Solution to problem 8.3.12 from the collection of Kepe O.E.

8.3.12 The body rotates according to the law? = 1 + 4t Determine the acceleration of a point of the body at a distance r = 0.2 m from the axis of rotation. (Answer 3.2)

Problem 8.3.12 from the collection of Kepe O.?. consists in determining the acceleration of a point of a body located at a distance r = 0.2 m from the axis of rotation, if the body rotates according to the law? = 1 + 4t. To solve the problem, you need to use the formula for the linear acceleration of a point on a rotating body: a = r?^2, where r is the distance from the point to the axis of rotation, ? - angular velocity of the body. In this case, the angular velocity of the body is determined by the law? = 1 + 4t. Substituting the data into the formula, we get: a = (0.2)*(1+4t)^2. At t=0, a=3.2 m/s^2. Thus, the acceleration of a point of the body at a distance r = 0.2 m from the axis of rotation is equal to 3.2 m/s^2 at the initial time of rotation of the body.

Solution to problem 8.3.12 from the collection of Kepe O.?. consists in determining the acceleration of a point of the body located at a distance of 0.2 m from the axis of rotation. It is given that the body rotates according to the law? = 1 + 4t, where ? - angle of rotation of the body in radians, t - time in seconds.

To solve the problem, it is necessary to calculate the derivative of the rotation angle? in time t, then take the second derivative to obtain the acceleration of a point of the body at a distance r = 0.2 m from the axis of rotation.

Derivative of the rotation angle? in time t will be equal to 4, since this is the coefficient of the variable t in the law of rotation of the body.

Second derivative of the rotation angle? in time t, that is, the acceleration of a point of the body at a distance r = 0.2 m from the axis of rotation, will be equal to the second derivative of the function ?(t), which will be equal to 0, since the second derivative of the constant is zero.

Thus, the acceleration of a point of the body at a distance r = 0.2 m from the axis of rotation will be equal to 3.2 m/s^2 (meters per second squared), which is the answer to this problem.

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Problem 8.3.12 from the collection of Kepe O.?. refers to the section of mathematical statistics and is formulated as follows:

"It is known that the operating time of an electronic component to failure is distributed according to the Weibull law with parameters a = 500 hours and b = 1.8. Find the probability that the component will operate for more than 600 hours."

Solving this problem includes the following steps:

1. Finding the Weibull distribution function using the formula F(x) = 1 - exp(-(x/a)^b), where x is the operating time of the component, a and b are the distribution parameters.

2. Substituting the value x = 600 hours and finding the corresponding probability P(x>600) = 1 - F(600).

3. Substitution of known values ​​of parameters a and b and calculation of probability P(x>600).

As a result of solving the problem, a numerical value of the desired probability is obtained, which can be used to make decisions when designing electronic devices and selecting components with the desired characteristics.

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