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Motion in A Plane

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Physics Part I

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Physics

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CBSE Gujarat Board Haryana Board

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Class 10 Class 12

What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop?

  • square root of 2 gR end root
  • square root of 3 gR end root
  • square root of 5 gR end root
  • square root of gR

C.

square root of 5 gR end root

The question is illustrated in the figure below,

Let, the tension at point A be TA.

Using Newton's second law, we have

straight T subscript straight A space minus space mg space equals space mv subscript straight c squared over straight R
Energy at point A = 1 half m v subscript o squared space space space space space space space space space... space left parenthesis i right parenthesis
Energy at point C is,

1 half m v subscript c squared space plus space m g space x space 2 R space space space space space space space space space space... space left parenthesis i i right parenthesis
At point C, using Newton's second law,

straight T subscript straight c space plus space mg space equals space mv subscript straight c squared over straight R
In order to complete a loop, Tc greater-than or slanted equal to space 0
So,
 mg space equals space mv subscript straight c squared over straight R
rightwards double arrow space straight v subscript straight c space equals space square root of gR space space space space space space space space... space left parenthesis iii right parenthesis

From equation (i) and (ii)

Using the principle of conservation of energy,

1 half m v subscript o squared space equals space 1 half m v subscript c squared space plus space 2 space m g R
rightwards double arrow space 1 half m v subscript o squared italic space italic equals italic space italic 1 over italic 2 m g R italic space italic plus italic space italic 2 m g R italic space x italic space italic 2
italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space open square brackets v subscript c italic space italic equals italic space square root of g R end root close square brackets
italic rightwards double arrow italic space v subscript o to the power of italic 2 italic space italic equals italic space g R italic space italic plus italic space italic 4 g R
italic rightwards double arrow italic space italic space v subscript o italic space italic equals italic space square root of italic 5 g R end root

The question is illustrated in the figure below,

Let, the tension at point A be TA.

Using Newton's second law, we have

straight T subscript straight A space minus space mg space equals space mv subscript straight c squared over straight R
Energy at point A = 1 half m v subscript o squared space space space space space space space space space... space left parenthesis i right parenthesis
Energy at point C is,

1 half m v subscript c squared space plus space m g space x space 2 R space space space space space space space space space space... space left parenthesis i i right parenthesis
At point C, using Newton's second law,

straight T subscript straight c space plus space mg space equals space mv subscript straight c squared over straight R
In order to complete a loop, Tc greater-than or slanted equal to space 0
So,
 mg space equals space mv subscript straight c squared over straight R
rightwards double arrow space straight v subscript straight c space equals space square root of gR space space space space space space space space... space left parenthesis iii right parenthesis

From equation (i) and (ii)

Using the principle of conservation of energy,

1 half m v subscript o squared space equals space 1 half m v subscript c squared space plus space 2 space m g R
rightwards double arrow space 1 half m v subscript o squared italic space italic equals italic space italic 1 over italic 2 m g R italic space italic plus italic space italic 2 m g R italic space x italic space italic 2
italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space open square brackets v subscript c italic space italic equals italic space square root of g R end root close square brackets
italic rightwards double arrow italic space v subscript o to the power of italic 2 italic space italic equals italic space g R italic space italic plus italic space italic 4 g R
italic rightwards double arrow italic space italic space v subscript o italic space italic equals italic space square root of italic 5 g R end root
5996 Views

Give three examples of vector quantities.

Force, impulse and momentum.
865 Views

What is a vector quantity?

A physical quantity that requires direction along with magnitude, for its complete specification is called a vector quantity.
835 Views

What is a scalar quantity?

A physical quantity that requires only magnitude for its complete specification is called a scalar quantity.
1212 Views

Give three examples of scalar quantities.

Mass, temperature and energy
769 Views

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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