Statics@ptu
PTU
Mechanical Engineering Division
Statics Tutorial Sheet 5.1 (for chapter 5)
PROBLEM 5.1
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 5.60 in, Y = 6.60 in.)
PROBLEM 5.2
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 96.4 mm, Y = 34.7 mm)
PROBLEM 5.3
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 12.00 in., Y = 16.00 in.)
PROBLEM 5.4
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 1.421 mm. Y = 12.42 mm.)
PROBLEM 5.5
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 49.4 mm, Y = 93.8 mm.)
PROBLEM 5.6
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 0.721 in., Y = 3.40 in..)
PROBLEM 5.7
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 31.1 mm. Y =X = 31.1 mm..)
PROBLEM 5.8
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 0., Y = 3.23 in.)
PROBLEM 5.9
For the area of Problem 5.8, determine the ratio r2/r1 so that y= 3r1/4.
[pic]
(Ans.: 16p2 + (16−9π)p+(16−9π)=0, p = r2/r1.)
PROBLEM 5.10
Show that as r1 approaches r2, the location of the centroid approaches that
of a circular arc of radius ( ) r1+r2 / 2.
[pic]
(Ans.: r2/r1.= 1.340)
PROBLEM 5.11
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 0., Y = 0.632 in..)
PROBLEM 5.12
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = −15.83 mm., Y = 3.34 mm.)
PROBLEM 5.13
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 32.0 mm, Y = 20.0 mm.)
PROBLEM 5.14
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 60.0 mm, Y = 129.6 mm.)
PROBLEM 5.15
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 5.95 in., Y = 14.41 in.)
PROBLEM 5.16
Locate the centroid of the plane area shown.
[pic]
(Ans.: X = 3.21 in., Y = 3.31 in.)
PROBLEM 5.17
The horizontal x axis is drawn through the centroid C of the area shown and divides the area into two component areas A1 and A2. Determine the first moment of each component area with respect to the x axis, and explain the results obtained
[pic]
(Ans.: Qx1 = 25.0×10 mm3, Qx2 = −25.0 × 10 mm3)
PROBLEM 5.18
The horizontal x axis is drawn through the centroid C of the area shown and divides the area into two component areas A1 and A2. Determine the first moment of each component area with respect to the x axis, and explain the results obtained.
[pic]
(Ans.: AI = 1.393 x106 mm3, AII = -1.393 x106 mm3)
PROBLEM 5.19
The first moment of the shaded area with respect to the x axis is denoted by Qx . (a) Express Qx in terms of r and θ . (b) For what value of θ is Qx maximum, and what is the maximum value?
[pic]
(Ans.: a. Qx = 2/3 r3 cos3 (, b. Qx = 2/3 r3)
PROBLEM 5.20
A composite beam is constructed by bolting four plates to four 2×2×3/8-in. angles as shown. The bolts are equally spaced along the
beam, and the beam supports a vertical load. As proved in mechanics of materials, the shearing forces exerted on the bolts at A and B is
proportional to the first moments with respect to the centroidal x axis of the shaded areas shown, respectively, in parts a and b of the figure. Knowing that the force exerted on the bolt at A is 70 lb, determine the
force exerted on the bolt at B.
[pic]
(Ans.: FB = 115.3 lb)
PROBLEM 5.21
A thin, homogeneous wire is bent to form the perimeter of the figure indicated. Locate the centre of gravity of the wire figure thus formed.
[pic]
(Ans.: X = 4.67 in, Y = 6.67 in.)
PROBLEM 5.22
A thin, homogeneous wire is bent to form the perimeter of the figure
indicated. Locate the centre of gravity of the wire figure thus formed.
[pic]
(Ans.: X = 92.2 mm, Y = 32.4 mm.)
PROBLEM 5.23
A thin, homogeneous wire is bent to form the perimeter of the figure
indicated. Locate the centre of gravity of the wire figure thus formed.
[pic]
(Ans.: X = 1.441 mm, Y = 12.72 mm)
PROBLEM 5.24
A thin, homogeneous wire is bent to form the perimeter of the figure indicated. Locate the centre of gravity of the wire figure thus formed.
[pic]
(Ans.: X = 0, Y = 2.94 in)
PROBLEM 5.25
A 750 = g uniform steel rod is bent into a circular arc of radius 500 mm as shown. The rod is supported by a pin at A and the cord BC. Determine :
a) the tension in the cord,
b) (b) the reaction at A.
[pic]
(Ans.: X = 1.441 mm, Y = 12.72 mm)
PROBLEM 5.26
The homogeneous wire ABCD is bent as shown and is supported by a pin at B. Knowing that l = 8 in., determine the angle θ for which portion
BC of the wire is horizontal.
[pic]
(Ans.: θ = 63.6°)
PROBLEM 5.27
The homogeneous wire ABCD is bent as shown and is supported by a pin at B. Knowing that θ = 30°, determine the length l for which portion CD of the wire is horizontal.
[pic]
(Ans. l = 12.77 in.)
PROBLEM 5.28
The homogeneous wire ABCD is bent as shown and is attached to a hinge at C. Determine the length L for which the portion BCD of the wire is horizontal.
[pic]
(Ans.: L=12.00 in)
PROBLEM 5.29
Determine the distance h so that the centroid of the shaded area is as close to line BB′ as possible when (a) k = 0.2, (b) k = 0.6.
[pic]
(Ans.: h=0.472a, h = 0.387a)
PROBLEM 5.30
Show when the distance h is selected to minimize the distance y from line BB′ to the centroid of the shaded area that y=h.
[pic]
PROBLEM 5.31
Determine by direct integration the centroid of the area shown. Express your answer in terms of a and h.
[pic]
(Ans.: x' = a/3, y' = 2h/3)
PROBLEM 5.32
Determine by direct integration the centroid of the area shown. Express your answer in terms of a and h.
[pic]
(Ans.: x' = 2a/5, y' = 4h/7)
PROBLEM 5.33
Determine by direct integration the centroid of the area shown. Express your answer in terms of a and h.
[pic]
(Ans.: x' = 16a/35, y' = 16h/35)
PROBLEM 5.34
Determine by direct integration the centroid of the area shown.
[pic]
(Ans.: x' = 0, [pic])
PROBLEM 5.35
Determine by direct integration the centroid of the area shown
[pic]
(Ans. X' = 0, [pic])
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