) = \(2 \times \text{Percentage error in } a + \text{Percentage error in } b + \frac{1}{2} \times \text{Percentage error in } c\)
= \(2 \times 1\% + 2\% + \frac{1}{2} \times 4\%\)
= \(2\% + 2\% + 2\% = 6\%\).
Therefore, the percentage error in \(X\) is 6%.

5. The density of a substance is given by the formula \(\rho = \frac{m}{V}\), where \(m\) is the mass and \(V\) is the volume. If the mass of a substance is \(15 \pm 0.2\) grams and the volume is \(5 \pm 0.1\) cubic centimeters, calculate the density and its absolute uncertainty.

• Density Calculation: \[ \rho = \frac{m}{V} = \frac{15}{5} = 3 \, \text{g/cm}^3 \]
• Absolute Uncertainty in Density: The formula for the absolute uncertainty in density is: \[ \Delta \rho = \rho \times \sqrt{\left(\frac{\Delta m}{m}\right)^2 + \left(\frac{\Delta V}{V}\right)^2} \] where \(\Delta m = 0.2 \text{ grams}\) and \(\Delta V = 0.1 \text{ cm}^3\). \[ \Delta \rho = 3 \times \sqrt{\left(\frac{0.2}{15}\right)^2 + \left(\frac{0.1}{5}\right)^2} \approx 3 \times \sqrt{0.000089 + 0.000040} \approx 3 \times \sqrt{0.000129} \approx 3 \times 0.0114 \approx 0.034 \, \text{g/cm}^3 \]

Therefore, the density of the substance is **3 g/cm³** with an absolute uncertainty of **±0.034 g/cm³**.

3-Mark Questions Solutions

1. State the difference between accuracy and precision. Why are both important in scientific measurements?

• Accuracy: Refers to how close a measured value is to the true value or accepted standard.
• Precision: Refers to how close repeated measurements are to each other, indicating the consistency of the measurements.

Both are important because accuracy ensures correctness, while precision ensures consistency. High accuracy and precision are necessary for reliable scientific results.

2. The diameter of a sphere is measured five times as 4.250 mm, 4.254 mm, 4.253 mm, 4.249 mm, and 4.256 mm. Calculate the mean diameter, the absolute error, and the percentage error.

• Mean diameter: \[ \text{Mean} = \frac{4.250 + 4.254 + 4.253 + 4.249 + 4.256}{5} = 4.254 \, \text{mm}. \]
• Absolute error: Errors: \(0.004, 0.000, 0.001, 0.005, 0.002 \, \text{mm}\). Mean absolute error = \[ \frac{0.004 + 0.000 + 0.001 + 0.005 + 0.002}{5} = 0.0024 \, \text{mm}. \]
• Percentage error: \[ \text{Percentage error} = \frac{0.0024}{4.254} \times 100 \approx 0.056\%. \]

Therefore, the mean diameter is **4.254 mm**, the absolute error is **0.0024 mm**, and the percentage error is **0.056%**.

3. A student measures the length of a rod several times and obtains the following values: 12.01 cm, 12.00 cm, 12.02 cm, 12.01 cm, and 12.00 cm. Calculate the mean length, the absolute error, and the percentage error if the true length is 12.00 cm.

• Mean Length: \[ \text{Mean} = \frac{12.01 + 12.00 + 12.02 + 12.01 + 12.00}{5} = 12.008 \, \text{cm}. \]
• Absolute Error: The absolute error for each measurement relative to the true value (12.00 cm) is \(0.01, 0.00, 0.02, 0.01, 0.00 \, \text{cm}\). Mean absolute error = \[ \frac{0.01 + 0.00 + 0.02 + 0.01 + 0.00}{5} = 0.008 \, \text{cm}. \]
• Percentage Error: \[ \text{Percentage error} = \frac{0.008}{12.00} \times 100 \approx 0.067\%. \]

Therefore, the mean length is **12.008 cm**, the absolute error is **0.008 cm**, and the percentage error is **0.067%**.

4. The mass of a body is measured as 15.000 grams with an uncertainty of ±0.005 grams. Find the absolute and percentage uncertainty in the measurement.

• Absolute Uncertainty: The absolute uncertainty is given as ±0.005 grams.
• Percentage Uncertainty: \[ \text{Percentage uncertainty} = \frac{0.005}{15.000} \times 100 \approx 0.033\%. \]

Therefore, the absolute uncertainty is ±0.005 grams and the percentage uncertainty is **0.033%**.

5. If a length is measured as \(3.45 \pm 0.02\) meters, and the width is measured as \(2.30 \pm 0.01\) meters, calculate the area of the rectangle and its absolute uncertainty.

• Area Calculation: \[ \text{Area} = \text{Length} \times \text{Width} = 3.45 \times 2.30 = 7.935 \, \text{m}^2. \]
• Absolute Uncertainty in Area: The absolute uncertainty in area can be calculated using: \[ \Delta \text{Area} = \text{Area} \times \sqrt{\left(\frac{\Delta L}{L}\right)^2 + \left(\frac{\Delta W}{W}\right)^2} \] where \(\Delta L = 0.02 \text{ m}\) and \(\Delta W = 0.01 \text{ m}\). \[ \Delta \text{Area} = 7.935 \times \sqrt{\left(\frac{0.02}{3.45}\right)^2 + \left(\frac{0.01}{2.30}\right)^2} \approx 7.935 \times \sqrt{0.000038 + 0.000091} \approx 7.935 \times \sqrt{0.000129} \approx 7.935 \times 0.0114 \approx 0.090 \, \text{m}^2. \]

Therefore, the area of the rectangle is **7.935 m²** with an absolute uncertainty of **±0.090 m²**.

Important Questions of Units and Measurement for CBSE Class XI Physics

The chapter Units and Measurement is the foundation of every physical calculation and analysis in Physics. This topic introduces students to the fundamental methods of measuring physical quantities, unit systems, dimensional analysis, and the concept of accuracy. It is essential for understanding how measurements influence scientific results, making it a key chapter in Class 11 Physics.

Overview of Units and Measurement

To effectively prepare for exams, students need to grasp the key concepts of this chapter. Below are some important areas to focus on:

• Fundamental and Derived Units: Understanding the SI units and their conversions.
• Measurement Errors: Systematic and random errors, accuracy, and precision.
• Dimensional Analysis: Verifying the consistency of physical equations.
• Significant Figures: Rules for identifying significant figures in a measurement.

Units and Measurement Class 11 Important Questions

Below are some important questions to help you prepare for your exams:

• Define the SI unit of length, mass, and time. Why are they called fundamental units?
• Explain the concept of dimensional analysis and how it helps in verifying equations.
• A physical quantity is given as \(X = a^2 b^3 / c\). Find the dimensions of \(X\).
• What are the different types of measurement errors? How can random errors be minimized?
• A measurement is recorded as \(5.20 \pm 0.05 \, \text{m}\). What is the percentage error in the measurement?

Units and Measurement Class 11 NCERT Solutions

To perform well in both school exams and competitive exams, students should regularly refer to the NCERT Class 11 Physics textbook solutions. These solutions provide step-by-step explanations to textbook problems, helping students understand key concepts more thoroughly.

Units and Measurement Notes for Quick Revision

Creating concise notes on Units and Measurement is highly recommended. Important topics to include in your notes:

• SI units and unit conversions.
• Formulae for dimensional consistency and analysis.
• Types of errors and how to report measurements with significant figures.
• Applications of dimensional analysis in real-world scenarios.

Conclusion

The chapter Units and Measurement forms the basis for solving advanced physics problems and ensures students grasp the importance of precise measurements. Practicing units and measurement important questions helps in mastering this topic for both board exams and competitive exams. You can enhance your preparation by using units and measurement class 11 notes and revisiting NCERT solutions regularly.