Question

Mary and her younger brother Alex decide to ride the carousel at the State Fair. Mary sits on one of the horses in the outer section at a distance of 2.0 m from the center. Alex decides to play it safe and chooses to sit in the inner section at a distance of 1.0 m from the center. The carousel takes 3.7 s to make each complete revolution.
a) What is Mary's angular speed %u03C9M and tangential speed vM?

b) What is Alex's angular speed %u03C9A and tangential speed vA?

Answer 1

Answer:

ωM = 1.7 rad/s

vM = 3.4 m/s

ωA = 1.7 rad/s

vA =  1.7  m/s

Explanation:

Calculation of the angular speed (ω)

We use the following formula for circular motion:

ω = θ / t Formula (1)

Where:

ω: angular speed (rad/s)

θ:angle that the particle travels (rad)

t: time interval (s)

Equivalence

1 revolution = 2π rad

Data

θ= 1 revolution = 2π rad

t = 3.7 s

We replace data in the formula (1):

ω = (2π rad) / (3.7s )= 1.7 rad/s

Because the angular speed depends only on the number of turns in a time interval then Mary's angular speed (ωM) is equal to Alex's angular speed (ωA).

ωM = ωA = 1.7 rad/s

Calculation of the  tangential speed (v)

We use the following formula for circular motion:

v = ω*r  Formula (2)

Where:

v : tangential speed (m/s)

ω: angular speed ( rad/s)

r  : radius of the circular path (m)

Data

ωM = ωM = ω= 1.7 rad/s

rM = 2m  : Mary's circular path radius

rA = 1m :  Alex's circular path radius

We apply the formula (2) to calculate v:

vM = ω* rM = (1.7) (2) = 3.4 m/s

vA = ω* rA = (1.7) (1) =  1.7  m/s