Free Printable Mechanics

1.

Which beam can carry more load? The left beam’s cross-section shows a maximum distance to the centroid of 25 mm, while the right beam’s cross-section shows a maximum distance to the centroid of 100 mm. Both beams are depicted under a downward load with end supports.

a)

The beam with maximum distance of 100 mm to the centroid (right)

b)

The beam with maximum distance of 25 mm to the centroid (left)

c)

Both beams carry the same load regardless of cross-section

d)

Neither; load capacity does not depend on section inertia

2.

Using I = , compute the area moment of inertia for a rectangular section oriented vertically with B = 1 and H = 4. Give the value in cm^4.

a)

0.33 cm^4

b)

1.33 cm^4

c)

5.33 $cm^4$

d)

16.00 $cm^4$

3.

Using I = , compute the area moment of inertia for a rectangular section oriented horizontally with B = 4 and H = 1. Give the value in cm^4.

a)

0.33 cm^4

b)

1.33 cm^4

c)

5.33 $cm^4$

d)

16.00 $cm^4$

4.

For the two rectangles of equal area shown (B = 1, H = 4) and (B = 4, H = 1), what is the ratio of their moments of inertia $I_{vertical} / I_{horizontal}$ based on $I = \frac{BH^3}{12}$ ?

a)

4

b)

8

c)

12

d)

16

5.

According to the general definition in Mechanics of Materials, which statement correctly characterizes the area moment of inertia for a sectional area?

a)

It is a load-dependent property varying with applied force.

b)

It is a geometrical property of a sectional area, denoted I (e.g., Ix).

c)

It is a material constant independent of shape.

d)

It is defined only for circular sections.

6.

In Mechanics of Materials, how does the magnitude of the area moment of inertia affect bending stresses in a beam of a given sectional area?

a)

Higher moment of inertia increases bending stresses.

b)

Higher moment of inertia lowers bending stresses due to bending.

c)

Moment of inertia has no effect on bending stresses.

d)

Lower moment of inertia lowers bending stresses due to bending.

7.

Which analytical expression defines Ix about a centroidal axis for an area A?

a)

Ix = $\iint A x^2 dA$

b)

Ix = $\iint A y^2 dA$

c)

Ix = ∬A xy dA

d)

Ix = $\iint A r^2 dA$

8.

For a rectangular section of width b and height h, what is the centroidal moment of inertia Ix (axis parallel to b)?

a)

$\frac{bh^3}{12}$

b)

$Ix = \frac{b^3h}{12}$

c)

Ix = bh/12

d)

$Ix = \frac{bh^2}{3}$

9.

For a solid circular section of radius R, what is the centroidal moment of inertia about the x-axis?

a)

$Ix = \pi R^4/4$

b)

$Ix = \pi R^3/3$

c)

$Ix = \pi R^2/2$

d)

Ix = $\pi R^4/2$

10.

For a solid circular section, which statement about the polar moment of inertia Ip over the area A is correct?

a)

Ip equals Ix only.

b)

Ip equals Iy only.

c)

Ip equals Ix + Iy.

d)

Ip equals Ix − Iy.

11.

Which relations are correctly recalled for polar coordinates in the context of the circular section? Select all that apply.

a)

$r^2 = x^2 + y^2$

b)

x = r cosθ

c)

y = r sinθ

d)

x = r sinθ

12.

For a circular section, what is the polar moment of inertia Jo about the centroid?

a)

Jo = $\pi r^4/4$

b)

Jo = $\pi r^4/3$

c)

Jo = $\frac{1}{2} \pi r^4$

d)

Jo = $2 \pi r^4$

13.

For a rectangular section with base b and height h, what is the centroidal polar moment Jc shown in the table?

a)

Jc = $(1/12) b h (b^2 + h^2)$

b)

Jc = $(\frac{1}{2}) b h (b^2 + h^2)$

c)

Jc = $(1/36) b h (b^2 + h^2)$

d)

$Jc = b h (b^2 + h^2)$

14.

For the right triangular area of base b and height h shown, with centroid marked at C located at b/3 and h/3 from the right-angle corner, what is the product moment of inertia about the centroidal x and y axes?

a)

$I_{xy}=-\dfrac{h^2 b^2}{72}$

b)

$I_{xy}=\dfrac{h^2 b^2}{72}$

c)

$I_{xy}=0$

d)

$I_{xy}=-\dfrac{hb}{12}$

15.

According to the definition shown, which integral gives the product moment of inertia of an area A about orthogonal axes x and y?

a)

$I_{xy}=\iint_A x\,y\,\mathrm{d}A$

b)

$I_{xy}=\iint_A x^2\,\mathrm{d}A$

c)

$I_{xy}=\iint_A y^2\,\mathrm{d}A$

d)

$I_{xy}=\iint_A (x+y)\,\mathrm{d}A$

16.

If a cross-section has at least one axis of symmetry (such as a rectangle or a circle), what is its product moment of inertia $I_{xy}$ about centroidal axes?

a)

$I_{xy}=0$

b)

$I_{xy}>0$

c)

$I_{xy}<0$

d)

$I_{xy}$ is undefined

17.

From the tabulated results for standard shapes, what is the product moment of inertia $I_{xy}$ for a rectangle about its centroidal axes?

a)

$I_{xy}=0$

b)

$I_{xy}=\dfrac{b^2 h^2}{24}$

c)

$I_{xy}=\dfrac{bh}{12}$

d)

$I_{xy}=-\dfrac{b^2 h^2}{72}$

18.

Using the parallel axis theorem, which expression correctly relates the moment of inertia about an axis x′ parallel to x and offset by $y_G$ from the centroid?

a)

$I_{x'}=I_x+y_G^{2}A$

b)

$I_{x'}=I_x+x_G^{2}A$

c)

$I_{x'}=I_x-y_G^{2}A$

d)

$I_{x'}=A+y_G^{2}I_x$

19.

In the derivation of the parallel axis theorem for $I_{x'}$ , the term involving $\iint_A y\,y_G'\,\mathrm{d}A$ evaluates to which value for a centroidal axis shift?

a)

0

b)

$y_G' A$

c)

$x_G' A$

d)

$I_x$

20.

For a composite section made of n sub-areas, which formula correctly computes $I_x$ about the composite centroid C?

a)

$I_x=\sum_{i=1}^{n}\left(I_{x c_i}+A_i\,(Y_C-Y_{c_i})^{2}\right)$

b)

$I_x=\sum_{i=1}^{n}\left(I_{y c_i}+A_i\,(X_C-X_{c_i})^{2}\right)$

c)

$I_x=\sum_{i=1}^{n}\left(A_i\,(X_C-X_{c_i})(Y_C-Y_{c_i})\right)$

d)

$I_x=\sum_{i=1}^{n} I_{x c_i}$

21.

For the same composite section, which expression is used to compute the product moment of inertia $I_{xy}$ about the composite centroid C?

a)

$I_{xy}=\sum_{i=1}^{n}\left(I_{xy c_i}+A_i\,(X_C-X_{c_i})(Y_C-Y_{c_i})\right)$

b)

$I_{xy}=\sum_{i=1}^{n}\left(I_{x c_i}+A_i\,(Y_C-Y_{c_i})^{2}\right)$

c)

$I_{xy}=\sum_{i=1}^{n}\left(I_{y c_i}+A_i\,(X_C-X_{c_i})^{2}\right)$

d)

$I_{xy}=\sum_{i=1}^{n} A_i$

22.

Using the composite-area centroid definition shown, which expression gives the X-coordinate of the centroid for n parts?

a)

$X_C=\dfrac{\sum_{i=1}^{n} x_{c_i}A_i}{\sum_{i=1}^{n} A_i}$

b)

$X_C=\dfrac{\sum_{i=1}^{n} y_{c_i}A_i}{\sum_{i=1}^{n} A_i}$

c)

$X_C=\dfrac{\sum_{i=1}^{n} A_i}{\sum_{i=1}^{n} x_{c_i}}$

d)

$X_C=\sum_{i=1}^{n} x_{c_i}$

23.

For a composite area made of n parts, select all correct expressions for the moments/products of inertia about the centroid (parallel-axis form).

a)

$I_x=\sum_{i=1}^{n}\big(I_{x c_i}+A_i\,(Y_C-Y_{c_i})^2\big)$

b)

$I_y=\sum_{i=1}^{n}\big(I_{y c_i}+A_i\,(X_C-X_{c_i})^2\big)$

c)

$I_{xy}=\sum_{i=1}^{n}\big(I_{xy c_i}+A_i\,(X_C-X_{c_i})(Y_C-Y_{c_i})\big)$

d)

$I_x=\sum_{i=1}^{n}\big(I_{x c_i}+A_i\,(X_C-X_{c_i})^2\big)$

24.

For a rectangular shape about its centroidal axes, what is the expression for $I_x$ ?

a)

$I_x=\dfrac{b h^3}{12}$

b)

$I_x=\dfrac{b^3 h}{12}$

c)

$I_x=\dfrac{b h}{12}$

d)

$I_x=\dfrac{h^3}{b}$

25.

For a rectangular shape about its centroidal axes, what is the expression for $I_y$ ?

a)

$I_y=\dfrac{b^3 h}{12}$

b)

$I_y=\dfrac{b h^3}{12}$

c)

$I_y=\dfrac{b h}{12}$

d)

$I_y=\dfrac{h^3}{b}$

26.

For the rectangular shape shown with an axis of symmetry, what is the centroidal product of inertia $I_{xy}$ ?

a)

$I_{xy}=0$

b)

$I_{xy}=\dfrac{b h^2}{12}$

c)

$I_{xy}=\dfrac{b^2 h}{12}$

d)

$I_{xy}=b h$

27.

In the stepped-triangle area shown, what is the total base length along the X-axis?

a)

$5\,\text{ft}$

b)

$6\,\text{ft}$

c)

$7\,\text{ft}$

d)

$4\,\text{ft}$

28.

In the stepped-triangle area shown, what is the maximum vertical height of the shape measured from the base?

a)

$1\,\text{ft}$

b)

$2\,\text{ft}$

c)

$3\,\text{ft}$

d)

$4\,\text{ft}$

29.

For a planar area with centroidal axes x and y rotated by an angle θ to new axes x′ and y′ as shown, which transformation correctly relates the coordinates of a differential area element?

a)

x′ = x cos θ + y sin θ, y′ = y cos θ − x sin θ

b)

x′ = x cos θ − y sin θ, y′ = y cos θ + x sin θ

c)

x′ = x sin θ + y cos θ, y′ = y sin θ − x cos θ

d)

x′ = x sin θ − y cos θ, y′ = y sin θ + x cos θ

30.

For the rotated axes x′ and y′ at angle θ, which expression gives the area moment of inertia about x′ in terms of the original moments about x and y and the product moment Ixy? Use standard trigonometric identities for cos 2θ and sin 2θ.

a)

Ix′ = (Ix + Iy)/2 + ((Ix − Iy)/2) cos 2θ − Ixy sin 2θ

b)

Ix′ = (Ix + Iy)/2 − ((Ix − Iy)/2) cos 2θ + Ixy sin 2θ

c)

Ix′ = $Ix cos^2 θ + Iy sin^2 θ + 2 Ixy sin θ cos θ$

d)

Ix′ = $Ix sin^2 θ + Iy cos^2 θ − 2 Ixy sin θ cos θ$

31.

Under principal axes for an area, which statement is correct regarding the product moment of inertia?

a)

Ixy′ = 0

b)

Ixy′ = Ixy

c)

Ixy′ = (Ix + Iy)/2

d)

Ixy′ = ((Ix − Iy)/2)

32.

For an area rotated by θ, which formula correctly transforms the product moment of inertia to the rotated basis?

a)

Ix′y′ = Ixy cos 2θ + ((Ix − Iy)/2) sin 2θ

b)

Ix′y′ = Ixy sin 2θ + ((Ix − Iy)/2) cos 2θ

c)

Ix′y′ = (Ix + Iy)/2 cos 2θ − Ixy sin 2θ

d)

Ix′y′ = (Ix − Iy) sin θ cos θ

33.

Which recap equation correctly gives Ix′ for a basis rotated by angle θ?

a)

Ix′ = (Ix + Iy)/2 + ((Ix − Iy)/2) cos 2θ − Ixy sin 2θ

b)

Ix′ = (Ix + Iy)/2 − ((Ix − Iy)/2) cos 2θ + Ixy sin 2θ

c)

Ix′ = (Ix − Iy)/2 − ((Ix + Iy)/2) cos 2θ + Ixy sin 2θ

d)

Ix′ = (Ix + Iy) cos 2θ − Ixy sin 2θ

34.

Which recap equation correctly gives Iy′ for a basis rotated by angle θ?

a)

Iy′ = (Ix + Iy)/2 − ((Ix − Iy)/2) cos 2θ + Ixy sin 2θ

b)

Iy′ = (Ix + Iy)/2 + ((Ix − Iy)/2) cos 2θ − Ixy sin 2θ

c)

Iy′ = (Ix − Iy)/2 + ((Ix + Iy)/2) cos 2θ − Ixy sin 2θ

d)

Iy′ = (Ix + Iy) cos 2θ + Ixy sin 2θ

35.

Which recap equation correctly states the product moment in the rotated basis?

a)

Ix′y′ = Ixy cos 2θ + ((Ix − Iy)/2) sin 2θ

b)

Ix′y′ = Ixy sin 2θ + ((Ix − Iy)/2) cos 2θ

c)

Ix′y′ = (Ix + Iy)/2 − Ixy cos 2θ

d)

Ix′y′ = ((Ix − Iy)/2) cos 2θ − Ixy sin 2θ

36.

For principal moments (max/min), the angle to principal axes satisfies which relation based on the derivative condition dIx′/dθ = 0?

a)

tan 2θ = Ixy / ((Ix − Iy)/2)

b)

tan 2θ = ((Ix − Iy)/2) / Ixy

c)

tan 2θ = Ixy / (Ix + Iy)

d)

tan 2θ = (Ix + Iy) / Ixy

37.

Which expressions correctly give the principal moments of inertia for a planar area?

a)

Imax = $(Ix + Iy)/2 + \sqrt{\left(\frac{(Ix − Iy)}{2}\right)^2 + Ixy^2}$

b)

Imin = $(Ix + Iy)/2 − sqrt(((Ix − Iy)/2)^2 + Ixy^2)$

c)

Imax = $(Ix − Iy)/2 + \sqrt{((Ix + Iy)/2)^2 + Ixy^2}$

d)

Imin = $(Ix − Iy)/2 − sqrt(((Ix + Iy)/2)^2 + Ixy^2)$

38.

When transforming to principal axes, which statement is consistent with the max/min moment results shown?

a)

Imax ≥ Imin and both equal (Ix + Iy)/2 only when Ixy = 0 and Ix = Iy

b)

Imax ≤ Imin and both equal (Ix + Iy)/2 only when Ixy ≠ 0

c)

Imax = Imin always and equals (Ix + Iy)/2

d)

Imax = $(Ix + Iy)/2 − sqrt(((Ix − Iy)/2)^2 + Ixy^2)$

Mechanics

Total questions: 38

Worksheet time: 19mins

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