Question

Which of the following have the same dimensional formula?
a) Tension and surface tension
b) Year and light year
c) Strain and angle
d) None of these

Answer 1

- Hint: This type of question must be solved by checking each option one by one because they do not have a direct answer. Find the expressions for each of these quantities in terms of known quantities and compare.

Complete step-by-step solution -
To solve the problem, we will first find the dimensional formula for both the quantities in each case and then compare both of them.
a) Tension and surface tension
Tension is a force experienced within a body to resist another force applied externally. For example, when we pull a rope or a rubber band, there is tension(force) generated within it which resists the pulling force. Since tension is a type of force, it will also have the same dimensional formula as the dimensional formula of force i.e. $\left[ \text{ML}{{\text{T}}^{\text{-2}}} \right]$.
Like rigid bodies, liquids also experience a tension on their surface due to the force of attraction from the inner molecules. But, here it is calculated for one unit length and is called surface tension. Surface tension is the ratio of the surface force applied on the liquid to the length along which the force acts. So, surface tension is force divided by length. Therefore, the dimensional formula will be $\dfrac{\left[ Force \right]}{\left[ length \right]}=\dfrac{\left[ \text{ML}{{\text{T}}^{\text{-2}}} \right]}{\left[ L \right]}=\left[ \text{M}{{\text{T}}^{\text{-2}}} \right]$
Now, if you compare both, the dimensional formulas are not the same.
b) Year and light year
We already know the dimensional formula of year i.e [T]. Light year is the distance the light travels within one year. Since light year is actually a distance, its dimensional formula will be [L]. Therefore, year and light year do not have the same dimensional formula.
c) Strain and angle
When an external force is applied on a body, there is some change in the shape and size of the body. Stain is the ratio of the amount of change of a particular component of the body to the original value of that component. For example, suppose we stretch a rubber band of length 2cm to 4cm then the strain in the length of the body will be 1, since the change of length is 2cm and the original length was also 2cm. As we can see, strain is a ratio of two similar quantities. The ratio will be of two same dimensional formulas which will be 1. Hence, strain is just a number without any dimensional formula.
We already know angle also does not have any dimensional formula; it is also just a number with no unit.
Hence the correct answer is (d)none of these, since the first pairs do not have the same dimensional formula and the in (c) both of them are dimensionless quantities,

Note: Sometimes it is misunderstood that angle has a dimension or unit of radian/degrees. Angle is the ratio of length of the arc to the radius of the circle and radius is also a length. So, it is the ratio of two same dimensions. Hence, it is a dimensionless quantity. Radian and degrees are symbols used to represent the angle so that it is not confused with the other numbers.