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Motion in A Plane

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Physics Part I

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Physics

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Class 10 Class 12

A particle  moves so that its position vector is given by r = cos ωt space straight x with hat on top space plus space sin space ωt space straight y with hat on top where straight omega is a cosntant. which of the following is true?

  • velocity and acceleration both are parallel to r

  • velocity is perpendicular to r and acceleration is directed towards the origin

  • velocity is perpendicular to r and acceleration is directed away from the origin

  • velocity and acceleration both are perpendicular to r


B.

velocity is perpendicular to r and acceleration is directed towards the origin

Position vector of the article is given by,
straight r space equals space cos space ωt space stack straight x space with hat on top space plus space sin space ωt space straight y with hat on top
where straight omega is a constant.
Velocity of the particle is,

straight v space equals space dr over dt space equals space straight d over dt open parentheses cos space straight omega space straight t space straight x with hat on top space plus space sin space ωt space straight y with hat on top close parentheses
straight v space equals space left parenthesis negative sinωt right parenthesis straight omega stack straight x space with hat on top space plus space left parenthesis cos space ωt right parenthesis straight omega straight y with hat on top

space space space equals space minus space straight omega space left parenthesis space sin space ωt space straight x with hat on top space minus space cos space ωt space straight y with hat on top right parenthesis

Acceleration space of space the space particle comma
straight a space equals space dv over dt
space space equals straight d over dt open square brackets negative straight omega space sin space ωt space straight x with hat on top space plus straight omega space cos space ωt space straight y with hat on top close square brackets
space equals space minus straight omega squared space cos space ωt space straight x with hat on top space minus space straight omega squared space sin space ωt space straight y with hat on top
space equals space minus straight omega squared space open parentheses cos space ωt space straight x with hat on top space space minus sin space ωt space straight y with hat on top space close parentheses
straight a space equals space minus straight omega squared space straight r space equals straight omega squared space left parenthesis negative straight r right parenthesis space
The particle is at a point P. That is, its position vector is directed as shown below:

Therefore, acceleration is directed towards -r, that is towards "O"

We have

straight v with rightwards harpoon with barb upwards on top. r with rightwards harpoon with barb upwards on top space equals space minus straight omega space left parenthesis sinω space straight t space straight x with hat on top space minus space cos space ωt space straight y with hat on top right parenthesis.
space space space space space space space space space space space space space space space space space space space left parenthesis cos space ωt space straight x with hat on top space plus space sin space ωt space straight y with hat on top right parenthesis 

       equals space minus straight omega space left square bracket space sin space straight omega. space cosωt space plus
space space space space space space space space space space space space 0 plus 0 space minus space sin space ωt. space cos space ωt right square bracket 

equals space minus straight omega space left parenthesis 0 right parenthesis space equals space 0

rightwards double arrow space straight v perpendicular space straight r

Position vector of the article is given by,
straight r space equals space cos space ωt space stack straight x space with hat on top space plus space sin space ωt space straight y with hat on top
where straight omega is a constant.
Velocity of the particle is,

straight v space equals space dr over dt space equals space straight d over dt open parentheses cos space straight omega space straight t space straight x with hat on top space plus space sin space ωt space straight y with hat on top close parentheses
straight v space equals space left parenthesis negative sinωt right parenthesis straight omega stack straight x space with hat on top space plus space left parenthesis cos space ωt right parenthesis straight omega straight y with hat on top

space space space equals space minus space straight omega space left parenthesis space sin space ωt space straight x with hat on top space minus space cos space ωt space straight y with hat on top right parenthesis

Acceleration space of space the space particle comma
straight a space equals space dv over dt
space space equals straight d over dt open square brackets negative straight omega space sin space ωt space straight x with hat on top space plus straight omega space cos space ωt space straight y with hat on top close square brackets
space equals space minus straight omega squared space cos space ωt space straight x with hat on top space minus space straight omega squared space sin space ωt space straight y with hat on top
space equals space minus straight omega squared space open parentheses cos space ωt space straight x with hat on top space space minus sin space ωt space straight y with hat on top space close parentheses
straight a space equals space minus straight omega squared space straight r space equals straight omega squared space left parenthesis negative straight r right parenthesis space
The particle is at a point P. That is, its position vector is directed as shown below:

Therefore, acceleration is directed towards -r, that is towards "O"

We have

straight v with rightwards harpoon with barb upwards on top. r with rightwards harpoon with barb upwards on top space equals space minus straight omega space left parenthesis sinω space straight t space straight x with hat on top space minus space cos space ωt space straight y with hat on top right parenthesis.
space space space space space space space space space space space space space space space space space space space left parenthesis cos space ωt space straight x with hat on top space plus space sin space ωt space straight y with hat on top right parenthesis 

       equals space minus straight omega space left square bracket space sin space straight omega. space cosωt space plus
space space space space space space space space space space space space 0 plus 0 space minus space sin space ωt. space cos space ωt right square bracket 

equals space minus straight omega space left parenthesis 0 right parenthesis space equals space 0

rightwards double arrow space straight v perpendicular space straight r
3747 Views

Give three examples of scalar quantities.

Mass, temperature and energy
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What is a vector quantity?

A physical quantity that requires direction along with magnitude, for its complete specification is called a vector quantity.
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What are the basic characteristics that a quantity must possess so that it may be a vector quantity?

A quantity must possess the direction and must follow the vector axioms. Any quantity that follows the vector axioms are classified as vectors. 


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What is a scalar quantity?

A physical quantity that requires only magnitude for its complete specification is called a scalar quantity.
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Give three examples of vector quantities.

Force, impulse and momentum.
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