The Auxiliary Law Of Power

Introduction:

We know that the rate at which an object works is called ability. To determine the power of an object we use the equation P = W ÷ t. If we know the momentum, mass and applied force over an object, we can determine the power of the object.

Abstract: 

The rate of object’s work is called power. Here we will discuss about the relationship between object’s momentum, mass and applied force. 

Method: 

We know that- v²=u²+2as

                     Or, v²= 2as          [u=0]

                     Or, 2as= v²          [Transforms]

                     Or, as= v²÷2.       [Transforms]

                    So, as= v²÷2__________(i)

Again, We know that- P=W÷t

                         Or, P= mas÷t

                         Or, P= (mv²÷2)×(1÷t)

                         Or, P= mv²÷2t

                          So, P= mv²÷2t

                                   = m²v²÷2mt 

                                   = (mv)²÷2mt

                                   = p²÷2mt.     [p=mv]

                                   = p×p÷2m×t

                                   = Fp÷2m

So, P= Fp÷2m_________(ii)

If Fp= 2Fp then-

P'= 2Fp÷2m

   = 2×(Fp÷2m)

   = 2P

So, P'= 2P

Again, if Fp= ½Fp then-

P'= ½Fp÷2m

   = ½×(Fp÷2m)

   = ½P

P'= ½P

So, P'= ½P

If the value of applied force and momentum increases, then the value of object’s power increases proportionally. If the value of applied force and momentum decreases, the value of object’s power decreases proportionally. 

So, P∞ Fp_________(iii)

If m= 2m then- 

P'= Fp÷2(2m)

   = Fp÷4m

   = ½×(Fp÷2m)

   = ½P

So, P'= ½P

Again, if m= ½m then

P'= Fp÷2(½m)

   = 2Fp÷2m

   = 2×(Fp÷2m)

   = 2P

So, P'= 2P

If the value of object’s mass increases, then the value of object’s power decreases disproportionally. If the value of object’s mass decreases, then the value of object’s power increases disproportionally. 

So, P∞ 1÷m_________(iv)

The graph of iii number equation- 

The graph of iv number equation- 

From iii and iv number equation, we get- 

P∞ Fp_________(iii)

P∞ 1÷m_________(iv)

      P∞ Fp×1÷m

Or, P∞ Fp÷m

Or, P= kFp÷m

So, P= kFp÷m

From ii number equation, we get-  P= Fp÷2m

So, k= ½. k- is a proportional constant. This is unit less. 

So, the main equation- P= Fp÷2m

Result: 

At least-

The value of object’s power is proportional to the value of applied force and momentum and disproportional to the value of object’s mass. 

Conditions of this law: 

There are some conditions of this law. They are-

i) The object should be a moving object. 

ii) i) This formula applies to moving objects in a plane. This formula does not apply to tilted surfaces, freely falling objects and stationary objects.

Conclusion: 

If we know the value of object’s applied force, momentum and mass, we can able to determine the value of object’s power.