4. A horse on a merry go round is 7 m from the center and travels at 10km/hr. What is its angular speed?

Answer 1

arc = rθ
in our case: arc = 7θ
d(arc)/dt = 7 dθ/dt
10000 m/hr = 7 m dθ/dt
dθ/dt = 10000/7 rad/hr or appr 1428.57 rad/hr = appr 23.8 rad/min

or , non-Calculus way

circumference = 14π = 43.98 m
speed = 10 km/h = 10000/60 m/min = 166.667 m/min
time to complete one revolution = distance/rate = 43.98/166.667 = appr .264 min

angular velocity = 2π rad/.264 min = 23.81 rad/min , same as above

You didn't say what units of time was required.

Answer 2

V = 10km/h = 10,000m/3600s = 2.78 m/s.
Cir. = pi * 2r = 3.14 * 14 = 44.0 m.
Va = 2.78m/s * 6.28rad/44m = 0.40 rad/s.