Question

Платформа в виде диска радиусом 1м вращается по инерции с частотой = 6 мин -1 . На краю платформы стоит человек массой 80 кг. С какой частотой 1 будет вращаться платформа, если человек перейдет в еѐ центр? Момент инерции платформы J 120 кг·м2 . Момент инерции человека рассчитывать как для материальной точки.

Answer 1

Воспользуемся законом сохранения момента импульса:
I1⋅ω1 = I2⋅ω2,где ω1 = 2π⋅ν1, ω2 = 2π⋅ν2, I1 = I + m⋅R2, I2 = I. Тогда
ν2 = (I + m⋅R2)⋅ν1/I,ν2 = 10 мин–1.

Answer 2

Calculation of the Angular Velocity of the Platform

To calculate the angular velocity of the platform when the person moves to its center, we can use the principle of conservation of angular momentum. The angular momentum of the system remains constant when no external torques act on it.

The initial angular momentum of the system is given by the product of the moment of inertia of the platform and its angular velocity:

L_initial = J_platform * ω_initial (Equation 1)

where: - L_initial is the initial angular momentum of the system, - J_platform is the moment of inertia of the platform, and - ω_initial is the initial angular velocity of the platform.

When the person moves to the center of the platform, the moment of inertia of the system changes. We can calculate the final angular velocity of the platform using the principle of conservation of angular momentum:

L_initial = L_final (Equation 2)

Substituting Equation 1 into Equation 2, we get:

J_platform * ω_initial = J_final * ω_final (Equation 3)

where: - J_final is the moment of inertia of the system after the person moves to the center, and - ω_final is the final angular velocity of the platform.

Calculation of the Moment of Inertia of the System

To calculate the moment of inertia of the system after the person moves to the center, we need to consider the moment of inertia of the platform and the person separately.

The moment of inertia of the platform is given as J_platform = 120 kg·m^2.

The moment of inertia of the person can be approximated as that of a point mass, given by I_person = m_person * r^2, where: - m_person is