Exercises for ENS 100 – Introduction to Engineering – Fusion of Engineering, Control, Coding, Machine Learning, and Science

Transform units:

a) $5 [\frac{miles}{hour}]$ to $[\frac{kilometer}{second}]$

b) $120 [\frac{centimeter}{second}]$ to $[\frac{milimeter}{hour}]$

c) $10^{6} [\frac{meter}{second}]$ to $\frac{miles}{hour}$

2. Interest rates. Initially, you borrowed $$ 10,000$ from a bank. The yearly interest rate is $7 \%$ . The loan is for $15$ years. After $15$ years, how much money do you have to return to the bank?

3. Simplify expression:

(1) $\begin{align*}\frac{5 \cdot 10^{3} \cdot 3 \cdot 100^{3} }{1000^{1}} = ? \cdot 10^{2}\end{align*}$

4. Find $x$ and $y$ such that the following equation holds true

(2) $\begin{align*}\frac{\frac{a\cdot b^2}{a^{5}}}{\frac{a^{7}\cdot b^{10}}{\sqrt{b}}} = ? \frac{a^{x}}{b^{y}}\end{align*}$

5. Sketch the functions:

(3) $\begin{align*}y=2x+5 \\y=-7x+2 \\y=x^{2}+5x\end{align*}$

6. Solve the quadratic equations

(4) $\begin{align*}x^{2}+6x+8=0 \\x^{2}+2x+1=0 \\x^{2}+9x+20=0\end{align*}$

7. Simplify the fractions:

(5) $\begin{align*}\frac{48}{100}= \\\frac{36}{96} = \\\frac{15}{90}=\end{align*}$

8. Two cars are approaching each other. The velocity of the first car is $20 [\frac{meter}{second}]$ and the velocity of the second car is $10 [\frac{meter}{second}]$ . If the initial distance between the cars is $5 [miles]$ , compute the time it takes until two cars reach each other. What is the distance the first car traveled?

9. A police car is chasing a criminal in another car. The velocity of the police car is $100 [\frac{miles}{hours}]$ and the velocity of the criminal’s car is $60 [\frac{kilimeters}{hour}]$ . Both cars are moving in the same direction along the straight line (the criminal is trying to escape ). If the initial distance between the cars was $5000 [meters]$ , compute the time it takes for the police car to reach the criminal’s car.

10. State the definition and graphically explain Newton’s third law.

11. An object is moving along a straight line. Its position that is a function of time is given by the following table.

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Estimate
a) Velocity at the time instant $5 [s]$
b) Velocity at the time instant $6 [s]$
c) Acceleration at the time instant $5 [s]$

12. Given two functions:

(6) $\begin{align*}y=2x-2 \\y=x^{2}-3\end{align*}$

a) Sketch these functions on a single graph
b) Numerically compute their intersection points

13. What is a free body diagram?

14. a) Sketch the following vectors

(7) $\begin{align*}F=\begin{bmatrix}2 \\ 3 \end{bmatrix}, F=\begin{bmatrix} -1 \\ 6 \end{bmatrix}, F=\begin{bmatrix} -4 \\ -3 \end{bmatrix}\end{align*}$

b) Calculate the magnitudes of these vectors and calculate their angles with respect to the positive x-direction.

15. If the length of the vector is $10$ and if its angle with respect to the positive $x$ direction is $30$ degrees, compute the $x$ and $y$ projections of this vector.

16. In the following figure (a)-d) panels), the vectors are defined with their magnitudes and angles. Calculate $x$ and $y$ projections of these vectors.

17. Sketch free body diagrams of the objects sketches in figures below

a) free body diagram of the car
b) free body diagram of the bar
c) free body diagram of the ball and the bar
d) free body diagram of the ball and the hanging cord

18. What is a moment of a force?

19. Given the force $F$ , angle, and distance $s$ , compute the moment of the force $F$ with respect to the point A for the cases given in the panels of the figure below.

20. What is a mechanical advantage? Give an example.

21. Explain Newton’s third law. Give a graphical as well as text explanation.

22. Explain a friction force. Give a graphical as well as text explanation.

23. What is a difference between object’s mass and object’s weight?

24. Consider a lever mechanism shown in the figure below. A box of the mass m=100 kg is being supported by one side of the lever.
a) Sketch free body diagrams of the box and the lever.
b) Calculate the magnitude of the force $F$ that needs to be applied to the lever, such that the lever mechanism is at rest at the horizontal position.
c) What is the mechanical advantage of the lever?

25. Solve the system of linear equations for $x,y,z$

(8) $\begin{align*}& x+2y=1 \\& z+x+y=5 \\& y+z=3\end{align*}$

26. See the figure below. The base length of the trapezoid is 10. The angle is 60 degrees and the length of the inclined side is 5. Compute the area of the trapezoid?

27. How many liters of water can a cylindrical barrel hold if the base of the barrel has a diameter of $0.5$ [m] and the height of $2.5 [m]$ ?

28. A force $F$ is acting on a box of the mass $m=100 [kg]$ (see the figure below). The coefficient of friction between the surface and the box is $\mu=0.4$ .

a) Sketch a free body diagram of the box?
b) What is the maximal intensity of the force $F$ that can be applied to the box such that the box remains still (at rest)?

29. A force $F$ is acting on a box of the mass $m=100 [kg]$ on an inclined plane with an angle of $30$ degrees. The inclined plane is without friction (no friction force).
a) Sketch a free body diagram of the box.
b) What is the force $F$ (the force is horizontal) that needs to be applied to the box such that the box does not move.

31. You are paining a room (see the figure below). The room dimensions are $8[m]\times 5[m] \times 2.5 [m]$ . You are painting all the inner walls of the room except the floor. You are also painting a ceiling. With a single container of paint, you are able to paint $1.5 [m^{2}]$ (1.5 square meters). How many containers of paint do you need to purchase to paint the room (everything except the floor)?

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