rotational motion

Question

A merry-go-round at an amusement park consists of an outer ring of horses and an inner ring of elephants.

The ring of eight horses has a radius of r_h = 8.7 m. Each of the eight horses has a mass m_h = 95 kg. The ring of four elephants has a radius of r_e = 6.8 m and each of the four elephants has a mass m_e = 195 kg. The platform of the merry-go-round itself has a mass of m_p = 1500 kg and a radius of r_p = 10.2 m.

At it’s operating speed, the merry-go-round makes one revolution every t = 12 s.

a) enter an expression for the total moment of inertia of the carousel about its axis, in terms of the defined quantities.

b) calculate the moment of inertia of the carousel about its axis, in units of kilogram meters squared.

c) enter an expression for the angular speed of the carousel, in terms of the defined quantities.

d) calculate the carousels angular speed, in radians per second.

Answer 1

a) The horses and elephants are considered to be point masses at their respective distances from the center. The platform is a disc, so its moment of inertia has a 1/2 factor.

Total I = 8 (m_h r_h²) + 4 (m_e r_e²) + 1/2 (m_p r_p²)

b) Plugging in numbers, I = 57,524 + 36,067 + 78,030 = 171,621 kg m² [1.7 x 10⁵ to correct # of sig figs]

c) Angular speed ω = 2π f = 2π / T, since frequency f = 1/T and the given 12s/rev is the period T.

d) ω = 2π /12 = 0.5236 rad/s [5.2 x 10^-1 to correct # of sig figs].