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
290.7k+ views
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°.
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°.
Last updated date: 03rd Jun 2023
•
Total views: 290.7k
•
Views today: 3.46k
Recently Updated Pages
Most eubacterial antibiotics are obtained from A Rhizobium class 12 biology NEET_UG

Salamin bioinsecticides have been extracted from A class 12 biology NEET_UG

Which of the following statements regarding Baculoviruses class 12 biology NEET_UG

Sewage or municipal sewer pipes should not be directly class 12 biology NEET_UG

Sewage purification is performed by A Microbes B Fertilisers class 12 biology NEET_UG

Enzyme immobilisation is Aconversion of an active enzyme class 12 biology NEET_UG

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
