Distance And Displacement

The angular momentum

%5Cvec%7BL%7D

of a particle about the origin is given by the cross product of the position vector

%5Cvec%7Br%7D

and the linear momentum

%5Cvec%7Bp%7D

%5Cvec%7BL%7D%20%3D%20%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7Bp%7D

The linear momentum

%5Cvec%7Bp%7D

is the product of mass

m

and velocity

%5Cvec%7Bv%7D

%5Cvec%7Bp%7D%20%3D%20m%5Cvec%7Bv%7D

Given

%5Cvec%7Bv%7D%20%3D%205%20%5Chat%7Bi%7D%20%2B%204%20%5Chat%7Bj%7D%20%2B%206%20%5Chat%7Bk%7D

and

%5Cvec%7Br%7D%20%3D%20-2%20%5Chat%7Bi%7D%20%2B%204%20%5Chat%7Bj%7D%20%2B%206%20%5Chat%7Bk%7D

%5Cvec%7Bp%7D%20%3D%20m(5%20%5Chat%7Bi%7D%20%2B%204%20%5Chat%7Bj%7D%20%2B%206%20%5Chat%7Bk%7D)

The cross product

%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7Bv%7D

is calculated as follows:

%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7Bv%7D%20%3D%20%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D%5Chat%7Bi%7D%20%26%20%5Chat%7Bj%7D%20%26%20%5Chat%7Bk%7D%20%5C%5C-2%20%26%204%20%26%206%20%5C%5C5%20%26%204%20%26%206%5Cend%7Barray%7D%5Cright%7C

Expanding the determinant, we get:

%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7Bv%7D%20%3D%20%5Chat%7Bi%7D(4%20%5Ccdot%206%20-%206%20%5Ccdot%204)%20-%20%5Chat%7Bj%7D(-2%20%5Ccdot%206%20-%206%20%5Ccdot%205)%20%2B%20%5Chat%7Bk%7D(-2%20%5Ccdot%204%20-%204%20%5Ccdot%205)

Simplifying further:

%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7Bv%7D%20%3D%20%5Chat%7Bi%7D(24%20-%2024)%20-%20%5Chat%7Bj%7D(-12%20-%2030)%20%2B%20%5Chat%7Bk%7D(-8%20-%2020)

%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7Bv%7D%20%3D%20%5Chat%7Bi%7D(0)%20%2B%20%5Chat%7Bj%7D(42)%20%2B%20%5Chat%7Bk%7D(-28)

%5Cvec%7Br%7D%20%5Ctimes%20%5Cvec%7Bv%7D%20%3D%2042%20%5Chat%7Bj%7D%20-%2028%20%5Chat%7Bk%7D

Multiplying by

m

, the angular momentum is:

%5Cvec%7BL%7D%20%3D%20m(42%20%5Chat%7Bj%7D%20-%2028%20%5Chat%7Bk%7D)

Therefore, the correct answer is:

m(42%20%5Chat%7Bi%7D%20-%2028%20%5Chat%7Bk%7D)

To determine the Vernier constant, we need to find the difference between one main scale division and one Vernier scale division.

Given:

1 main scale division (MSD) = 1 mm
100 Vernier scale divisions (VSD) = 99 main scale divisions

We need to find the length of 1 Vernier scale division (1 VSD):

$1%20%5Ctext%7B%20VSD%7D%20%3D%20%5Cfrac%7B%5Ctext%7Bvalue%20of%2099%20MSDs%7D%7D%7B100%7D%20%3D%20%5Cfrac%7B99%20%5Ctext%7B%20mm%7D%7D%7B100%7D%20%3D%200.99%20%5Ctext%7B%20mm%7D$

Therefore, the difference between one main scale division and one Vernier scale division (the Vernier constant) is:

$%5Ctext%7BVernier%20Constant%7D%20%3D%201%20%5Ctext%7B%20MSD%7D%20-%201%20%5Ctext%7B%20VSD%7D$

$%3D%201%20%5Ctext%7B%20mm%7D%20-%200.99%20%5Ctext%7B%20mm%7D%20%3D%200.01%20%5Ctext%7B%20mm%7D$

Since 0.01 mm is equivalent to 10 micrometers (µm), the Vernier constant is:

10 µm

About Distance And Displacement - OJEE

Distance And Displacement is a vital chapter for OJEE aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

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Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Distance And Displacement carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

About Distance And Displacement - OJEE

Frequently Asked Questions

Why focus on Distance And Displacement PYQs?

How to best use this analysis?

Distance And Displacement

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

About Distance And Displacement - OJEE

Frequently Asked Questions

Why focus on Distance And Displacement PYQs?

How to best use this analysis?

Chapter Questions
2 MCQs