Solution to problem 11.2.5 from the collection of Kepe O.E.

In the problem it is necessary to determine the absolute speed of point M at time t = 1 s. The movement of point M along square plate 1 is described by the equation BM = 0.1t2. Cranks AB = CD = 0.5 m rotate according to the law of angular velocity? = 0.25?t.

To solve the problem, we use the formula for the absolute speed of a point located on a rigid body moving translationally and rotating at the same time:

VM = Vp + Vvr,

where Vп is the speed of point M relative to the center of the plate, Vвр is the speed of the center of the plate relative to the fixed coordinate system.

Let us find the speed of the center of the plate relative to the fixed coordinate system:

Vвр = R x ?,

where R is the radius of the crank, ? - angular velocity of the crank.

Since the cranks are the same, the speed of the center of the plate is equal to:

Vвр = 0,5 x 0,25 ?t = 0,125 ?t.

Let's find the speed of point M relative to the center of the plate:

Vп = d(BM)/dt,

where BM is the distance between the center of the plate and point M.

Let's differentiate the equation VM = 0.1t2:

VМ = d(0,1t2)/dt = 0,2t.

Then:

BM = a/2 + ?(VМt)^2,

where a is the length of the side of the plate.

At t = 1 with:

BM = 0.5/2 + ?(0.2)^2 = 0.55 м.

Now we can find the absolute speed of point M:

VM = Vn + Vvr = 0.2 - 0.125 = 0.075 m/s.

Answer: 0.075 m/s.

Solution to problem 11.2.5 from the collection of Kepe O.?.

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In this digital product you will find a complete and detailed solution to Problem 11.2.5, which describes the motion of point M on a square plate given by the equation BM = 0.1t2. Cranks AB = CD = 0.5 m rotate according to the law of angular velocity? = 0.25?t. The solution was completed by a professional mathematician and presented in a convenient format.

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The product description is an electronic version of the solution to problem 11.2.5 from the collection of Kepe O.?. The task is to determine the absolute speed of point M at time t = 1 s, when moving along a square plate 1 with the equation BM = 0.1t2. Cranks AB = CD = 0.5 m rotate according to the law of angular velocity? = 0.25?t. The solution to the problem was completed by a professional mathematician and presented in a convenient format. By purchasing this product, you get access to an accurate and high-quality solution to the problem, which will help you better understand the topic and successfully solve kinematics problems. The answer to the problem is 0.438 m/s.

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Solution to problem 11.2.5 from the collection of Kepe O.?. consists in determining the absolute speed of point M at time t = 1 s, if its movement along square plate 1 is given by the equation BM = 0.1t2. Cranks AB = CD = 0.5 m rotate according to the law? = 0.25?t.

To solve the problem, you need to use the formula to find the absolute speed of a point on the crank:

v(abs) = v(otn) + R * w,

where v(rel) is the relative speed of point M relative to the crank, R is the radius of the crank, w is the angular velocity of the crank.

The first step is to find the angular velocity of the crank, which is given by the law of rotation? = 0.25?t. Substituting t = 1 s, we get:

? = 0.25 * 1 = 0.25 rad/s.

Then we determine the relative speed of point M relative to the crank. To do this, it is necessary to express the coordinates of point M through the angle of rotation of the crank:

x = AB + BMcos(?), y = BMsin(?),

where BM is the distance from the center of the crank to point M.

Differentiating these expressions with respect to time, we obtain the speed of point M relative to the crank:

vx = -BM*?son(?), vy = BM?*cos(?).

Substituting values? and BM, we get:

vx = -0,50,25sin(0.25) = -0.054 m/s, vy = 0.50,25cos(0.25) = 0.473 m/s.

Finally, we find the absolute speed of point M using the formula:

v(abs) = v(otn) + R * w,

where R = AB = 0.5 m - crank radius. Substituting the values, we get:

v(abs) = sqrt(vx^2 + vy^2) + R * ? = sqrt(0.054^2 + 0.473^2) + 0.5 * 0.25 = 0.438 m/s.

Thus, the absolute speed of point M at time t = 1 s is equal to 0.438 m/s.

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Problem 11.2.5 from the collection of Kepe O.E. Perfect for preparing for exams and tests.

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