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Free FE Mechanical Example Practice Problems

We've selected 10 diverse practice problems from our question bank that you can use to review for the Mechanical engineering FE exam and give you an idea about some of the content we provide.

1) Given the values below, what is nearest the total power dissipated for this circuit? $R_1 = \SI{10}{\Omega}$, $R_2 = \SI{3}{\Omega}$, $R_3 = \SI{8}{\Omega}$ $V_1 = \SI{5}{V}$, $V_2 = \SI{8}{V}$

2) The following three lines form a triangle with vertices at what points? $$x+3y=1$$ $$2x+y=0$$ $$-2x+y=0$$

3) A car accelerates at $a(t) = 2t^2 + 2$ with an initial velocity of 10 m/s. How fast is it going after 3 s?

4) The plot below represents the movement of a block on a rough surface due to an external force. Select all that represent the line segment $b$.

5) An aluminum rod with diameter of $d = \SI{12}{\milli\meter}$ is loaded axially in compression as shown. Using a Young's modulus of $\SI{70}{\giga\pascal}$ and Poisson ratio of 0.35, what is most nearly the increase in diameter of the rod at the midspan of the rod? Assume the internal loading at the midspan of the rod is -500 kN.

6) What is approximately the velocity of the incompressible flow at $x_2$ if the flow exits the pipe at atmospheric pressure and $ h = \SI{0.5}{m}$? Ignore friction losses

7) A parallel flow heat exchanger operates with water ($c_{p,w} = \SI{4.186}{\kilo\joule\per\kilogram\per\kelvin}$) and oil ($c_{p,o} = \SI{3.4}{\kilo\joule\per\kilogram\per\kelvin}$). Water, the cold fluid, enters the inner tube at $\SI{300}{\kelvin}$ and leaves at $\SI{400}{\kelvin}$. What is the log mean temperature difference if oil, the hot fluid, enters the heat exchanger at $\SI{550}{\kelvin}$ and leaves the system at $\SI{450}{\kelvin}$?

8) A total of 50 valve pressure measurements were taken but only 9 of those measurements will be utilized. What is most nearly the standard deviation of the sample measurements? $$ \begin{array}{c|c} \text{Pressure} & \text{# of Measurements} \\ \hline 20 & 3 \\ 22 & 4 \\ 25 & 2 \\ \end{array} $$

9) In the stress-strain diagram, select all that point A describes.


Solutions

1) Given the values below, what is nearest the total power dissipated for this circuit? $R_1 = \SI{10}{\Omega}$, $R_2 = \SI{3}{\Omega}$, $R_3 = \SI{8}{\Omega}$ $V_1 = \SI{5}{V}$, $V_2 = \SI{8}{V}$


A.1.1 W
B.3.0 W
C.7.1 W
D.21 W
The correct answer is B.

Explanation:

Refer to the Power Absorbed by a Resistive Element section in the Electrical and Computer Engineering chapter of the FE Reference Handbook.$$ P = \frac{V^2}{R} $$ $$ = \frac{(V_2-V_1)^2}{R_3} + \frac{(V_1)^2}{R_1+R_2} $$ $$ = \frac{(\SI{8}{V}-\SI{5}{V})^2}{\SI{8}{\Omega}} + \frac{(\SI{5}{V})^2}{\SI{13}{\Omega}} $$ $$ \approx \SI{3.05}{W} $$

2) The following three lines form a triangle with vertices at what points? $$x+3y=1$$ $$2x+y=0$$ $$-2x+y=0$$

A.$ (- \frac{1}{5},\frac{2}{5}), (0,0), (\frac{1}{7}, \frac{2}{7}) $
B.$ (- \frac{1}{5},\frac{2}{5}), (1,1), (\frac{1}{7}, \frac{2}{7}) $
C.$ (- \frac{1}{5},-\frac{2}{5}), (0,1), (\frac{1}{7}, \frac{2}{7}) $
D.$ (\frac{1}{5},\frac{2}{5}), (0,0), (\frac{1}{7}, \frac{2}{7}) $
The correct answer is A.

Explanation:

Find the intersection for each pair of lines. Using the initial two equations, let's try substituting the second into the first by solving the second for y: $$ y = -2x \\ \Rightarrow x+3(-2x)=1 \\ -5x = 1 \\ x = - \frac{1}{5} $$ Then plug that back into either of the first two equations to find the y-coordinate of the intersection. $$ y = -2(- \frac{1}{5}) = \frac{2}{5} \\ \rightarrow (x,y)=(- \frac{1}{5},\frac{2}{5}) $$ Now we do the same thing twice for the second and third equations and then the first and third equations. For the second and third equations, we see that $y=-y$, so the second intersection we know that only (0,0) will work. For the last pair: $$ y = 2x \\ \Rightarrow x+3(2x)=1 \\ 7x = 1 \\ x = \frac{1}{7} \\ \rightarrow (x,y) = (\frac{1}{7}, \frac{2}{7}) $$ We have: $$ (- \frac{1}{5},\frac{2}{5}), (0,0), (\frac{1}{7}, \frac{2}{7}) $$

3) A car accelerates at $a(t) = 2t^2 + 2$ with an initial velocity of 10 m/s. How fast is it going after 3 s?

A.34 m/s
B.97 m/s
C.60 m/s
D.50 m/s
The correct answer is A.

Explanation:

Refer to the Particle Rectilinear Motion section in the Dynamics chapter of the FE Reference Handbook. $$ a = \mathrm{d}v/\mathrm{d}t $$ $$ v = \int a\,\mathrm{d}t $$ $$ v = \frac{2}{3}t^3 + 2t + v_0 $$ $$ = \frac{2}{3}(3)^3 + 2(3) + 10 $$ $$ = \SI{34}{m/s} $$

4) The plot below represents the movement of a block on a rough surface due to an external force. Select all that represent the line segment $b$.


The correct answers are A, B, and D.

Explanation:

The block begins moving and remains in motion during $b$ because $x(t)$ is increasing. This is where the coefficient of kinetic friction applies. Work is being done because a force is acting over a distance.

Also, because there is motion but no acceleration on the block, it's also in dynamic equilibrium.

5) An aluminum rod with diameter of $d = \SI{12}{\milli\meter}$ is loaded axially in compression as shown. Using a Young's modulus of $\SI{70}{\giga\pascal}$ and Poisson ratio of 0.35, what is most nearly the increase in diameter of the rod at the midspan of the rod? Assume the internal loading at the midspan of the rod is -500 kN.


A.$\SI{49}{\micro\meter}$
B.0.31 mm
C.$\SI{265}{\micro\meter}$
D.230 nm
The correct answer is C.

Explanation:

Refer to the Uniaxial Stress-Strain section in the Mechanics of Materials chapter of the FE Reference Handbook.

Ultimately, we must determine the increase in rod diameter after the rod is compressed. We can determine the increase in length of a member from its engineering strain. $$ \epsilon=\Delta L/L_0 $$ Which can be interpreted as $$ \epsilon_{lateral}=\Delta L_{lateral}/L_{0 lateral}=\Delta d/d_0 $$ Solve for the unknowns. Ultimately, we must solve for $\Delta d$.
$ d_0=12\si{mm}=0.012\si{m} \\ \Delta d=? \\ \epsilon_{lateral}=? $

Side note: lateral strain refers to strain that occurs to the sides of the rod (cross section) and NOT strain that occurs along the length of the rod, that would be longitudinal strain. Therefore the equation above was set up in terms of lateral strain because this problem asks for the change in length laterally (at the cross section). Solve for lateral strain using Poisson's ratio. $$ \nu=\frac{-\text{lateral strain}}{\text{longitudinal strain}}=0.35=\frac{-\text{lateral strain}}{\text{longitudinal strain}} $$ Solve for longitudinal strain using Hooke's law, so we can then come back and solve for lateral strain. $$ E=\sigma/ \epsilon_{longitudinal}\rightarrow E=(P/A)/\epsilon_{longitudinal} $$ Solve for the unknowns.
$ E= 70\times 10^9\si{Pa} \\ A=\pi/4\cdot d^2=\pi/4\cdot (0.012\si{m})^2=0.000113097 \si{m^2} \\ P=\text{internal loading}=500\si{kN}=500,000\si{N} $ $$ 70\times 10^9\si{Pa}=(-500,000\si{N}/0.000113097 \si{m^2})/\epsilon_{longitudinal} \\ \therefore \epsilon_{longitudinal}=-0.0631569 $$ Now we can go back and solve for lateral strain using Poisson's ratio. $$ 0.35=\frac{-\text{lateral strain}}{\text{longitudinal strain}}=\frac{-\epsilon_{lateral}}{-0.0631569} $$ $$ \therefore \epsilon_{lateral}=0.022104915 $$ Now we can go back to the first equation and solve for the change in diameter. $$ \epsilon_{lateral}=\Delta d/d_0 \\ 0.022104915=\Delta d/0.012\si{m};\text{solve for } \Delta d \\ \Delta d = 0.000265259\si{m}=\SI{265}{\micro\meter} $$ Some of the takeaways from this problem are:
Strain and deformation to a rod occur longitudinally (along the length of the rod) or laterally (along the cross section diameter length). Therefore, you can use any of the Hooke's Law equations and modify them to be in terms of lateral deformation or longitudinal deformation. In this problem, the rod got shorter and the diameter got larger as the two 500 kN pushed into the rod axially.

6) What is approximately the velocity of the incompressible flow at $x_2$ if the flow exits the pipe at atmospheric pressure and $ h = \SI{0.5}{m}$? Ignore friction losses


A.3.1 m/s
B.6.1 m/s
C.7.0 m/s
D.11 m/s
The correct answer is A.

Explanation:

Refer to the Principles of One-Dimensional Fluid Flow section in the Fluid Mechanics chapter of the FE Reference Handbook.

Ultimately, we must solve for the velocity at the outlet of the system, $v_2$. Use Bernoulli's equation to solve for $v_2. $ $$ \frac{p_2}{\gamma} +\frac{v_2^2} {2g} +z_2 = \frac{p_1}{\gamma} + \frac{v_1^2} {2g} +z_1 $$ Calculate the unknowns:

$p_1=0 \si{Pa}$ because the water is exposed to the atmosphere, which is 0 gauge pressure or 101.3 kPa absolute pressure
$\gamma_{water}=9.81\si{kN/m^3}$
$z_1=0.5\si{m}$
$g=9.81\si{m/s^2}$
$v_1=0$ since the water at $x_1$ is just sitting there, not moving
$v_2=?$
$p_2=0 \si{Pa}$ because the water is released to the atmosphere, which is 0 gauge pressure or 101.3 kPa absolute pressure
$z_2=0$ $$ \frac{0}{9.81\si{kN/m^3}} +\frac{v_2^2} {2(9.81\si{m/s^2})} +0 = \frac{0}{9.81\si{kN/m^3}} \\ + \frac{0^2} {2(9.81\si{m/s^2})} +0.5\si{m}; \text{solve for }v_2 $$ $$ v_2=3.1\si{m/s} $$

7) A parallel flow heat exchanger operates with water ($c_{p,w} = \SI{4.186}{\kilo\joule\per\kilogram\per\kelvin}$) and oil ($c_{p,o} = \SI{3.4}{\kilo\joule\per\kilogram\per\kelvin}$). Water, the cold fluid, enters the inner tube at $\SI{300}{\kelvin}$ and leaves at $\SI{400}{\kelvin}$. What is the log mean temperature difference if oil, the hot fluid, enters the heat exchanger at $\SI{550}{\kelvin}$ and leaves the system at $\SI{450}{\kelvin}$?

A.$\SI{240}{\kelvin}$
B.$\SI{234}{\kelvin}$
C.$\SI{108}{\kelvin}$
D.$\SI{124}{\kelvin}$
The correct answer is D.

Explanation:

According to the FE Reference Handbook, the log mean temperature difference for parallel flow heat exchangers is defined as $$ \Delta T_{lm} = {(T_{Ho} - T_{Co}) - (T_{Hi} - T_{Ci}) \over ln \left ( T_{Ho} - T_{Co} \over T_{Hi} - T_{Ci} \right )} $$ where $T_{Hi}$ the hot fluid inlet temperature, $T_{Ho}$ the hot fluid outlet temperature, $T_{Ci}$ the cold fluid inlet temperature and $T_{Co}$ the cold fluid outlet temperature. In our problem water is the cold fluid and oil the hot one, therefore we have $$ \Delta T_{lm} = {(T_{oil,out} - T_{water,out}) - (T_{oil,in} - T_{water,in}) \over ln \left ( T_{oil,out} - T_{water,out} \over T_{oil,in} - T_{water,in} \right )} \\ = {(\SI{450}{\kelvin} - \SI{400}{\kelvin}) - (\SI{550}{\kelvin} - \SI{300}{\kelvin}) \over ln \left (\SI{450}{\kelvin} - \SI{400}{\kelvin} \over \SI{550}{\kelvin} - \SI{300}{\kelvin} \right )} \\ = \SI{124.3}{\kelvin}$$

8) A total of 50 valve pressure measurements were taken but only 9 of those measurements will be utilized. What is most nearly the standard deviation of the sample measurements? $$ \begin{array}{c|c} \text{Pressure} & \text{# of Measurements} \\ \hline 20 & 3 \\ 22 & 4 \\ 25 & 2 \\ \end{array} $$

A.0.51
B.2.18
C.1.93
D.1.20
The correct answer is C.

Explanation:

Refer to the Dispersion, Mean, Median, and Mode Values section in the Engineering Probability and Statistics chapter of the FE Reference Handbook.

The population is 50, but the sample is 9. Since we only have pressure values for 9 measurements out of the 50 total measurements, we must use the sample standard deviation formula. Ultimately, we must solve for: $$ s = \sqrt{ \frac{1}{(n-1)} \sum_{i=1}^n \left(X_i-\bar{X}\right)^2 } $$ Solve for the unknowns.
The total number of samples is $n=9$.
Calculate the arithmetic mean, $\bar{X}.$ $$ \bar{X}=(1/n)(X_1+X_2+...+X_n) \\ \bar{X}=(1/9)(20+20+20+22+ \\ 22+22+22+25+25) $$ $$\bar{X} = 22$$ Now that we have all the unknowns, solve for standard deviation. $$ s = \sqrt{ \frac{1}{9-1} \left[ 3(20-22)^2 + 4(22-22)^2 + 2(25-22)^2 \right] } $$ $$ \approx 1.93 $$
Note: You can use your TI-36X pro calculator's standard deviation function to solve this problem more quickly.

9) In the stress-strain diagram, select all that point A describes.


The correct answers are A and B.

Explanation:

The graph above shows a material's reaction to increasing axial tensile stress.

Along A, a material's behavior is elastic - its deformation is not permanent. The slope of A is used to find Young's modulus.