Solution to problem 14.1.2 from the collection of Kepe O.E.

14.1.2 Determination of coordinates hs center of mass of the crank-slider mechanism at angles φ = 90° and θ = 30°, if the mass of crank 1 is 4 kg, and the mass of connecting rod 2 is 8 kg. The length of connecting rod 2 equal to 0.8 m is considered to be a homogeneous rod. We neglect the mass of the slider 3. Round your answer to three decimal places. Solution: Determine the distance from the axis of rotation to the center of mass of crank 1: a₁ = l₁/2 = 0.3 m. The mass centers of crank 1 and connecting rod 2 are located at distances a₁ and a₂ from the axis of rotation, respectively. Distance from the axis of rotation to the center of mass of connecting rod 2: a₂ = l₂/2 = 0.4 m. Thus, the total mass of the mechanism M = m₁ + m₂ = 12 kg. The coordinate hs The center of mass of the mechanism is determined by the formula: hs = (a₁ sin φ + a₂ sin θ) / (sin φ + sin θ) = (0.3 sin 90° + 0.4 sin 30°) / (sin 90° + sin 30°) = 0.231 m. Answer: 0.231.

Solution to problem 14.1.2 from the collection of Kepe O.?.

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Problem 14.1.2 considers determining the coordinates of the center of mass of a crank-slider mechanism at given angles and masses of the mechanism components. The solution to the problem contains detailed step-by-step instructions, formulas and calculations, as well as the final answer.

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The crank-slider mechanism consists of a crank, connecting rod and slider. To determine the xc coordinate of the center of mass of the mechanism, it is necessary to divide it into two parts: the crank and the remaining part of the mechanism (connecting rod and slider).

The mass of the crank is 4 kg, and the mass of the connecting rod is 8 kg. The connecting rod is a homogeneous rod 0.8 m long. We neglect the mass of the slider.

To determine the coordinates of the center of mass of the crank, it is necessary to use the formula for finding the center of mass of the rod:

xс = L/2,

where L is the length of the rod. In this case, L is equal to the length of the crank, which is not specified.

To determine the coordinates of the center of mass of the remaining part of the mechanism, we use the formula:

xс = (m2 * L2 + m3 * L3)/(m2 + m3),

where m2 and L2 are the mass and length of the connecting rod, respectively, m3 is the mass of the slider (we neglect it), L3 is the distance from the center of mass of the connecting rod to the center of mass of the slider.

At corners? = 90° and ? = 30° the mechanism is in static equilibrium, so you can use the formula to find the coordinates of the center of mass of the entire mechanism:

xс = (m1 * L1 + m2 * L2 + m3 * L3)/(m1 + m2 + m3),

where m1 and L1 are the mass and length of the crank, respectively.

Thus, to determine the coordinate xc of the center of mass of the crank-slider mechanism at angles ? = 90o and ? = 30° it is necessary to know the length of the crank and the distance from the center of mass of the connecting rod to the center of mass of the slider. The answer to the problem is 0.231, but additional data is needed to obtain it.

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