Are Linear and Tangential Acceleration Just Angular Values Times the Radius?

[forty]

Question:
A fighter pilot is being trained in a centrifuge of radius 15m. it rotates according to theta=0.25(t^3) + ln(t+1) befor it stabilises (theta in radians). what are the magnitudes of the pilots:

a) angular velocity: d(theta)/dt at t = 4 (12.2 rad/sec)

b) linear velocity: 12.2 * 15

c radial acceleration: 2nd derivative of theta=0.25(t^3) + ln(t+1) at t = 4 (5.96rad/sec^2)

d) tangential acceleration: 5.96 * 15

Are b and d just the angular values times the radius that's if a and c are right in the first place??


Thanks

 

[tiny-tim]

Are b and d just the angular values times the radius that's if a and c are right in the first place??

Yes!

(v = rω and dv/dt = (d/dt)(rω) = r(dω/dt), since r is constant.)

But why was that worrying you?

 

[forty]

Because all the angular equivalents scare me!

 

[tiny-tim]

… just keep differentiating rθ …

Hi forty!

Just remember the definition of a radian: the angle whose arc-length = r.

And therefore generally:

tangential length = rθ.​

So (if r is constant), differentiate once for:

tangential speed v = rθ´ = rω

tangential acceleration a = rθ´´ = rω´. ​

(and of course radial acceleration = -rω² = -vω = -v²/r.)