0

I am reading kleppner,it said when change in a vector is always perpendicular to itself then vector's magnitude doesn't change only direction changes.

I'm having a contradiction with my thought process here's how....

In limit Δt tends to zero if vectors change is perpendicular to itself then there would be a infinitesimal change in magnitude of vector (visualize triangle law of vector addition) a finite time interval is made up of infinite such infinitesimal intervals then wouldn't this infinitesimal change in magnitude would add up to a finite change in a finite time interval??? which means that radius of a uniform circular motion should change in time???

Chh Unnu
  • 1

2 Answers2

1

You are multiplying an infinitesimal by infinity and expecting the result to be meaningful when it is actually undefined. This is leading you astray.

There are various ways to see that the magnitude of the velocity is unchanged. My favourite is to write the magnitude as the dot product:

$$ |v|^2 = \mathbf v \cdot \mathbf v $$

If we differentiate this wrt time we get:

$$ 2 |v| \frac{d|v|}{dt} = 2 \mathbf v \cdot \mathbf a $$

If $\mathbf a$ is normal to $\mathbf v$ then the dot product on the right hand side is zero. Since $|v| \ne 0$ this means:

$$ \frac{d|v|}{dt} = 0 $$

John Rennie
  • 355,118
0

Usually, when we talk in physics, all our 'basis' vectors are unit length. If we have a set of three orthonormal basis vectors of form $\{ e_1, e_2 , e_3\}$ then a vector in can be written as:

$$ V= \sum_{i=0}^3 V^i e_i$$

Now suppose you were to differentiate this vector with time, let's say this was a velocity vector.

$$ \frac{dV}{dt} = \sum_{i=0}^3 \frac{dV^i}{dt} e_i + V^i \frac{de_i}{dt}$$

So, consider the $ \frac{de_i}{dt}$ term, clearly since it's unit length by definition, it's length can not increase. This means that the only way that it can change with time is rotations.

These type of transformations are useful when you have rotating coordinate systems

See this answer and linked post to question:

tryst with freedom
  • 7,940