Question

Платформа в виде горизонтально расположенного диска может вращаться вокруг вертикальной оси, проходящей через центр платформы. На платформе находится человек, которого в условии задачи можно рассматривать как материальную точку. Расходом энергии на преодоление сил трения пренебречь. Человек массой 60 кг стоит на краю неподвижной платформы. С какой скоростью (относительно платформы) должен пойти человек вдоль края платформы, чтобы она начала вращаться со скоростью, соответствую шей 3,0 об/мин? Масса платформы 120 кг, ее радиус 2.0 м.

Answer 1

Обозначим m₁ = 60 кг, m₂ = 120 кг, v₁ -- искомая скорость человека, v₂ -- линейная скорость платформы на её краю, R = 20 м, $\nu = 3$ об/мин = 0. 05 c^-1
Так как скорость платформы и человека направлены по касательной к платформе и противоположны, то по закону сохранения импульса, получим
$m_1v_1=m_2v_2$
Откуда
$v_2= \frac{m_1v_1}{m_2}$
Линейную скорость платформы на её краю можно представить так
$v_2=2\pi\nu R$
$\frac{m_1v_1}{m_2}=2\pi\nu R$
Откуда искомая скорость человека
$v_1 = 2\pi\nu R * \frac{m_2}{m_1} \approx2*3,14*0,05* \frac{120}{60} = 0,628$ м/с

Answer 2

Problem Analysis

We are given a platform in the form of a horizontally placed disk that can rotate around a vertical axis passing through its center. A person of mass 60 kg is standing on the edge of the stationary platform. We need to find the speed at which the person should walk along the edge of the platform so that it starts rotating at a speed of 3.0 rev/min. The mass of the platform is 120 kg and its radius is 2.0 m.

Solution

To solve this problem, we can use the principle of conservation of angular momentum. The initial angular momentum of the system (platform + person) is zero since the platform is stationary. When the person starts walking along the edge of the platform, the angular momentum of the system will be conserved.

The angular momentum of an object rotating around an axis is given by the product of its moment of inertia and its angular velocity. The moment of inertia of a disk rotating around its central axis is given by (1/2) * mass * radius^2.

Let's denote the initial angular velocity of the platform as ω_initial and the final angular velocity as ω_final. The moment of inertia of the platform is I_platform and the moment of inertia of the person is I_person.

According to the conservation of angular momentum, we have:

I_platform * ω_initial + I_person * ω_initial = I_platform * ω_final + I_person * ω_final

Since the initial angular velocity of the platform is zero, the equation simplifies to:

I_person * ω_initial = I_platform * ω_final

We can rearrange the equation to solve for ω_initial:

ω_initial = (I_platform * ω_final) / I_person

Now, let's calculate the moment of inertia of the platform and the person.

The moment of inertia of a disk rotating around its central axis is given by (1/2) * mass * radius^2. For the platform, the mass is 120 kg and the radius is 2.0 m. Therefore, the moment of inertia of the platform (I_platform) is:

I_platform = (1/2) * 120 kg * (2.0 m)^2

Now, let's calculate the moment of inertia of the person. Since the person can be considered as a point mass, the moment of inertia of the person (I_person) is equal to the mass of the person multiplied by the square of the distance between the person and the axis of rotation. In this case, the distance is equal to the radius of the platform, which is 2.0 m. Therefore, the moment of inertia of the person is:

I_person = 60 kg * (2.0 m)^2

Now, we can substitute the values of I_platform, I_person, and ω_final into the equation to calculate ω_initial.

Calculation

Let's calculate the values:

I_platform = (1/2) * 120 kg * (2.0 m)^2 I_person = 60 kg * (2.0 m)^2

ω_initial = (I_platform * ω_final) / I_person

Substituting the values:

I_platform = 120 kg * (2.0 m)^2 = 480 kg·m^2 I_person = 60 kg * (2.0 m)^2 = 240 kg·m^2

ω_initial = (480 kg·m^2 * (3.0 rev/min)) / 240 kg·m^2

Note: We need to convert the final angular velocity from rev/min to rad/s. Since 1 rev = 2π rad, we have:

ω_final = 3.0 rev/min * (2π rad/1 rev) * (1 min/60 s) = 3.0 * 2π/60 rad/s

Now, let's calculate ω_initial:

ω_initial = (480 kg·m^2 * (3.0 * 2π/60 rad/s)) / 240 kg·m^2

Answer

The person should walk along the edge of the platform with a speed of ω_initial rad/s (relative to the platform) in order for the platform to start rotating with a speed of 3.0 rev/min.

Let's calculate the value of ω_initial:

ω_initial = (480 kg·m^2 * (3.0 * 2π/60 rad/s)) / 240 kg·m^2

After performing the calculation, we find that ω_initial is approximately equal to 2π/5 rad/s.

Therefore, the person should walk along the edge of the platform with a speed of approximately 2π/5 rad/s (relative to the platform) in order for the platform to start rotating with a speed of 3.0 rev/min.

References

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