Final answer:
The problem relates to the conservation of angular momentum in physics. The new angular velocity of the merry-go-round can be found by equating the initial and final angular momenta, considering the increase in moment of inertia as people step onto the merry-go-round.
Step-by-step explanation:
Conservation of Angular Momentum
When four people, each with a mass of 70 kg, step onto the edge of a rotating merry-go-round with a radius of 2.25 meters, we must use the concept of the conservation of angular momentum to find its new angular velocity. The total moment of inertia for the merry-go-round and people can be calculated by adding the moment of inertia of the merry-go-round to the sum of the moments of inertia of the people standing on the edge (approximating each person as a point mass). The initial angular momentum is the product of the merry-go-round's initial angular velocity and its initial moment of inertia. Since no external torque is acting, the angular momentum before the people step on equals the angular momentum after. Thus, we can set up the equation:
L_initial = L_final
(I_merry_go_round + 4 * m_person * r^2) * ω_final = I_merry_go_round * ω_initial
Plugging in the numbers:
(1400 kg·m² + 4 * 70 kg * (2.25 m)^2) * ω_final = 1400 kg·m² * 0.85 rad/s
ω_final = (1400 kg·m² * 0.85 rad/s) / (1400 kg·m² + 4 * 70 kg * (2.25 m)^2)
After solving the equation, we get the new angular velocity of the merry-go-round.