Which of the following physical quantities are dimensionless, that is, have no fundamental physical dimensions?

Introduction / Context:
Physical quantities are often classified by their dimensions and units. Some quantities have dimensions of length, mass or time, while others are ratios of similar quantities and therefore dimensionless. This question asks which of the listed quantities angle, strain and specific gravity are dimensionless in the sense that they do not have fundamental physical dimensions, even though they may have units for convenience.

Given Data / Assumptions:

• Angle can be measured in radians or degrees.
• Strain is defined as change in length divided by original length.
• Specific gravity is defined as a ratio of densities.
• We examine each quantity in terms of base dimensions like length, mass and time.

Concept / Approach:
A dimensionless quantity is one that can be expressed as a pure number, usually as a ratio of two quantities that share the same dimensions. Strain is change in length divided by original length, so both numerator and denominator have the dimension of length, which cancel out. Specific gravity is the ratio of the density of a substance to the density of a reference substance, usually water, so mass per volume in numerator and denominator cancel. Angle in radians is defined as arc length divided by radius, again a ratio of lengths. As a result, all three quantities angle, strain and specific gravity are dimensionless, even though we sometimes attach words like radian to angle or specify conditions for specific gravity.

Step-by-Step Solution:
Step 1: Consider strain, which equals change in length divided by original length. The dimension of both lengths is L, so strain has dimension L divided by L, which equals 1, making it dimensionless. Step 2: Consider specific gravity, defined as density of a substance divided by density of water. Each density has dimension mass per length cubed, or M divided by L^3, so the ratio again has dimension 1. Step 3: Consider angle in radians, defined as arc length divided by radius. Both arc length and radius have dimension L, so their ratio has dimension 1. Step 4: Since all three quantities are ratios of similar dimensional quantities, they all are dimensionless. Step 5: Therefore the correct option is that all of these quantities are dimensionless.

Verification / Alternative check:
Dimensional analysis tables in physics books usually list examples of dimensionless quantities. Strain is always listed with no dimensions, just a number. Specific gravity is given without fundamental dimensions, though it may sometimes be described as a number such as 0.8 or 1.2. Angle in radians is treated as a pure number, because one full circle is 2 times pi radians, and pi is a dimensionless constant. These references confirm that all three quantities are dimensionless as far as physical dimensions are concerned.

Why Other Options Are Wrong:
Angle only, strain only or specific gravity only each wrongly suggest that the other two quantities have dimensions, which is not correct.
None of these is clearly false because standard definitions show all three as ratios of like quantities and therefore dimensionless.

Common Pitfalls:
Students sometimes think that angle has a dimension because it is measured in radians or degrees, but these are just convenient labels for a pure ratio. Similarly, they may treat specific gravity as having mass or volume units, forgetting that it is a ratio of two densities. To avoid confusion, always check whether a quantity definition involves a ratio of identical dimensions; if so, the resulting quantity is dimensionless.

Final Answer:
All three listed quantities angle, strain and specific gravity are dimensionless, so the correct choice is that all of these are dimensionless.