Question

please here is the figure !!!​

Answer 1

The magnitude of the centripetal force on an object of mass m moving at tangential speed v along a path with radius of curvature r is:[6]

{\displaystyle F_{c}=ma_{c}={\frac {mv^{2}}{r}}} {\displaystyle F_{c}=ma_{c}={\frac {mv^{2}}{r}}}

{\displaystyle a_{c}={\frac {v}{t}}{\hat {r}}={\frac {r\omega }{t}}{\hat {r}}=v\omega ={\frac {v^{2}}{r}}} {\displaystyle a_{c}={\frac {v}{t}}{\hat {r}}={\frac {r\omega }{t}}{\hat {r}}=v\omega ={\frac {v^{2}}{r}}}

where {\displaystyle a_{c}} a_c is the centripetal acceleration. The direction of the force is toward the center of the circle in which the object is moving, or the osculating circle (the circle that best fits the local path of the object, if the path is not circular).[7] The speed in the formula is squared, so twice the speed needs four times the force. The inverse relationship with the radius of curvature shows that half the radial distance requires twice the force. This force is also sometimes written in terms of the angular velocity ω of the object about the center of the circle, related to the tangential velocity by the formula

{\displaystyle v=\omega r} v = \omega r

so that

{\displaystyle F_{c}=mr\omega ^{2}\,.} {\displaystyle F_{c}=mr\omega ^{2}\,.}

Expressed using the orbital period T for one revolution of the circle,

{\displaystyle \omega ={\frac {2\pi }{T}}\,} {\displaystyle \omega ={\frac {2\pi }{T}}\,}

the equation becomes

{\displaystyle F_{c}=mr\left({\frac {2\pi }{T}}\right)^{2}.} {\displaystyle F_{c}=mr\left({\frac {2\pi }{T}}\right)^{2}.}[8]

In particle accelerators, velocity can be very high (close to the speed of light in vacuum) so the same rest mass now exerts greater inertia (relativistic mass) thereby requiring greater force for the same centripetal acceleration, so the equation becomes:

{\displaystyle F_{c}={\frac {\gamma mv^{2}}{r}}} {\displaystyle F_{c}={\frac {\gamma mv^{2}}{r}}}

where

{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}} \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}

is called the Lorentz factor.

More intuitively:

{\displaystyle F_{c}=\gamma mv\omega } {\displaystyle F_{c}=\gamma mv\omega }

which is the rate of change of relativistic momentum ( {\displaystyle \gamma mv} {\displaystyle \gamma mv})