Need help finding tangential and angular velocities

Question

You are swinging on a swing. If the ropes holding the swing are 6 ft. long,
and you swing through a 6 ft. arc every half second, what is your tangential velocity? What is your angular
velocity?

If the swing slows down exponentially after you jump off, it’s swinging through a 3 ft. arc when you jump off, and
the size of the motion decays at a steady rate of 5% per second, how long
will it be before the swing is only moving through a 1 inch arc?

Answer 1

Hi Jennifer,

The tangential velocity is V = 6 ft/sec

and

ω = V/R , R = 6ft

ω = 6ft/sec/6ft = 1 rad/sec 

X(t) = A e^-γt cos(ωt +φ),  A cos(ωt +φ)= 3 ft,  γ = 5% = 0.05, X(t) = 1 ft

1    = 3 e^-0.05t

(1/3)= e^-0.05t   take the natural log of both sides

ln(1/3) = -0.05t

-20ln(1/3) = t

    21.97   = t

It will take 21.97 seconds