Question

Однородный диск массой 3 кг и радиусом 20 см, вращающийся с угловой скоростью 16π рад/с, начинает тормозиться касательной силой 0,05 Н, приложенной к его ободу. Через какое время диск остановится?

Answer 1

Уравнение движения:
$\frac{ MR^2}{2}\dot\omega=-FR$
Кинематика:
$t=\frac{\omega_0}{|\dot\omega|}$.
Подставим первое уравнение во второе, найдем ответ:
$\boxed{t=\frac{MR\omega_0}{2F}}=\frac{3\cdot 0.2\cdot 16\pi}{2\cdot 0.05}=96\pi\mathrm{\ s.}$

Answer 2

Calculation of Stopping Time for a Rotating Disk

To calculate the time it takes for the disk to come to a stop, we can use the equation for rotational motion:

τ = Iα

where τ is the torque applied to the disk, I is the moment of inertia of the disk, and α is the angular acceleration of the disk.

In this case, the torque applied to the disk is equal to the product of the tangential force and the radius of the disk:

τ = F * r

Given that the tangential force is 0.05 N and the radius of the disk is 20 cm (or 0.2 m), we can calculate the torque:

τ = 0.05 N * 0.2 m = 0.01 Nm

The moment of inertia of a solid disk rotating about its central axis is given by the equation:

I = (1/2) * m * r^2

where m is the mass of the disk and r is the radius of the disk.

Given that the mass of the disk is 3 kg and the radius is 20 cm (or 0.2 m), we can calculate the moment of inertia:

I = (1/2) * 3 kg * (0.2 m)^2 = 0.06 kgm^2

Now, we can rearrange the equation τ = Iα to solve for α:

α = τ / I = 0.01 Nm / 0.06 kgm^2 = 0.1667 rad/s^2

The angular acceleration of the disk is 0.1667 rad/s^2.

To find the time it takes for the disk to come to a stop, we can use the equation:

ω = ω0 + αt

where ω is the final angular velocity of the disk, ω0 is the initial angular velocity of the disk, α is the angular acceleration of the disk, and t is the time.

In this case, the initial angular velocity of the disk is 16π rad/s, and the final angular velocity is 0 rad/s (since the disk comes to a stop).

Plugging in the values, we get:

0 = 16π rad/s + (0.1667 rad/s^2) * t

Solving for t, we find:

t = -16π rad/s / (0.1667 rad/s^2) ≈ -302.4 s

The negative sign indicates that the time is negative, which doesn't make physical sense in this context. Therefore, we can conclude that the disk will not come to a complete stop under the given conditions.

Please note that the above calculations assume ideal conditions and do not take into account factors such as friction or air resistance, which may affect the actual stopping time of the disk.