Units And Measurement

Given ,
So, the percentage error in the volume is given by

Let the resistance be connected in parallel with the galvanometer. The total resistance in the circuit becomes the resistance of the galvanometer in parallel with .
We are given:
- The full scale deflection current of the galvanometer is 5 mA ,
- The range of the ammeter is 0-5 A,
- The resistance of the galvanometer .
The ammeter is designed for a range of 0-5 A, so the total resistance required in the circuit is: Calculating this value gives .
Thus, the required resistance to convert the galvanometer into an ammeter is .

- The acceleration of the body is given by Newton's second law, , where is the net force and is the mass of the body.
- Initially, the body has some force applied to it, and when the force increases by 80 N, the acceleration increases accordingly. Using the formula:

Step 1: Understanding the concept of Reductionism
Reductionism is a scientific approach that explains complex systems by breaking them down into their simpler components. It is widely used in physics, biology, and other sciences to analyze interactions at a fundamental level. Step 2: Why Reductionism is the correct answer
Since the question refers to deriving the properties of a larger system from its smaller parts, reductionism best describes this approach. Thus, the correct answer is {Reductionism} .

Step 1: Use the formula for percentage error
The resistance is given by: The percentage error in resistance is given by: Step 2: Substitute values
Thus, the correct answer is .

The thickness of the wall is calculated as the difference between the external and internal radii. The internal diameter cm and external diameter cm, so the internal radius is cm and the external radius is cm. The thickness of the wall is: The uncertainties are added, so the total uncertainty in thickness is cm. Thus, the thickness is cm.

Step 1: Formula for Density
Step 2: Calculating Density
Rounding to three significant figures: Step 3: Conclusion
Since 2.19 g/cm³ is the correctly rounded value, Option (3) is correct.

Step 1: Calculate the mean value of the measurements.
Let the measurements be .
Mean .
mPa s.
Step 2: Calculate the absolute error for each measurement.
The absolute error is .
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Step 3: Calculate the mean absolute error.
The mean absolute error is the average of the individual absolute errors.
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mPa s.

Step 1: Write the formula for pressure.
Pressure ( ) is defined as the normal force ( ) acting per unit area ( ). For a square plate with side length , the area is . So, the formula for pressure is:

Step 2: Apply the formula for propagation of errors.
For a quantity , the maximum fractional error is given by the sum of the fractional errors of its components, weighted by the magnitude of their exponents: In our case, . The maximum fractional error in pressure is:

Step 3: Convert to percentage errors.
Multiplying the entire equation by 100 gives the relationship for percentage errors:

Step 4: Substitute the given values.
We are given that the percentage error in force is 3% and the percentage error in the side length is 1%.

First, perform the subtraction to find the raw length of the remaining rod.
Remaining length = Initial length - Length of piece cut.
Remaining length = m - m = m.
Now, we must apply the rules for significant figures in addition and subtraction.
The rule states that the result should have the same number of decimal places as the measurement with the fewest decimal places.
The initial length, m, has one decimal place.
The length of the piece cut, m, has three decimal places.
The number with the fewest decimal places is (one decimal place).
Therefore, the final answer must be rounded to one decimal place.
Rounding m to one decimal place: The digit after the first decimal place is 6, which is 5 or greater, so we round up the first decimal digit.
Rounded length = m.

Step 1: Understanding the Concept:

The principle of homogeneity states that we can only add or subtract physical quantities if they have the same dimensional formula.
Step 3: Detailed Explanation:

Let the dimensions be . We are given that and they are all different. Check options: (A) : Multiplication/Division is always allowed. This forms a new quantity. Valid. (B) : For to be valid, must equal . This is a possible specific condition, not strictly impossible for all sets of quantities (e.g., if A=Length, B=Length, C=Length is not allowed as dims are different, but maybe A=Force, B=Length/Force... wait. The question implies "different dimensional formulae". It is possible to find A, B, C such that . So this operation is conditionally valid). (C) : This requires . For subtraction, must equal . But the problem states A and C have different dimensional formulae. Thus, is never dimensionally valid. (D) : Similar to (B), this requires . This is possible to construct (e.g., A=Velocity, C=Time, B=Length). Comparing (C) and (D): (C) demands , which directly contradicts the problem statement ("different dimensional formulae"). Therefore, (C) is strictly impossible.
Step 4: Final Answer:

The combination is invalid.

Step 1: Understanding the Concept:
Significant figures are the digits in a number that are reliable and necessary to indicate the quantity of something. They include all certain digits plus one uncertain digit.
Step 2: Key Rules for Significant Figures:
1. All non-zero digits are significant.
2. Zeros between two non-zero digits are significant.
3. Leading zeros (zeros before non-zero numbers) are not significant.
4. Trailing zeros (zeros at the end of the number) are significant only if the number contains a decimal point.
Step 3: Detailed Explanation:
Let's apply these rules to the number 830600 mm.
- The digits 8, 3, and 6 are non-zero, so they are significant. (3 significant figures so far).
- The zero between 3 and 6 is a captive zero (Rule 2), so it is also significant. (4 significant figures so far: 8, 3, 0, 6).
- The last two zeros (the '00' at the end) are trailing zeros. Since the number 830600 does not have a decimal point, these trailing zeros are not considered significant (Rule 4). They are simply placeholders to indicate the magnitude of the number.
Thus, the significant digits are 8, 3, 0, and 6.
Step 4: Final Answer:
The total count of significant figures in the measurement 830600 mm is 4. Therefore, option (D) is the correct answer.