Concept explainers
(a)
The angular velocity of the sheet metal component at time
Answer to Problem 18.78P
The angular velocity of the sheet metal component at time
Explanation of Solution
Given information:
The total mass is
Write the expression for the angular velocity of sheet metal component.
Here, the angular velocity of the sheet metal component is
Calculation:
Substitute
Conclusion:
The angular velocity of the sheet metal component at time
(b)
The dynamic reactions at
The dynamic reactions at
Answer to Problem 18.78P
The dynamic reactions at
The dynamic reactions at
Explanation of Solution
Write the expression for the sum of the moment acting on the body along x -direction.
Here, the product of the moment of the inertia of
Write the expression for the sum of the moment acting on the body along y -direction.
Write the expression for the sum of the moment acting on the body along z -direction.
Here, the moment of the inertia along the z -direction is
Draw the diagram for the for the sheet metal component.
Figure-(1)
Write the expression for the area of the section 1 shown in the Figure-(1).
Here, the constant dimension is
Write the expression for the area of the section 2 shown in the Figure-(1).
Write the expression for the area of the section 3 shown in the Figure-(1).
Write the expression for the total area of the sheet.
Substitute
Write the expression of mass per unit area of the system.
Here, the mass of the sheet metal component is
Write the expression for the variation of the
Here, the coordinate of the considered point is
The below figure represent the schematic diagram of the elemental strip of section 1.
Figure-(2)
Write the expression for the distance of the centroid of the element from the
Write the expression for the mass of the elemental strip.
Here, the area of the elemental strip is
Write the expression for the moment of inertia of the element with respect to z- axis.
Write the expression for the moment of the inertia of the section 1.
Write the expression for the product of moment of inertia of the plane
Write the expression for the product of moment of inertia of the plane
Write the expression for the variation of the
The below figure represent the schematic diagram of the elemental strip of section 2.
Figure-(3)
Write the expression for the mass of the elemental strip of section 2.
Write the expression for the moment of the inertia of the section 2.
Write the expression for the product of moment of inertia of the plane
Write the expression for the product of moment of inertia of the plane
Write the expression of mass per unit area of the section3 in Figure-(1).
Here, the mass of the rectangular sheet metal component is
Write the expression for the moment of the inertia of the section 3.
The product moment of the inertia for the plane
Write the expression for the moment of the inertia of the whole system.
Write the expression for the product of moment of inertia of the whole system.
Write the expression for the product of moment of inertia of the whole system.
Draw the diagram for the system to shows the action of forces on the system.
Figure-(4)
Here, the reaction on the point
Write the expression for the dynamic reaction at point
Write the expression for the dynamic reaction at point
Write the expression for the reaction forces along the y- direction.
Write the expression for the reaction forces along the x- direction.
Write the expression for the sum of the moment acting on the body along x -direction.
Here, distance between the point
Write the expression for the sum of the moment acting on the body along y -direction.
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Conclusion:
The dynamic reactions at
The dynamic reactions at
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Chapter 18 Solutions
Vector Mechanics For Engineers
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