Question

Грузик, подвешенный на нити равномерно вращается в горизонтальной плоскости так, что расстояние от точки подвеса до плоскости, в которой происходит вращение, равно 10 см. При каком ускорении грузика частота его вращения будет равна 1 Гц? Угол между нитью и вертикалью составляет 45 градусов.

Answer 1

Из подобия соответствующих треугольников подвеса и сил следует
tgα = ma/mg = R/h
значит, искомая h равна:
h = Rg/a
здесь m - масса грузика
g=9,8 м с⁻² - ускорение свободного падения
a  - центростремительное ускорение
a = ω²R
ω - круговая частота
R - радиус траектории грузика
h - расстояние от подвеса до плоскости вращения
ω = 2пf
f = 0.5 Гц - частота вращения грузика
Откуда, после элементарных преобразований получается:
h = g/(4·п²·f²) = 9.8/(4*3.14²0.5²) = 1 м

Answer 2

Problem Analysis

We are given a scenario where a weight is suspended on a string and is rotating uniformly in a horizontal plane. The distance from the point of suspension to the plane of rotation is 10 cm. We need to determine the acceleration of the weight when its rotational frequency is 1 Hz. The angle between the string and the vertical is 45 degrees.

Solution

To solve this problem, we can use the formula for centripetal acceleration:

a = ω²r

where: - a is the centripetal acceleration, - ω is the angular velocity, and - r is the radius of rotation.

In this case, the radius of rotation is given as 10 cm, which is equal to 0.1 meters. We need to find the angular velocity when the rotational frequency is 1 Hz.

The relationship between angular velocity and frequency is given by:

ω = 2πf

where: - ω is the angular velocity, and - f is the frequency.

We are given that the frequency is 1 Hz, so we can substitute this value into the equation to find the angular velocity.

Finally, we can substitute the values of angular velocity and radius into the formula for centripetal acceleration to find the answer.

Let's calculate the acceleration step by step.

Calculation

Given: - Radius of rotation (r) = 0.1 m - Frequency (f) = 1 Hz

Step 1: Calculate the angular velocity (ω) using the formula ω = 2πf. - ω = 2π * 1 = 2π rad/s

Step 2: Calculate the centripetal acceleration (a) using the formula a = ω²r. - a = (2π)² * 0.1 = 4π² * 0.1 ≈ 12.566 m/s²

Therefore, the acceleration of the weight when its rotational frequency is 1 Hz is