Question 294: Units , Dimensions and Error Analysis

Question

Planck's constant h, speed of light c and gravitational constant G are used to form a unit of length L and a
unit of mass M. Then the correct option(s) is(are)

Accepted Answer

The dimensions of Planck's constant is $$h = [{M^1}{L^3}{T^{ - 2}}]$$
Speed of light is $$c = [{L^1}{T^{ - 1}}]$$
Gravitational constant is $$G = [{M^{ - 1}}{L^3}{T^{ - 2}}]$$
Let $$L \propto {h^x}{c^y}{G^z}$$.
$$[L] = {[{M^1}{L^2}{T^{ - 1}}]^x}{[{L^1}{T^{ - 1}}]^y}{[{M^{ - 1}}{L^3}{T^{ - 2}}]^z}$$
Comparing, we get
$$\left. {\matrix{
   \hfill {x - z = 0} \cr 
   \hfill {2x + y + 3z = 1} \cr 
   \hfill { - x - y - 2z = 0} \cr

} } \right\}$$
Solving, we get
$$x = z$$
$$y + 5x = 1$$
$$ - y - 3x = 0$$
or, $$2x = 1$$
$$x = {1 \over 2} = z$$
$$y =  - {3 \over 2}$$
Therefore,
$$L \propto {h^{1/2}}{C^{ - 3/2}}{G^{1/2}}$$
$$L \propto \sqrt h $$
$$L \propto \sqrt G $$
Let $$M \propto {h^a}{C^b}{G^c}$$
$$[M] = {[{M^1}{L^2}{T^{ - 1}}]^a}{[{L^1}{T^{ - 1}}]^b}{[{M^{ - 1}}{L^3}{T^{ - 2}}]^c}$$
Comparing, we get $$a - c = 1$$.
$$\left. \matrix{
  2a + b + 3c = 0 \hfill \cr 
   - a - b - 2c = 0 \hfill \cr}  \right\}a + c = 0$$
Therefore,
$$2a = 1$$
$$a = {1 \over 2}$$
$$c =  - {1 \over 2}$$
$$b = {1 \over 2}$$
Therefore,
$$M \propto {h^{1/2}}{c^{1/2}}{G^{ - 1/2}}$$
$$M \propto \sqrt c $$

Answer

$$M \propto \sqrt c $$

Answer

$$M \propto \sqrt G $$

Answer

$$L \propto \sqrt h $$

Answer

$$L \propto \sqrt G $$

Units , Dimensions and Error Analysis

Planck's constant h, speed of light c and gravitational constant G are used to form a unit of length L and a unit of mass M. Then the correct option(s) is(are)

Choose the correct answer: