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The Centripetal Force Requirement

Uniform circular motion can be described as the motion of an object in a circle at a constant speed. As an object moves in a circle, it is constantly changing its direction. Because of this direction change, you can be certain that an object undergoing circular motion is accelerating (even if it is moving at constant speed). And in accord with Newton's second law of motion, an accelerating object must be acted upon by an unbalanced force. This unbalanced force is in the same direction as the direction of the acceleration. For objects in uniform circular motion, the net force and subsequent acceleration is directed inwards. It is often said that circular motion requires an inward (or "centripetal") force.

Without a centripetal force, an object cannot travel in circular motion. In fact, if the forces are balanced, then an object in motion continues in motion in a straight line at constant speed. This can be demonstrated by carrying a tennis ball upon a flat, level board. Once the tennis ball and the board are in motion, they will continue in motion in the same direction at the same speed unless acted upon by an unbalanced force.

This is in accord with Newton's first law of motion. But if an unbalanced force is applied to the flat board, then the flat board will accelerate. If the force is continually directed towards a point at the center of the circle, then the flat board will round the corner in a circular-like path. The ball on the other hand will continue to move in the same direction since there is no unbalanced force acting upon it. The board will move out from under the tennis ball. This is illustrated in the animation on the left below.

Now if a block is secured to the board in such a manner that the block applies an unbalanced force to the ball that is directed towards the center of the circle, then quite another phenomenon will be observed. With the block providing a normal force directed inward, the ball can round the corner in a circular-like path. The block supplies the centripetal force required for circular motion. With the centripetal motion requirement met, uniform circular motion can occur. This is illustrated in the animation on the right below.

Without a centripetal force, an object in motion continues along a straight-line path.

With a centripetal force, an object in motion will be accelerated and change its direction.

The tendency of a moving object to continue in a straight line in the absence of an unbalanced force and to turn in a circle in the presence of a inward-directed force (i.e., centripetal force) has been experienced by any passenger in an automobile. When the car makes a sudden turn, the passengers tend to continue in their straight line path. This straight line motion continues until the presence of a side door or another passenger pushes upon the passenger in order to accelerate him/her towards the center of the turn. The force experienced by the passenger is an inward force; without it, the passenger would slide out of the car.

May be many of u would be knowing this.... incase u dnt check this out...

Let A = [a_ij]

then adj (A) = [C_ij] whr C_ij is the cofactor of a_ij

now let A is given by | a₁₁ a₁₂ a₁₃ .......... a_1n |

| a₂₁ a₂₂a₂₃.......... a_2n |

| ..................................|

|...................................|

| a_n1 a_n2a_n3.......... a_nn |

adj (A) = | C₁₁ C₁₂ C₁₃ .......... C_1n|

| C₂₁ C₂₂C₂₃.......... C_2n|

| ................................ ..|

|.................................. .|

| C_n1 C_n2C_n3.......... C_nn |

now we have a property in detrminants which states that "if all the elemnts of a prticular row or coloumn is multiplied with its cofactor then thr algebraic sum will be same as determinant's and if they are multiplied by cofactor of any other's row (or coloumn's ) elemnt, then its value will be 0"

using this property...

we'll be having

adj (A) = | |A| 0 0 .......... 0|

| 0 |A| 0......... 0|

| ...........................|

|............................|

| 0 0 0.........|A| |

here the diagonal elemts will be |A| and rest 0 (by the above property)

thrfore adj (A) = |A| I_n

also thr's this property if all the entries of a determinant of order n is replaced by thr corresponding cofactors then value of new-determinant D is D = |A|^n-1[power cofactor formula]

thrfore using this, here

|adj (A)| = |A|^n-1

this can be genralised to |adj (adj (adj..... (n tymes)..(adj (A))))|= |A|^{[n-1)^n]}

Hopefully It helps.....

Dnt go much in proof... just rember the formula...and the property... that will do

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The Centripetal Force Requirement