How much work is required to uniformly accelerate a merry-go-round of mass 1490 kg and a radius of 7.80 m from rest to a rotational rate of 1 revolution per 8.33 s? Model the merry-go-round as a solid cylinder (I = ½ MR2). How much power is required to accelerate the merry-go-round to that rate in 8.33 s?

Given:

Time, t = 8.33 s

Mass, M = 1490 kg

Angular velocity, w = 1 rev in 8.33 s

= 0.12 rev/s

Converting rev/s to rad/s,

1 rev = 2pi rad

0.12 rev/s * 2pi rad/1 rev

= 0.754 rad/s

Radius, r = 7.8 m

Inertia, I = 1/2 * M * r^2

= 1/2 * 1490 * 7.8^2

= 45325 kgm^2

Kinetic energy, Uk = 1/2 * I * w^2

= 1/2 * 45325 * 0.754^2

= 12884.5 J

= 12.8 kJ

Power = energy/time

Kinetic energy, Uk = energy

Power = 12884.5/8.33

= 1546.725 W

= 1.55 kW.

W = 12,884.221 J

Explanation:

I = ½ MR² (given)

I = 0.5 * 1490 kg * (7.8 m) ²

I = 45,325.8 Kg m²

and ω = 2π / T = 2 * 3.14 / 8.33 s

ω = 0.754 radians/s

now W = Δ K. E. = 1/2 Iω² = 0.5 * 45,325.8 Kg m² * (0.754 radians/s) ²

W = 12,884.221 J