Solution to problem 13.3.7 from the collection of Kepe O.E.

13.3.7 Solution of the problem of the movement of a material point along a curvilinear trajectory Given: $m = 5\text{ kg}$, $F_{\text{tan}} = 7\text{ N}$, $F_{\text{n}} = 0.1t^2$ at $t = 12\text{ s}$ Find: acceleration modulus of a point Solution: Acceleration modulus of a point is determined by the formula: $a = \sqrt{(F_{\text{tan}}/m)^2 + (F_{\text{n }}/m)^2}$ Substituting the known values, we get: $a = \sqrt{(7/5)^2 + (0.1\cdot 12^2/5)^2} \approx \boxed{3 .20}$ Answer: 3.20.

Solution to problem 13.3.7 from the collection of Kepe O.?. This digital product is a solution to one of the problems in the collection of Kepe O.?. in physics. In particular, we consider the problem of the movement of a material point along a curvilinear trajectory under the action of a force specified by its projections onto the tangent and normal to the trajectory. The solution to the problem is presented in the form of an HTML document with a beautiful design, which you can purchase in our digital goods store. By purchasing this product, you receive a ready-made solution to the problem with a step-by-step explanation and answer, which can be used as a sample when performing similar tasks.

This product is a solution to problem 13.3.7 from the collection of Kepe O.?. in physics. The problem describes the movement of a material point along a curvilinear trajectory under the influence of a force specified by its projections on the tangent and normal to the trajectory. To solve the problem, it is necessary to find the acceleration modulus of the point, which is determined by the formula: $a = \sqrt{(F_{\text{tan}}/m)^2 + (F_{\text{n}}/m)^2} $. In solving the problem, known values ​​are substituted, the resulting acceleration of the point is rounded to two decimal places and given in the answer. By purchasing this product, the buyer receives a ready-made solution to the problem in the form of a beautifully designed HTML document with a step-by-step explanation and answer, which can be used as a sample when performing similar tasks.

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The book "Collection of problems for the course of higher mathematics" by Kepe O.?. contains problems and their solutions in various branches of mathematics. Problem 13.3.7 from this collection belongs to Chapter 13 “Differential Equations”, section 13.3 “Linear Differential Equations of the nth Order with Constant Coefficients”. To solve it, it is necessary to use the method of indefinite coefficients. The solution to a problem is a sequence of mathematical operations leading to finding a general solution to the equation. The solution to this problem can be useful for students and teachers of higher mathematics studying this branch of mathematics.

Problem 13.3.7 from the collection of Kepe O.?. is formulated as follows:

A material point with a mass of 5 kg moves along a curved path under the influence of a force, the projection of which on the tangent is 7 N, and on the normal - 0.1t². It is necessary to find the acceleration modulus of a point at time t = 12 s.

To solve this problem it is necessary to use Newton's laws. Since the force is decomposed into projections onto the tangent and normal, the acceleration of the point consists of the tangent and normal components. The tangential acceleration can be found using Newton's second law: F = ma, where F is the tangential component of the force, m is the mass of the point, a is the tangential acceleration.

Thus, the tangential acceleration of a point at time t can be found using the formula: a? =F? /m where is F? = 7 N - projection of force onto the tangent.

The normal acceleration can be found knowing that it is equal to the product of the curvature of the trajectory and the square of the speed of the point. The curvature of the trajectory can be found using the derivative of the angle of inclination of the tangent to the trajectory. Thus, the normal acceleration of a point at time t can be found by the formula: an = v² / R, where v is the speed of the point, R is the radius of curvature of the trajectory.

Since the speed of the point is unknown, it can be found using the equation of motion relative to the coordinate. Then you can find the curvature of the path and the radius of curvature. The curvature of the trajectory is equal to the second derivative of the y coordinate with respect to x: k = |y''| / (1 + y'²)^(3/2), where y' and y'' are the first and second derivatives of the y coordinate with respect to x.

The radius of curvature of the trajectory can be found using the formula: R = 1/k.

Thus, after finding the speed, curvature of the trajectory and radius of curvature, you can find the normal acceleration of the point. Then you can find the total acceleration of the point as the vector sum of the tangent and normal accelerations and find its modulus, which is the desired answer.

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