1 Marks Question

Question 11 Mark

Give an example of:
A constant which has no unit.

Answer

Reynold's number is a constant which has no unit.

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Question 21 Mark

Give an example of:
A constant which has a unit.

Answer

Gravitational constant $(\text{G})=6.67\times10^{-11}\text{N-m}^2/\text{kg}^2$

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Question 31 Mark

If velocity of light $c,$ Planck’s constant h and gravitational contant $G$ are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.

Answer

We have to apply principle of homogeneity to solve this problem.
Principle of homogeneity states that in a correct equation, the dimensions of each term added or subtracted must be same,
i.e., dimensions of $\text{LHS}$ and $\text{RHS}$ should be equal, We know that, dimensions of,$[\text{h}]=[\text{ML}^2\text{T}^{-1}],[\text{c}]=[\text{LT}^{-1}],\text[{G}]=[\text{M}^{-1}\text{L}^3\text{T}^{-2}]$

Let $\text{m}\propto\text{c}^\text{x}\text{h}^\text{v}\text{G}^\text{z}$

$\Rightarrow\text{m}=\text{kc}^\text{a}\text{h}^\text{b}\text{G}^\text{c}\ \ \ \ ...(\text{i})$
Where, $k$ is a dimensionless constant of proportionality.
Substituting dimensions of each term in Eq. $(i)$, we get,
$[\text{ML}^0\text{T}^0]=[\text{LT}^{-1}]^\text{a}\times[\text{ML}^2\text{T}^{-1}]^\text{b}[\text{M}^{-1}\text{L}^3\text{T}^{-2}]^\text{c}$
Comparing powers of same terms on both sides, we get,
$\text{b}-\text{c}=1\ \ \ ...(\text{ii})$
$\text{a}+2\text{b}+3\text{c}=0\ \ \ \ ...(\text{iii})$
$-\text{a}-\text{b}-2\text{c}=0\ \ \ \ ...(\text{iv})$
Adding Eqs. $(ii), (iii)$ and $(iv),$ we get,
$2\text{b}=1\Rightarrow\text{b}=\frac{1}{2}$
Substituting value of $b$ in Eq. $(ii),$ we get,
$\text{c}=-\frac{1}{2}$
From Eq. $(iv)$
$\text{a}=-\text{b}-2\text{c}$
Substituting values of $b$ and $c,$ we get,
$\text{a}=-\frac{1}{2}-2\Big(-\frac{1}{2}\Big)=\frac{1}{2}$
Putting values of $a, b$ and $c$ in Eq. $(i)$, we get,
$\text{m}=\text{kc}^\frac{1}{2}\text{h}^\frac{1}{2}\text{G}^{-\frac{1}{2}}=\text{k}\sqrt{\frac{\text{ch}}{\text{G}}}$

Let $\text{L}\propto\text{c}^\text{a}\text{h}^\text{b}\text{G}^\text{c}$

$\Rightarrow\text{L}=\text{kc}^\text{a}\text{h}^\text{b}\text{G}^\text{c}\ \ \ \ ...(\text{v})$
Where $k$ is a dimensionless constant.
Substituting dimensions of each term in Eq. $(v)$, we get
$[\text{M}^0\text{LT}^0]=[\text{LT}^{-1}]^\text{a}\times[\text{ML}^2\text{T}^{-1}]^\text{b}\times[\text{M}^{-1}\text{L}^3\text{T}^{-2}]^\text{c}$
$=[\text{M}^{\text{b}-\text{c}\ }\text{L}^{\text{a}+2\text{b}+3\text{c}}\ \text{T}^{-\text{a}-\text{b}-2\text{c}}]$
On comparing powers of same terms, we get,
$\text{b}-\text{c}=0\ \ \ ...(\text{vi})$
$\text{a}+2\text{b}+3\text{c}=1\ \ \ ...(\text{vii})$
$-\text{a}-\text{b}-2\text{c}=0\ \ \ \ ...(\text{viii})$
Adding Eqs. $(vi), (vii)$ and $(viii),$ we get,
$2\text{b}=1\Rightarrow\text{b}=\frac{1}{2}$
Substituting value of $b$ in Eq. $(vi),$ we get,
$\text{c}=\frac{1}{2}$
From Eq. $(viii), \text{a}=-\text{b}-2\text{c}$
Substituting values of $b$ and $c,$ we get,
$\text{a}=-\frac{1}{2}-2\Big(\frac{1}{2}\Big)=-\frac{3}{2}$
Putting values of $a, b$ and $c$ in Eq. $(v)$, we get,
$\text{L}=\text{kc}^{-\frac{3}{2}}\text{h}^\frac{1}{2}\text{G}^\frac{1}{2}=\text{k}\sqrt{\frac{\text{hG}}{\text{c}^3}}$

Let $\text{T}\propto\text{c}^\text{a}\text{h}^\text{b}\text{G}^\text{c}$

$\Rightarrow\text{T}=\text{c}^\text{a}\text{h}^\text{b}\text{G}^\text{c}\ \ \ \ ...(\text{ix})$
Where, $k$ is a dimensionless constant.
Substituting dimensions of each term in Eq. $(ix),$ we get
$[\text{M}^0\text{L}^0\text{T}^1]=[\text{LT}^{-1}]^\text{a}\times[\text{ML}^2\text{T}^{-1}]^\text{b}\times[\text{M}^{-1}\text{L}^3\text{T}^{-2}]^\text{c}$
$=[\text{M}^{\text{b}-\text{c}}\text{ L}^{\text{a}+2\text{b}+3\text{c}}\ \text{T}^{-\text{a}-\text{b}-2\text{c}}]$
On comparing powers of same terms, we get,
$\text{b}-\text{c}=0 \ \ \ ...(\text{x})$
$\text{a}+2\text{b}+3\text{c}=1\ \ \ ...(\text{xi})$
$-\text{a}-\text{b}-2\text{c}=1\ \ \ \ ...(\text{xii})$
Adding Eqs. $(x), (xi)$ and $(xii)$, we get,
$2\text{b}=1\Rightarrow\text{b}=\frac{1}{2}$
Substituting the value of $b$ in Eq. $(x)$, we get,
$\text{c}=\text{b}=\frac{1}{2}$
From Eq. $(xii),$
$\text{a}=-\text{b}-2\text{c}-1$
Substituting values of $b$ and $c,$ we get,
$\text{a}=-\frac{1}{2}-2\Big(\frac{1}{2}\Big)-1=-\frac{5}{2}$
Putting values of $a, b$ and $c$ in Eq. $(ix),$ we get,
$\text{T}=\text{kc}^{\frac{-5}{2}}\text{h}^\frac{1}{2}\text{G}^\frac{1}{2}=\text{k}\sqrt{\frac{\text{hG}}{\text{c}^5}}$

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Question 41 Mark

Calculate the length of the arc of a circle of radius 31.0cm which subtends an angle of $\frac{\pi}{6}$ at the centre.

Answer

$\text{Angle}=\frac{\text{length of arc}}{\text{radius of arc}}$$\frac{\pi}{6}=\frac{\text{x}}{31}$
$\text{x}=\frac{31\times\pi}{6}=\frac{31\times3.14}{6}=16.22\text{cm}.$

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Question 51 Mark

Give an example of:
A physical quantity which has neither unit nor dimensions.

Answer

$\text{Specific density}=\frac{\text{Density of medium}}{\text{Density of water at}\ 4^\circ\text{C}}$It is a ratio of two same quantities. So, it is a unitless and dimensionless constant.

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Question 61 Mark

Give an example of:
A physical quantity which has a unit but no dimensions.

Answer

Solid angle $\Omega=\frac{\text{A}}{\text{r}^2}$ steradian and a plane angle $\theta=\frac{\text{L}}{\text{r}}$ radian. Both are dimensionless but have units.

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