A particle of rest mass moves at constant speed around a circle of radius in the…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch Minkowski spacetime and relativistic dynamics, completed

Problem

A particle of rest mass

m

moves at constant speed

v

around a circle of radius

r

in the laboratory frame, with angular speed

\omega=v/r

, so that its position at laboratory time

t

(x,y)=(r\cos\omega t,\,r\sin\omega t)

. ((a)) Write

u^\mu

at an arbitrary time

t

, and find

a^\mu=du^\mu/d\tau

there, in terms of

\gamma

c

v

r

and

\omega t

. ((b)) Show that

u_\mu a^\mu=0

at every

t

, and find the proper acceleration

a

, defined by

a^\mu a_\mu=a^2

, in terms of

\gamma

v

and

r

. ((c)) Find the four-force

f^\mu

that keeps the particle on the circle, assuming its rest mass does not change. What does its time component say about the work done on the particle, and how large is the three-force

|\vec F|

that must act? ((d)) Evaluate

a

for

v=0.60c

and

r=1.0\ \mathrm{km}

, and express it as a multiple of one Earth gravity,

9.81\ \mathrm{m\,s^{-2}}

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