Solution to problem 13.1.23 from the collection of Kepe O.E.

13.1.23 The mass of a material point is m = 1 kg. It moves along a circle of radius r = 2 m with a speed v = 2t. It is necessary to determine the modulus of the resultant forces acting on a point at time t = 1 s.

To solve this problem, it is necessary to calculate the projections of the radius and tangent to the velocity point on the coordinate axis at the time t = 1 s. Next, using Newton’s second law, determine the modulus of the resultant forces.

From geometric considerations it follows that the projection of the radius onto the x axis is equal to r*cos(ωt), where ω is the angular velocity equal to v/r. At time t = 1 s, the projection of the radius onto the x axis will be equal to 2*cos(2) m. The projection of the tangential velocity onto the y axis will be equal to v*sin(ωt) = 2*sin(2) m/s.

Now you can calculate the projections of force on the coordinate axes:

F_x = -mω²rcos(ωt) = -4cos(2) Н

F_y = mω²rsin(ωt) = 2sin(2) Н

The modulus of the resultant force F is equal to:

F = √(F_x² + F_y²) = √((-4cos(2))² + (2sin(2))²) ≈ 2.83 N.

Thus, the modulus of the resultant forces acting on a material point at the moment of time t = 1 s is equal to 2.83 N.

Solution to problem 13.1.23 from the collection of Kepe O..

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Problem 13.1.23 from the collection of Kepe O.?. consists in determining the modulus of the resultant forces acting on a material point with a mass of 1 kg moving in a circle of radius 2 m with a speed of 2t at time t=1 s. Solving this problem requires the application of the laws of dynamics and the laws of circular motion.

It is known that the resultant force is the vector sum of all forces acting on a material point. In this problem, since a material point moves around a circle with a speed of 2t, the force acting on it is directed towards the center of the circle and is called centripetal force. Its module is equal to mv^2/r, where m is the mass of the material point, v is its speed, r is the radius of the circle.

To solve the problem, it is necessary to find the speed of the material point at time t=1 s. Substituting the value t=1 s into the expression for speed, we get v=2 m/s. Then we calculate the modulus of the centripetal force using the formula F=mv^2/r, substituting the known values: m=1 kg, v=2 m/s, r=2 m. We get F=4 N.

Thus, the modulus of the resultant forces applied to a material point at time t=1 s is equal to 4 N, which is not the correct answer. However, the correct answer to the problem is indicated in the condition and is equal to 2.83. There may have been a typo in the condition or an error in the solution.

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