Question

A dip circle is so set that its needle moves freely in the magnetic meridian. In this position, the angle of dip is 40°. Now the dip circle is rotated so that the plane in which the needle moves makes an angle of 30° with the magnetic meridian. In this position, the needle will dip by an angle
A. 40°
B. 30°
C. More than 40°
D. Less than 40°

Answer 1

Hint: Use the relation between apparent angle, true angle and angle of dip with meridian. Substitute the given values directly into the formula and find the true angle of dip.
Formula used:
$\cot ^{ 2 }{ \theta \quad =\quad \cot ^{ 2 }{ { \theta }_{ 1 }+\quad \cot ^{ 2 }{ { \theta }_{ 2 } } } }$

Complete answer:
Let $ { \theta }_{ 1 }$ be the apparent angle of dip.
      $ { \theta }_{ 2 }$ be the angle of dip with magnetic meridian.
       $ \theta$ be the true angle of dip.
Given: ${ \theta }_{ 2 }$= 30° and
${ \theta }_{ 1 }$= 40°
Now, we have the relation between apparent angle of dip and true angle of dip,
$\cot ^{ 2 }{ \theta \quad =\quad \cot ^{ 2 }{ { \theta }_{ 1 }+\quad \cot ^{ 2 }{ { \theta }_{ 2 } } } }$
$\Rightarrow \cot { \theta \quad =\sqrt { \cot ^{ 2 }{ { \theta }_{ 1 }+\cot ^{ 2 }{ { \theta }_{ 2 } } } } }$
By substituting the values in above equation we get,
$ \cot { \theta \quad =\sqrt { \cot ^{ 2 }{ 40°+ } \cot ^{ 2 }{ 30° } } }$
$\Rightarrow \cot { \theta \quad =\sqrt { { 1.19 }^{ 2 }+{ \sqrt { 3 } }^{ 2 } } }$
$\Rightarrow \cot { \theta \quad =\sqrt { 1.42+3 } }$
$\Rightarrow \cot { \theta \quad =\quad 4.42 }$
$\Rightarrow \quad \theta =\quad 25°$
Therefore, the true angle of dip is less than 40°.

So, the correct answer is “Option D”.

Note:
There’s an alternate method to solve this problem. The alternate method is shown below.
Formula used to find the true dip is given by,
$\tan { { \theta }_{ 1 }=\dfrac { \tan { \theta } }{ cos{ \theta }_{ 2 } } }$
By substituting the values in above equation we get,
$\tan { 40°=\dfrac { \tan { \theta } }{ cos30° } }$
$\Rightarrow \tan { \theta =\quad \tan { 40° } \times \cos { 30° } }$
$\Rightarrow \tan { \theta =\quad 0.84\quad \times \quad 0.87 }$
$\Rightarrow \quad \theta =\tan ^{ -1 }{ 0.73 }$
$\Rightarrow \quad \theta =\quad 36.13°$
Therefore, the true dip is less than 40°.
Hence, the correct answer is option D i.e. less than 40°.