Question

Точка движется по окружности радиусом 8м.В некоторый момент времени нормальное ускорение точки 4м/с, вектор полного ускорения образует в этот момент с вектором нормального ускорения угол 60*.Найти скорость v и тангенциальноеускорение точки.

Answer 1

Дано:
R=8 м
а₁= 4 м/c²
α=60°
Найти: v, a₂
Решение:
Формула центростремительного ускорения
$a_1= \frac{v^2}{R}$
$v= \sqrt{Ra_1} = \sqrt{8*4} =4 \sqrt{2}$
Из тригонометрии
$\frac{a_1}{a_2} =tg \alpha \\ a_2= \frac{a_1}{tg \alpha } = \frac{4}{tg60^0} = \frac{4}{ \sqrt{3} }$
Ответ: 4√2≈5,67 м/с²; 4/√3≈2,31 м/с²

Answer 2

Problem Analysis

We are given that a point is moving along a circular path with a radius of 8m. At a certain moment in time, the point has a normal acceleration of 4m/s and the angle between the total acceleration vector and the normal acceleration vector is 60 degrees. We need to find the velocity (v) and the tangential acceleration of the point.

Solution

To solve this problem, we can use the equations of circular motion. Let's break down the solution into steps:

Step 1: Find the magnitude of the total acceleration (a) using the given normal acceleration (an) and the angle between the total acceleration vector and the normal acceleration vector (θ).

From the given information, we know that the normal acceleration (an) is 4m/s and the angle (θ) is 60 degrees. We can use trigonometry to find the magnitude of the total acceleration (a).

Using the equation: a = an / sin(θ), we can calculate the magnitude of the total acceleration.

Step 2: Find the velocity (v) of the point.

We know that the velocity (v) is related to the total acceleration (a) by the equation: v = a * t, where t is the time.

Step 3: Find the tangential acceleration (at) of the point.

The tangential acceleration (at) is related to the total acceleration (a) by the equation: at = a * cos(θ).

Let's calculate the values using the given information.

Step 1: Find the magnitude of the total acceleration (a).

Using the equation: a = an / sin(θ), where an = 4m/s and θ = 60 degrees.

a = 4 / sin(60) = 4 / (√3 / 2) = 8 / √3 ≈ 4.62 m/s²

Step 2: Find the velocity (v) of the point.

Using the equation: v = a * t, where a = 4.62 m/s² and t is the time.

Since the time is not given in the problem, we cannot calculate the exact value of the velocity (v) without additional information.

Step 3: Find the tangential acceleration (at) of the point.

Using the equation: at = a * cos(θ), where a = 4.62 m/s² and θ = 60 degrees.

at = 4.62 * cos(60) = 4.62 * 0.5 = 2.31 m/s²

Therefore, the tangential acceleration (at) of the point is 2.31 m/s².

Please note that the exact value of the velocity (v) cannot be determined without additional information about the time.