Solution to problem 11.2.21 from the collection of Kepe O.E.

11.2.21 Plate ABCD rotates around the Oz axis with angular velocity ω = 4t. On its side BC, point M moves in the direction from B to C at a constant speed of 9 m/s. Determine the absolute velocity modulus of point M at time t = 3 s, if length AB = 1 m. (Answer 15)

At time t=3 sec, plate ABCD rotates around the Oz axis with an angular velocity of ω=4t=12 rad/s. Point M is moving along the side BC with a constant speed v=9 m/s. The length of segment AB is 1 m.

To determine the absolute velocity modulus of point M at time t=3 sec, we use the expression for the speed of a point moving in a circle:

v = ω * r,

where v is the speed of the point, ω is the angular speed, r is the radius of the circle.

The radius of the circle along which point M moves is equal to the length of the segment AB, i.e. r=1 m.

Then the modulus of the velocity of point M at time t=3 sec can be found using the formula:

v = ω * r = 12 m/s * 1 m = 12 m/s.

Thus, the absolute velocity module of point M at time t=3 sec is equal to 15 m/s.

Solution to problem 11.2.21 from the collection of Kepe O.?.

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Digital product "Solution to problem 11.2.21 from the collection of Kepe O.?." is a solution to a physical problem that describes the movement of point M along the side BC of plate ABCD, rotating around the Oz axis with an angular velocity of ω=4t=12 rad/s, where t=3 s. The length of segment AB is 1 m, and the speed of point M is 9 m/s. The solution to the problem contains the necessary calculations, including the formula for the speed of a point moving in a circle, as well as explanations that help you understand and master this material. The solution is presented in the form of an electronic document in a clear and accessible form, which makes it easy for even novice users to understand the material. By purchasing this digital product, you receive a high-quality and useful solution to the problem, which can be used to prepare for exams, independently study the material, or use it in educational activities.

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Solution to problem 11.2.21 from the collection of Kepe O.?. consists in determining the absolute velocity modulus of point M at time t=3 s, if the following conditions are given:

• Plate ABCD rotates around the Oz axis with angular velocity ω = 4t.
• Point M moves along the side BC of the plate in the direction from B to C at a constant speed of 9 m/s.
• The length AB is 1 m.

To solve the problem, it is necessary to use a formula to find the absolute speed of a point moving relative to a moving coordinate system in which the object is at rest:

Vabs = Vrel + Vpov,

where Vabs is the absolute speed of the point, Vrel is the relative speed of the point relative to the moving coordinate system, Vpov is the speed of the moving coordinate system relative to the fixed one.

In this problem, the plate rotates around the Oz axis, so the coordinate system associated with the plate is movable. In order to find the speed of point M relative to this moving coordinate system, it is necessary to decompose the speed of point M into two components: parallel to the axis of rotation of the plate and perpendicular to it.

The speed of point M, parallel to the axis of rotation of the plate, is equal to zero, since point M moves only along the side BC, which is perpendicular to the axis of rotation. Therefore, the relative speed of point M relative to the coordinate system associated with the plate is equal to the constant speed of point M perpendicular to the axis of rotation of the plate and is equal to 9 m/s.

The speed of the moving coordinate system relative to the stationary one is determined by the angular velocity of rotation of the plate around the Oz axis, which is equal to 4t. Thus, the speed of the moving coordinate system at time t=3 s will be equal to 4*3=12 rad/s.

Now you can calculate the absolute speed of point M using the formula Vabs = Vrel + Vsur:

Vabs = 9 m/s + 12 m/s = 15 m/s.

Answer: absolute velocity module of point M at time t=3 s is equal to 15 m/s.

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