Question

The only mechanical quantity which has a negative dimension of mass is

• A. Angular momentum
• B. Torque
• C. Coefficient of thermal conductivity
• D. Gravitational constant

Answer 1

The correct option is D

Gravitational constant

The explanation for the correct option:

In the case of option D,

1. The gravitational constant often denoted by G is an empirical physical constant used in the computation of gravitational effects in both Albert Einstein's theory of general relativity and Sir Isaac Newton's law of universal gravitation.
2. The general formula of gravitational constant is $G=\frac{F{r}^2}{m_1{m}_2}$ where $F=$Force, $r=$the distance between two objects and $m_1,m_2=$mass.
3. The dimensional formula for force is calculated using the formula of force $F=ma=\left[ML{T}^{-2}\right]$
4. The dimensional formula of $G$ is $\frac{\left[ML{T}^{-2}\right]\left[L^2\right]}{\left[M\right]\left[M\right]}=\left[M^{-1}{L}^3{T}^{-2}\right]$
5. So, from the dimension, It is clear that the gravitational constant has the negative dimension of mass.

Explanation of the incorrect options:

In the case of option A,

1. Angular momentum is calculated using the formula $L=mvr$.
2. Where $m$ is the mass, $v$ is the velocity, and $r$ is the radius.
3. The dimension can be calculated as- $\left[M\right]\left[L{T}^{-1}\right]\left[L\right]$.
4. Thus, The dimension of angular momentum is $\left[M{L}^2{T}^{-1}\right]$.
5. So, it can be seen that it has a positive dimension of mass.

In the case of option B,

1. Torque is defined as $\tau =F\times r=\left[ML{T}^{-2}\right]\left[L\right]$ .
2. Thus, the torque has the positive dimension of the mass because its dimension is $\left[M{L}^2{T}^{-2}\right]$.

In the case of option C,

1. From the formula heat conduction is defined as- $\frac{dQ}{dt}=-kA\frac{dT}{dx}$where, $\frac{dQ}{dt}$ is heat transfer, $\frac{dT}{dx}$ is temperature gradient, $A$is area perpendicular to the direction of heat transfer
2. The dimension of heat $Q=work=F\times d=\left[M{L}^2{T}^{-2}\right]$.
3. The dimension for temperature is $\left[K\right]$.
4. The dimension of the coefficient of thermal conductivity will be calculated as- $k=\frac{Qdx}{AdtdT}=\frac{\left[M{L}^2{T}^{-2}\right]\left[L\right]}{\left[L^2\right]\left[K\right]\left[T\right]}$
5. Thus, the dimension of thermal conductivity is obtained as $\left[ML{T}^{-3}{K}^{-1}\right]$.
6. So, it is clear that it has a positive dimension of mass.

Hence, option D is the correct answer.