448 CHAPTER 24. MOVING COORDINATE SYSTEMS

the velocity and acceleration in terms of er and eθ ? Using the chain rule,

der

dt=

der

dθθ′ (t) ,

deθ

dt=

deθ

dθθ′ (t)

and so from 24.1,der

dt= θ

′ (t)eθ ,deθ

dt=−θ

′ (t)er (24.2)

Using 24.2 as needed along with the product rule and the chain rule,

r′ (t) = r′ (t)er + r (t)ddt

(er (θ (t)))

= r′ (t)er + r (t)θ′ (t)eθ .

Next consider the acceleration.

r′′ (t) = r′′ (t)er + r′ (t)der

dt+ r′ (t)θ

′ (t)eθ + r (t)θ′′ (t)eθ + r (t)θ

′ (t)ddt

(eθ )

= r′′ (t)er +2r′ (t)θ′ (t)eθ + r (t)θ

′′ (t)eθ + r (t)θ′ (t)(−er)θ

′ (t)

=(

r′′ (t)− r (t)θ′ (t)2

)er +

(2r′ (t)θ

′ (t)+ r (t)θ′′ (t)

)eθ . (24.3)

This is a very profound formula. Consider the following examples.

Example 24.1.1 Suppose an object of mass m moves at a uniform speed v, around a circleof radius R. Find the force acting on the object.

By Newton’s second law, the force acting on the object is mr′′. In this case, r (t) = R, aconstant and since the speed is constant, θ

′′ = 0. Therefore, the term in 24.3 correspondingto eθ equals zero and mr′′ =−Rθ

′ (t)2er. The speed of the object is v and so it moves v/Rradians in unit time. Thus θ

′ (t) = v/R and so

mr′′ =−mR( v

R

)2er =−m

v2

Rer.

This is the familiar formula for centripetal force from elementary physics, obtained as avery special case of 24.3.

Example 24.1.2 A platform rotates at a constant speed in the counter clockwise directionand an object of mass m moves from the center of the platform toward the edge at constantspeed along a line fixed in the rotating platform. What forces act on this object?

Let v denote the constant speed of the object moving toward the edge of the platform.Then

r′ (t) = v, r′′ (t) = 0, θ′′ (t) = 0,

while θ′ (t) = ω , a positive constant. From 24.3

mr′′ (t) =−mr (t)ω2er +m2vωeθ .

Thus the object experiences centripetal force from the first term and also a funny force fromthe second term which is in the direction of rotation of the platform. You can observe thisby experiment if you like. Go to a playground and have someone spin one of those merrygo rounds while you ride it and move from the center toward the edge. The term 2mvωeθ

is called the Coriolis force.