WORK, ENERGY AND POWER

WORK, ENERGY AND POWER

In your daily life you have seen many things like person pushing heavy box , a teacher teaching students , mom cooking food , all are said to be working . And the person who have capacity to do heavy work is said to be having high stamina or energy. In boxing or in many games you have seen power full punches delivered by the player at a higher speed.

Before we go further detail, we have to learn some important things like multiplication of two vectors. We have two ways. Dot product and cross product in this chapter we talk only about dot product.

Dot product (Scaler product) of two vectors

The dot product of two vectors ​\( \vec{A} \)​ and ​\( \vec{B} \)​ represented as ​\( \vec{A}{.}\vec{B} \)​ read as (vector A dot vector B) which is equal to the product of magnitudes of ​\( \vec{A} \)​ and ​\( \vec{B} \)​ with the cosine of the angle between them.

\[ \vec{A}{.}\vec{B}=\lvert{\vec{A}}\rvert\lvert{\vec{B}}\rvert~{cos~\theta} \]

Since, A , B and cos ​\( \theta \)​ all are scalers that’s why their product is also called scalar product.

 

SOME SPECIAL CASES :

CASE-1 .  When two vectors are parallel to each other.

Then the ​\( \theta \)​ is 0° , ​\( cos~0°=1 \)​ , ​

\[ \vec{A}{.}\vec{B}=\lvert{\vec{A}}\rvert\lvert{\vec{B}}\rvert{cos0°}=\lvert{\vec{A}}\rvert\lvert{\vec{B}}\rvert{(1)} \]

For unit vector 

\[ \hat{i}{.}\hat{i}=\hat{j}{.}\hat{j}=\hat{k}{.}\hat{k}=1 \]

 

 

CASE-2 .  When two vectors are perpendicular to each other.

Then the ​\( \theta \)​ is 90° , ​\( cos~90°=1 \)​ , ​

\[ \vec{A}{.}\vec{B}=\lvert{\vec{A}}\rvert\lvert{\vec{B}}\rvert{cos90°}=\lvert{\vec{A}}\rvert\lvert{\vec{B}}\rvert{(0)}=0 \]

For unit vector 

\[ \hat{i}{.}\hat{j}=\hat{j}{.}\hat{k}=\hat{k}{.}\hat{i}=0 \]

 

CASE-2 .  When vectors are antiparallel to each other.

Then the ​\( \theta \)​ is 180° , ​\( cos~180°=-1 \)​ , ​

\[ \vec{A}{.}\vec{B}=\lvert{\vec{A}}\rvert\lvert{\vec{B}}\rvert{cos180°}=\lvert{\vec{A}}\rvert\lvert{\vec{B}}\rvert{(-1)}=-AB \]

PROPERTIES OF DOT PRODUCT OF TWO VECTORS

(1) Dot product or Scalar product holds commutative property.
i.e. ​\( \vec{A}{.}\vec{B}=\vec{B}{.}\vec{A} \)

(2) Dot product or Scalar product holds distributive property.
i.e. ​\( \vec{A}{.}(\vec{B}+\vec{C})=\vec{A}{.}\vec{B}+\vec{A}{.}\vec{C} \)

(3) Dot product or Scalar product of  a vector to itself gives square of its magnitude. 

i.e. ​​\( \vec{A}{.}\vec{A}=\lvert{\vec{A}}\rvert\lvert{\vec{A}}\rvert~{cos}~0°=A^{2} \)

(4) Also , ​\( \vec{A}{.}(\lambda\vec{B})=\lambda{.}(\vec{A}{.}\vec{B}) \)​ where ​\( \lambda \)​ is a real number.

 

Dot product in cartesian cordinates

Let ​\( \vec{A}=A_{x}\hat{i}+A_{y}\hat{j}+A_{z}\hat{k} \)​ and  ​\( \vec{B}=B_{x}\hat{i}+B_{y}\hat{j}+B_{z}\hat{k} \)

\[ \vec{A}{.}\vec{B}=(A_{x}\hat{i}+A_{y}\hat{j}+A_{z}\hat{k}){.}(B_{x}\hat{i}+B_{y}\hat{j}+B_{z}\hat{k})\\\vec{A}{.}\vec{B}=A_{x}B_{x}(\hat{i}{.}{\hat{i}})+A_{y}B_{y}(\hat{j}{.}{\hat{j}})+A_{z}B_{z}(\hat{k}{.}{\hat{k}})\\\vec{A}{.}\vec{B}=A_{x}B_{x}(1)+A_{y}B_{y}(1)+A_{z}B_{z}(1)\\\vec{A}{.}\vec{B}=A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z} \]

WORK

Work is defined as the

 

CONSERVATIVE AND NON-CONSERVATIVE FORCES

Conservative forces – A force is said to be conservative if the force is path independent

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