Question

Шкив радиусом 0,1 м приводится во вращение грузом, подвешенным на нити и постепенно сматывающимся со шкива. В начальный момент шкив неподвижен. Когда груз прошел расстояние 1 м, его скорость стала 1 м/с. Определить угловое ускорение шкива. Записать закон изменения угловой скорости шкива от времени.

Answer 1

1) Рассмотрим движение груза
1. Груз движется равноускорено, уравнение движения: 
$s= \frac{ at^{2} }{2}$ (начальная скорость и координата равны нулю)
С другой стороны, ускорение груза:
$a= \frac{ v-v_{0} }{t}$
Путь тела $s$ равен 1 м. Составим и решим систему уравнений:
$\left \{ {{s= \frac{at^{2} }{2} } \atop {a= \frac{v- v_{0} }{t} }} \right.$
$v- v_{0} =1-0=1$
Решая систему уравнений методом подстановки найдем, что $t=2$с
$a=0.5$м/с^2
2. Так как тело опустилось на 1 метр, то и веревка размоталась на 1 метр. Найдем угол, на который повернулся шкив:
$l= \frac{ \pi R}{180}* \alpha$
$\alpha = \frac{180}{ \pi } *\frac{l}{R}$
$[tex] \alpha = \frac{180}{ \pi }*\frac{1}{0.1} = 10$[/tex]рад
3. Угловая скорость:
$w= \frac{ \alpha }{t} = \frac{10}{2}= 5$ рад/с
Как найти угловое ускорение не знаю, не имею нужных знаний для вычислений.

Answer 2

Problem Analysis

We are given that a pulley with a radius of 0.1 m is being rotated by a weight hanging from a string. The weight gradually unwinds the string from the pulley. Initially, the pulley is stationary, and when the weight has traveled a distance of 1 m, its speed is 1 m/s. We need to determine the angular acceleration of the pulley and express the law of change of its angular velocity with respect to time.

Solution

To solve this problem, we can use the following equations:

1. The linear velocity of the weight is given by the equation: v = rω where v is the linear velocity, r is the radius of the pulley, and ω is the angular velocity of the pulley.

2. The linear acceleration of the weight is given by the equation: a = rα where a is the linear acceleration and α is the angular acceleration.

3. The relationship between linear acceleration and angular acceleration is given by: a = rα

4. The relationship between linear velocity and angular velocity is given by: v = rω

5. The relationship between linear displacement and angular displacement is given by: s = rθ where s is the linear displacement and θ is the angular displacement.

Using these equations, we can solve the problem step by step.

Step 1: Finding the angular velocity

We are given that when the weight has traveled a distance of 1 m, its speed is 1 m/s. Since the weight is unwinding from the pulley, the linear displacement of the weight is equal to the linear displacement of the pulley. Therefore, we can write: s = rθ where s is the linear displacement of the weight and θ is the angular displacement of the pulley.

In this case, s = 1 m and r = 0.1 m. Therefore, we have: 1 = 0.1θ Solving for θ, we find: θ = 10 radians

Now, we can use the relationship between linear velocity and angular velocity to find the angular velocity ω when the weight has traveled a distance of 1 m: v = rω Substituting the given values, we have: 1 m/s = 0.1 m * ω Solving for ω, we find: ω = 10 rad/s

Step 2: Finding the angular acceleration

To find the angular acceleration α, we can use the relationship between linear acceleration and angular acceleration: a = rα Since the linear acceleration a is not given, we need to find it. We can use the equation of motion: v^2 = u^2 + 2as where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.

In this case, the initial velocity u is 0 m/s, the final velocity v is 1 m/s, and the displacement s is 1 m. Substituting these values into the equation, we have: 1^2 = 0^2 + 2a(1) Simplifying, we find: 1 = 2a Solving for a, we get: a = 0.5 m/s^2

Now, we can use the relationship between linear acceleration and angular acceleration to find the angular acceleration α: a = rα Substituting the given values, we have: 0.5 m/s^2 = 0.1 m * α Solving for α, we find: α = 5 rad/s^2

Step 3: Writing the law of change of angular velocity

The law of change of angular velocity with respect to time is given by: ω = ω0 + αt where ω is the final angular velocity, ω0 is the initial angular velocity, α is the angular acceleration, and t is the time.

In this case, the initial angular velocity ω0 is 0 rad/s, the angular acceleration α is 5 rad/s^2, and the time t is not given. Therefore, the law of change of angular velocity can be written as: ω = 0 + 5t Simplifying, we have: ω = 5t rad/s

Summary

To summarize, the angular acceleration of the pulley is 5 rad/s^2, and the law of change of angular velocity with respect to time is ω = 5t rad/s.

Note: The above solution is based on the given information and the assumption that the pulley is ideal and there are no external forces acting on it.