Question

Даю 100 баллов. После включения электродвигателя его вал, вращаясь равноускорено за 30 с, сделал 150 об. Определить с каким угловым ускорением вращается вал в этот период и какое число об/мин, он приобрел в конце периода.Каковы будут скорость и ускорение точек обода шкива, насаженного на вал в конце 30-й секунды?Диаметр шкива 540 мм

Answer 1

Дано:

$t=30~\text c\\N=150\\D=540~_{\text {MM}}=0,54~_{\text M}\\\omega_0=0~\text c ^{-1}$

Найти:

$\varepsilon-?\\\omega-?\\v-?\\a-?$

Решение:

за один оборот вал проходит угол $2\pi$.

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Т. к. начальная угловая скорость отсутствовала, то формула приобретает вид

$\varphi=\frac{\varepsilon t^2}{2}$

Отсюда $\varepsilon=\frac{2\phi}{t^2} =\frac{4\pi N}{t^2} =\frac{4\times 3,14 \times 150}{30^2} =\frac{157}{75} \approx 2,1~\text c ^{-2}$

$\boxed{w=w_0+\varepsilon t }$

${w=w_0+\varepsilon t }=\varepsilon t=\frac{4\pi N}{t} =62,8 ~\text {c}^{-1}\approx 1,05$ мин⁻¹

$\boxed {a=\varepsilon R=\frac{\varepsilon D}{2} }$

$a=\frac{2\pi ND}{t^2} =0,5652\approx0,6$ м/с²

$\boxed {v=\omega R=\frac{\omega D}{2} }$

$v=\frac{2\pi ND}{t} =16,956 \approx 17$ м/с

Ответ: ε=2,1 с⁻², ω=1,05 мин⁻¹, a=0,6 м/с², υ=17 м/с

Answer 2

Calculation of Angular Acceleration and Final Angular Velocity

To determine the angular acceleration and final angular velocity of the shaft, we can use the formula:

Angular acceleration (α) = (Change in angular velocity) / (Time taken)

Given that the change in angular velocity is 150 revolutions and the time taken is 30 seconds, we can calculate the angular acceleration as follows:

Angular acceleration (α) = 150 rev / 30 s = 5 rev/s².

To find the final angular velocity, we can use the formula:

Final angular velocity (ω) = Initial angular velocity + (Angular acceleration * Time taken)

Since the initial angular velocity is zero (as the shaft starts from rest), the final angular velocity can be calculated as:

Final angular velocity (ω) = 0 + (5 rev/s² * 30 s) = 150 rev/s.

Calculation of Linear Velocity and Linear Acceleration at the Rim of the Pulley

To calculate the linear velocity and linear acceleration at the rim of the pulley, we need to know the diameter of the pulley. Given that the diameter of the pulley is 540 mm, we can calculate the radius (r) as half of the diameter:

Radius (r) = Diameter / 2 = 540 mm / 2 = 270 mm = 0.27 m.

The linear velocity (v) at the rim of the pulley can be calculated using the formula:

Linear velocity (v) = Angular velocity (ω) * Radius (r)

Substituting the values, we get:

Linear velocity (v) = 150 rev/s * 0.27 m = 40.5 m/s.

The linear acceleration (a) at the rim of the pulley can be calculated using the formula:

Linear acceleration (a) = Angular acceleration (α) * Radius (r)

Substituting the values, we get:

Linear acceleration (a) = 5 rev/s² * 0.27 m = 1.35 m/s².

Therefore, at the end of the 30th second, the linear velocity of the points on the rim of the pulley is 40.5 m/s and the linear acceleration is 1.35 m/s².

Please let me know if there is anything else I can help you with!