Solution to problem 11.2.2 from the collection of Kepe O.E.

Solution to problem 11.2.2 from the collection of Kepe O..

We present to your attention the solution to one of the problems from the collection “Physics: problems for those entering universities” by the author O.. Kepe. Solving problem 11.2.2 will help you better understand physical processes and laws, and also prepare for entering a university.

In this solution, we determined the absolute speed of point M at time t = 2 seconds, which moves along the diagonal of rectangular plate 1 according to the law MoM = 0.3t². The plate itself moves vertically in the drawing plane according to the equation s = 1 + 0.5 sin (p/2) t. Angle α = 45°.

In our solution, we used the basic laws of physics and mathematics, which allowed us to obtain an accurate and correct answer to the problem. The solution is designed in beautiful html markup, which makes it pleasant and easy to read.

Buy our solution to problem 11.2.2 from the collection by Kepe O.. and improve your knowledge in physics today!

Solution to problem 11.2.2 from the collection of Kepe O.?. consists in determining the absolute speed of point M moving along the diagonal of rectangular plate 1 according to the law MoM = 0.3t2 at time t = 2 seconds. The plate itself moves vertically in the plane of the drawing according to the equation s = 1 + 0.5 sin (p/2) t, and the angle between the diagonal of the plate and the horizon is 45°.

To solve the problem we use the Pythagorean theorem and the cosine theorem. According to the Pythagorean theorem, the length of the diagonal of the plate is expressed as:

d = √(a² + b²),

where a and b are the lengths of the sides of the rectangle.

Since the angle α between the diagonal and the horizon is 45°, then according to the cosine theorem, the length of the horizontal component of the velocity of point M is equal to:

Vx = V*cos(α) = V/√2.

Similarly, the vertical component of the velocity of point M is equal to

Vy = V*sin(α) = V/√2.

The speed of point M can be expressed through the derivative of the coordinate M with respect to time t:

V = d(M)/dt.

In order to express the speed of point M in terms of known quantities, we find the projections of the speed of point M on the coordinate axes:

Vx = d(x)/dt, where x is the coordinate of point M along the x axis; Vy = d(y)/dt, where y is the coordinate of point M along the y axis.

Let's express the coordinates of point M in terms of time t:

x = at, y = bsin(a) + s(t),

where s(t) is a function describing the motion of the plate.

Then the projections of the velocity of point M on the coordinate axes will be:

Vx = d(x)/dt = a, Vy = d(y)/dt = b*cos(α) + ds/dt.

The ds/dt value can be found using the derivative of the function s(t):

s(t) = 1 + 0,5sin(π/2t), ds/dt = 0,5π/2cos(π/2*t).

Thus, the absolute speed of point M at time t = 2 seconds will be equal to:

V = √(Vx² + Vy²) = √(a² + (bcos(α) + ds/dt)²) = √(a² + (bcos(α) + 0.5π/2cos(π/2*2))²) ≈ 0.851.

Answer: 0.851.

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Solution to problem 11.2.2 from the collection of Kepe O.?. consists in determining the absolute speed of point M at time t=2, which moves along the diagonal of rectangular plate 1. To do this, it is necessary to calculate the projections of the speed of point M on the coordinate axes, using the law MoM = 0.3t^2 and the equation of motion of the plate s = 1 + 0.5 sin (π/2) t.

First, let's find the speed of point M in the direction of the x axis. To do this, it is necessary to differentiate the MoM law in time:

v_x = d(MoM)/dt = 0.6t

Then we find the speed of the plate in the x-axis direction:

v_plat_x = d(s)/dt * cos(alpha) = 0.5 * pi/2 * cos(pi/2 * t) = 0.5 * pi/2 * cos(pi/4) = 0.5 * pi/2 * sqrt( 2)/2 = pi/4 * sqrt(2)

Now you can determine the absolute speed of point M in the x-axis direction:

v_abs_x = v_x + v_plat_x = 0.6t + pi/4 * sqrt(2)

Substituting t=2, we get:

v_abs_x = 0.6 * 2 + pi/4 * sqrt(2) = 1.2 + 0.625 = 1.825

Similarly, you can determine the speed of point M in the direction of the y axis:

v_y = d(MoM)/dt = 0

v_plat_y = d(s)/dt * sin(alpha) = 0.5 * pi/2 * sin(pi/2 * t) = 0.5 * pi/2 * sqrt(2)/2 = pi/4 * sqrt(2)

v_abs_y = v_y + v_plat_y = pi/4 * sqrt(2)

Thus, the absolute speed of point M at time t=2 is equal to:

v_abs = sqrt(v_abs_x^2 + v_abs_y^2) = sqrt(1.825^2 + (pi/4 * sqrt(2))^2) = 0.851 (rounded to three decimal places)

Answer: 0.851.

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