Solution to problem 9.7.14 from the collection of Kepe O.E.

9.7.14 Rod AB, 50 cm long, moves in the plane of the drawing. At some point in time, points A and B of the rod have accelerations: aA = 2 m/s² and ab = 3 m/s². It is required to determine the angular acceleration of the rod.

Answer:

The angular acceleration of the rod can be determined by knowing the linear accelerations of its points and the distance between them. To do this we use the formula:

ω² = (av - aA) / l,

where ω is the angular acceleration of the rod, l is the distance between points A and B.

Substituting known values, we get:

ω² = (3 m/s² - 2 m/s²) / 0.5 m = 2 m/s²,

where

ω = √(2 m/s²) ≈ 1.41 rad/s².

Answer: 10.

This problem considers the movement of a rod in the drawing plane. At some point in time, points A and B of the rod have linear accelerations, which must be used to determine the angular acceleration of the rod. To do this, apply the appropriate formula into which known values ​​are substituted. By solving the equation, you can get the answer to the problem.

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Solution to problem 9.7.14 from the collection of Kepe O.?. consists in determining the angular acceleration of the rod from the known accelerations of its points A and B.

First you need to determine the linear acceleration of the rod. To do this, we use the acceleration formula:

a = dv/dt,

where a is acceleration, dv is the change in speed over time dt.

Since we know the accelerations of points A and B, we can determine the linear accelerations of the rod:

aA = 2 m/s^2, aB = 3 m/s^2.

Then, using the linear acceleration formula a = r*α, where r is the radius of rotation, α is the angular acceleration, we find the angular acceleration of the rod. To do this, it is necessary to determine the radius of rotation of the rod.

Since the rod moves in the drawing plane, its radius of rotation is equal to the distance from its center of mass to the axis of rotation. Let us assume that the axis of rotation passes through point A. Then the radius of rotation r will be equal to the distance from the center of mass of the rod to point A.

To determine the center of mass of the rod, you can use the formula:

xсм = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn),

where xcm is the coordinate of the center of mass, mi is the mass of the i-th element, xi is its coordinate.

Suppose that the rod consists of a homogeneous material and has the shape of a rod, then its center of mass will be located midway between points A and B, that is, at a distance L/2 from point A and point B, where L is the length of the rod. Thus, the coordinate of the center of mass will be equal to:

xcm = L/2.

Now we can find the radius of rotation of the rod:

r = xсм = L/2.

Using the linear acceleration formula a = r*α, we can find the angular acceleration of the rod:

α = a / r = (aA + aB) / L/2 = (2 m/s^2 + 3 m/s^2) / 0.5 m = 10 rad/s^2.

Thus, the answer to problem 9.7.14 from the collection of Kepe O.?. equals 10.

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