Units And Measurement

The correct option is (D) : Watt/m²K⁴

Stefan's law is E=σ(T⁴ )

σ=

where, E= =Watt/ m² σ= Watt/m²k⁴

Dimensional Analysis for Length

The question asks for the combination of the fundamental constants: Planck's constant (h), the speed of light (c), and Newton's gravitational constant (G) that gives the dimension of length.

Dimensional Formulae of Constants:

• (Planck's constant):
• (speed of light):
• (Newton's gravitational constant):

Finding the Combination of Constants with Dimension of Length:

We want to find the correct combination that results in the dimension of length (L).

Consider the following combination:

Combination:

Let's calculate the dimensional formula for this combination:

The dimensional formula for each constant is:

Now calculate the dimensional formula for :

For : =

For :

Now, combining the two parts:

\frac{M⁰ L².5 T⁻1.5}{L^1.5 T⁻1.5}$ =\)

Therefore, the combination gives the dimension of length.

Conclusion:

The correct combination of constants that has the dimension of length is: .

To determine the diameter of the wire using the screw gauge readings, we need to understand the relationship between the main scale and the circular scale.

1. The Main Scale Reading (MSR) is given as .
2. The Circular Scale Reading (CSR) is given as divisions.
3. It's also provided that on the main scale corresponds to divisions on the circular scale. Hence, each division on the circular scale corresponds to:
4. The total reading on the screw gauge is calculated as follows:
5. Substituting the values, we have:
6. This simplifies to:
7. Convert the measurement from millimeters to centimeters, since the options are in centimeters:
8. Therefore, the diameter of the wire is: 0.052 cm.
9. Comparing with the given options, the correct answer is 0.052 cm. This matches with the provided correct answer choice.

This clear step-by-step computation explains why the diameter of the wire measures 0.052 cm.

To find the dimensions of energy using the given fundamental physical quantities: Force [F], Acceleration [A], and Time [T], we need to express energy in terms of these quantities.

Energy is typically defined as work done, which can be calculated using the formula:

We know that:

• Force is already given as a fundamental quantity.
• Distance can be expressed as the product of Velocity and Time:

Also, Velocity can be expressed in terms of Acceleration and Time:

Substituting this into the distance formula:

Thus, Energy becomes:

Therefore, the dimensions of energy in terms of [F], [A], and [T] are:

This matches the provided correct answer option: [F] [A] [T²].

To find the maximum possible percentage error in the measurement of density, we use the formula for density, which is given by:

Density ( ) =

The volume of the wire, which is a cylinder, is calculated by:

where is the radius and is the length of the wire.

The percentage error in density, using error propagation, can be calculated by:

Given:

Mass, g, g

Radius, mm = 0.03 cm, mm = 0.0001 cm

Length, cm, cm

Calculate percentage errors:

Percentage error in density:

Rounding to nearest significant digits, the maximum possible percentage error is about 1.6%.

To solve the problem of finding the area of the rectangular field and rounding it to the correct number of significant figures, follow these steps:

1. Calculate the area using the formula for the area of a rectangle:

Area = Length × Breadth

2. Given:

Length = 55.3 m (3 significant figures), Breadth = 25 m (2 significant figures)

3. Calculate the area:

Area = 55.3 m × 25 m = 1382.5 m²

4. The result should be rounded to the least number of significant figures in the given dimensions, which is 2.

5. Rounding 1382.5 to 2 significant figures, we get 1400 m², which can be expressed in scientific notation as 1.4 × 10³.

6. To match the options presented, this is equivalent to 14 × 10².

Therefore, the correct answer is: 14 × 10²

• Random Errors: These are caused by unpredictable variations in measurement conditions. When parameters like temperature and voltage fluctuate unexpectedly, they can induce variability in measurements, leading to errors. Such alterations happen without any regular pattern or fixed cause, thus being termed random.
• Other Types of Errors:
  • Instrumental Errors: These result from malfunctions or limitations of the measuring instruments themselves, not typically due to environmental changes.
  • Personal Errors: These occur due to the observer's subjective judgement or error in reading measurements, unrelated to external fluctuations.
  • Least Count Errors: These arise when the measurement device cannot read smaller divisions beyond certain limits, a fixed characteristic rather than an environmental variation.

Least count (LC) of screw gauge
Main scale reading (MSR) = 1 mm
Circular scale reading (CSR) = 63 divisions
Final reading = MSR + CSR LC =