The equation,
$$ a_c=\frac{4\pi^2r}{T^2} $$
Can be expressed as:
$$ F_c=\frac{m4\pi^2r}{T^2} $$
I am confused as to how to arrive at this second equation, and the relationship between these two equations.
centripetal-forcenewtonian-mechanics
The equation,
$$ a_c=\frac{4\pi^2r}{T^2} $$
Can be expressed as:
$$ F_c=\frac{m4\pi^2r}{T^2} $$
I am confused as to how to arrive at this second equation, and the relationship between these two equations.
Best Answer
If you have some object moving in a circle with a velocity $v$, then as any Physics textbook will tell you the acceleration towards the centre is:
$$a = \frac{v^2}{r}$$
To get the velocity we note that the circumference of the circle is $2\pi r$, so if the object takes a time $T$ to go round the circle the velocity is just distance divided by time:
$$ v = \frac{2\pi r}{T}$$
so
$$ v^2 = \frac{4\pi^2 r^2}{T^2}$$
and if you put this expression for $v^2$ in the first equation it gives:
$$ a = \frac{4\pi^2 r}{T^2}$$
The way to get the force is to note that Newton's first law tells us:
$$ F = ma $$
where $m$ is the mass of the moving object. Put our expression for $a$ into this equation and we get;
$$ F = m\frac{4\pi^2 r}{T^2}$$
or as you have written it:
$$ F = \frac{m4\pi^2 r}{T^2}$$