Oral Boards
Draw and explain a typical Carnot cycle.
- 1:2 isothermal expansion (T1=T2); 2:3 adiabatic (heat does not enter or leave the system) expansion; 3:4 isothermal compression; 4:1 adiabatic compression -1:2- Ideal gas absorbs q(in) at a high temperature and does work on its surroundings. -2:3- Reversible adiabatic expansion process. The system is thermally insulated. The gas continues to expand and do work on its surroundings, which allows the system to cool to T3 -3:4- Reversible compression process. The surroundings do work to the gas at the lower temperature, which causes a loss of heat q(out) -4:1- Reversible adiabatic gas compression process. The system is thermally insulated. Surroundings continue to do work to the gas, which causes the temperature to rise back to the higher temperature
What is the Ideal Gas Law?
- Describes ideal gases to relate properties of the gas to describe how gases will behave in ideal conditions -To be ideal: Volume of particles are negligible, particles are equal sized, particles move randomly, no energy loss - PV=nRT (pressure)(volume) = (moles)(gas constant)(temperature)
A sliding block slows from 16 m/sec to 8 m/sec in two seconds. If it weighs 500 kg, what is the coefficient of sliding friction?
- Only moving in the x direction ( y forces cancel) -Find acceleration -Write force equations ( F = mu * W = m*a) -Solve for friction force (mu = a/g) -mu = 0.408
Describe the forces on a block in water and pulled from the tension of two ropes at an angle.
- Tension up, buoyancy force up, force of gravity
List and discuss the laws of thermodynamics.
-0th Law : States that if two systems are in thermodynamic equilibrium with a third system, the two original systems are in thermal equilibrium with each other. Basically, if system A is in thermal equilibrium with system C and system B is also in thermal equilibrium with system C, system A and system B are in thermal equilibrium with each other. -1st : States that energy can be converted from one form to another with the interaction of heat, work and internal energy, but it cannot be created nor destroyed, under any circumstances. 2nd : States that the state of entropy of the entire universe, as an isolated system, will always increase over time. The second law also states that the changes in the entropy in the universe can never be negative. -3rd : Essentially allow us to quantify the absolute amplitude of entropies. It says that when we are considering a totally perfect (100% pure) crystalline structure, at absolute zero (0 Kelvin), it will have no entropy (S). Note that if the structure in question were not totally crystalline, then although it would only have an extremely small disorder (entropy) in space, we could not precisely say it had no entropy.
Find the area of A and B using Calculus. Derive the formulas for the volumes of C and D using Calculus. A. Triangle B. Circle C. Pyramid D. Cone
-A and B will just be cone from 0 to h b/hx dx -Circle: intregral from 0 to r sqrt(r^2-x^2) dx -
What is an acid? What is a buffer solution?
-Acid = any compound that yields hydrogen ions when dissolved in water. Has a pH level < 7. Names end in -ic. Examples of acidic solutions are vinegar, lemon juice, hydrochloric acid, lactic, and citric acid -Buffer = Can resist pH change upon the addition of acidic or basic components. Can also neutralize small amounts of added acid or base, thus maintaining the pH of the solution relatively stable. Contains a wear acid-base pair. The base will most likely be a salt containing the conjugate acid. Examples include acetic acid and sodium acetate, pyridine and pyridinium chloride or ammonia and ammonium hydroxide
If a 50 lb block is attached on a pulley by a 35 lb block, what is the velocity of the 50 lb block? The 50 lb block is hanging off the side.
-Assuming no air resistance or friction and massless pulley with no friction -Velocities are similar in both (can look at system as the same) -Solve for forces in x and y. - a = sum forces/mass -a is constant based on 50 lb block. -solve for v1 using kinematics (vo = 0)
You are building a raft out of two by fours. How many two by fours do you need in your raft in order for you to float? Make assumptions and calculate a buoyant force.
-Assuming that each board is standard 2*4*10. These are 21 lbs or 9.5 kg (1 lb/2.205 = kg) -Buoyant force = W - rogV = 9.81(x*9.5+68)
What is a derivative? How is it used? What is a differential? What is the significance of the first and second derivative?
-Derivative : Rate of change of a function with respect to a variable -First : Slope of tangent line to the function at the point x -Second : Derivative of derivative of function. Can tell if the function is increasing or decreasing
How far will the water shoot out? If it shoots out of a container that has a height difference of h1. The container is h2 off the ground. What is the distance traveled?
-Determined by Bernoulli equation Ep = Ek mgh = v^2/2 Vo = sqrt(2gh) h = vot^2 + gt^2/2 t = sqrt(2h/g) d = vot + at^2/2 d = 2 sqrt (h1h2)
Find the final velocity of M for both an elastic collision and an inelastic collision. mass M - Vo mass m - Vo = 0
-Elastic : MVo = mv1 + Mv2 -Inelastic : MVo = (M+m)V
What is the difference between an elastic collision and an inelastic collision? In which cases can you use Conservation of Energy? Conservation of Momentum?
-Elastic collision : After collision, objects separate and do not lose any energy. Can use conservation of momentum -Inelastic collision : After collision, objects stick together. Kinetic energy is not conserved, due to heat loss. Can use conservation of momentum
Draw and explain a stress-strain curve for steel. Where is the elastic limit? The yield point? What are stress and strain? How would work be defined on the stress-strain curve?
-Elastic limit ends where the line on the curve starts to curve instead of being linear . Elastic modulus can also be found using this slope. -Yield point describes the point at which the material can no longer return to its original length. This is the transition from the elastic region into the plastic region. -Stress is the y-axis. It is the quotient of load and area (usually psi). Causes a deformation. Strain is the x-axis. It compares the elongation of a material to its original length. Has no units, but can be written as in/in or ft/ft. -Work in this case will be required to strain a bar to the engineering strain. The area under the curve is a direct measure of the work per unit volume needed to effect the strain.
What is enthalpy? How is it measured? How is it used to calculate entropy?
-Enthalpy : A thermodynamic quantity equivalent to the total heat content of a system. It is equal to the internal energy of the system plus the product of pressure and volume. -Measured as a dimension of energy (measured in joules). Value determined by temperature, pressure, or composition of system -If transferring to entropy, take enthalpy and divide it by the temperature.
How much would the 1 kg mass raise the 50 kg mass in the figure below? Assuming the 50 kg mass is attached to a pulley with 1 kg mass attached at an angle.
-FBD for 1g -2Tsin theta = 1g -FBD for 50g -T = 50g -substitute T --> theta = 0.57 degrees with horizontal
How far does the man have to walk down the beam in order to tip the beam off fulcrum a? There is a 50 lb mass at 5 ft R to A, B is 10 ft to right of A. The entire beam is 15 ft at 25 lb/ft. The person is 150 lbs.
-FBD of all forces. -To tip off from end A, Ra = 0. -Sum moments from b = 0 : 50(10-5) + (25*15*(10-7.5)) = 150* (x-10) x = 17.9166 from a
Find the tensions in the two ropes at an angle
-Forces in x : T1sin(a) + T2sin(b) = mg -Forces in y : T1cos(a) = T2cos(b) -Worth noting : Higher the angle of suspension, the higher the tension will be. Maximum tension at 90 degrees. -Used during rock climbing, suspension bridge, attached to sails of a ship
Draw and explain a typical Rankine cycle. Include a discussion of enthalpy and entropy changes.
-Idealized thermodynamic cycle of a constant pressure heat engine that converts part of heat into mechanical work -1:2 - Isentropic compression. Liquid compressed adiabatically by centrifugal pumps and pumped into high pressure boiler. Increases enthalpy and compressing liquid (increasing pressure) -2:3 - Isobaric heat addition. Constant pressure heat transfer to the liquid condensate from external force. Feedwater heated to boiling point (2-3a) and then evaporated in boiler. Entropy does not change. Enthalpy is the flow process work done on the system -3:4 - Isentropic expansion. Steam from boiler expands adiabatically to produce work and discharged to the condenser. Steam works on surroundings. Loses enthalpy equal to the work that leaves the system. -4:1 - Isobaric heat rejection. Cycle completes by constant-pressure process from the partially condensed steam. Vapor condenses and the temperature increases. Entropy does not change. Enthalpy is the flow process work done on the system
What is an integral? How is it used? What is the difference between a definite and an indefinite integral?
-Integral : Corresponding to the fundamental object of calculus corresponding to infinitesimal pieces to find the content of a continuous region. -Indefinite integral has no limits of integration -Definite integral : Contains limits of integration
Given an I beam, what types of forces are acting at the point of load on this beam? How do you find these forces?
-Load of weight -Weight of beam -Normal of R1 -Normal of R2 Find using equilibrium conditions of force and moment
What is a logarithm? How is e, the natural logarithm base, defined?
-Logarithm : Exponent by which another fixed value, the base, has to be raised to produce that number -e is about 2.718. Derived by the infinite series of 1 + 1/1! + 1/2! + 1/3!. When defined as the base for a logarithm, it is written as ln(x). -Worth noting, ln(e) = 1 ln(1) = 0
Why is pH important in materials selection?
-Metals are attacked by acid, leading to corrosion -Aluminum and zinc are attacked by both acid and base -Cooler and heating water must be delicately adjusted for pH to avoid chemical and galvanic corrosion of metal containment
Describe how to classify differential equations
-Ordinary (ODE) : Classified by the order that is the highest derivative in the equation -Partial (PDE) : Taking partial derivatives of functions with respect to one variable -Differential-algebraic (DAE) : System of equations that govern phenomena and have a combination of differential equations and algebraic equations
What type of smooth curve would go through these points: (0,4), (2,0) and (-2,0)? What would its equation be?
-Parabola. One half of the parabola is a mirror of the other -y = -x^2 + 4
There is a system where area A (9 in^2) is attached to an area B (900 in^2) and is compressed by 9000 lbs. The height difference is 1 ft. What is force F applied at point A
-Pascal's Law -Pressure = F/A -F1/A1 = F2/A2 -P2 = Pa + hrog -Fa = 2898 - 2.0125 ro
If the population doubles in two years, how long does it take to triple?
-Population growth equation: P = P(o)e^rt -P/P(o) = 2 solve for r -New P/P(o) = 3
Draw the shear and moment diagrams for this uniformly loaded beam.
-Shear will be negative, crossing 0 at center line. wl/2 is max at x = 0 -Moment will be upside down parabola with max at wl^2/8
What geometric surface encloses the maximum volume with the minimum surface area? How would you prove it?
-Sphere has the maximum volume with the minimum surface area. -Prove using : CHEGG
Derive the quadratic equation
-Start with y = ax^2 + bx + c (y=0) -Solve for c and divide by a -Divide the x term by 2 and square it. Add to both sides -Take sqrt of both sides -Solve for x
There is r1 and r2 on either side of a beam with length L. There is a weight w at L/6 and force F at 2/3 L. What is the reactions of R1 and R2
-Sum of forces (R1+R2 = W + F) -Moments about R1 = 0 = (F*2/3*L + W*L/12) - R2L R2 = (8F+W/12) -Solve for R1 (R1 = F+ W - 8F+W/12) R1 = (4F+ 11W/12)
What is a Laplace transform, a Fourier transform or a Taylor series? How are each used?
-Taylor series : Used for approximations of functions -Laplace transform : Linear transformation used to convert differential equations into purely algebraic equations -Fourier transform : Used to transform a function of time to a function of frequency
A wooden block with a mass of 1000 grams is suspended by a rope from a tree. You shoot a bullet with a mass of 10 grams and a velocity of 1000 m/s. The bullet imbeds in the block causing it to swing upwards. How high in the vertical direction will the block swing?
-Use law of conservation of momentum : mv + Mv = (m+M)V V = 9.9 m/s 1/2(m+M)V^2 = (m+M)gh h = 5 m
Prove that the derivative of x2 is 2x
-Use product rule of x^2)' = x' * x + x*x' = 2*x -Use the difference quotient
Find the maximum or minimum of a parabola and determine if it is a maximum or minimum
-Use quadratic formula -Use x = -b/2a and the value is f(x). If a > 0 then the parabola opens up and is a minimum. If a < 0 then the parabola opens down
When do you use L'Hopital's Rule?
-Use whenever direct substitution of a limit yields an indeterminate form
Derive the equation of a circle around any point.
-Using distance formula : d = sqrt(x2-x1)^2 + (y2 - y1)^2 -Substitute (x1,y1) with (h,k) -Square each side : (x-h)^2 + (y-k)^2 = r^2
A tank of N2 is at 2000 psig and 200K. If the temperature of the tank rises to 400K, what is the new pressure in the tank?
-Using gay-lussac's law : P1/T1 = P2/T2 (Temperature in K)
Given the following diagram, calculate the distance traveled by the ball being thrown off the monument. The ball is thrown from 500 ft high at a velocity of 50 ft/s
-Using kinematic equations -d = 1/2at^2
A ball of diameter 10 cm and mass 10 grams is dropped in a container of water. The cross-sectional area of the container is 100 cm2 . What is the change in the height of the water column?
-Volume increased = volume of ball -Volume of a sphere = 4/3pir^3 -Volume of water = A * h -h = 4pir^3/3*A -h = 5.24 cm
In the figure below, you can walk 3 mph through the woods. You can walk 5 mph on the trail. What is the quickest that you can get from point A to point B?
-Walk partly through the woods and rest on trail -Path through woods = sqrt(x^2 + 25) / 3 -Path on trail = 10-x/5 -Take first derivative to find minimum , set = 0 -x = 3.75 -Find t using equation above
State the mathematical expression for the pH factor of a solution. What is the pH of pure water? What happens to pH if the hydrogen ion concentration increases?
-pH factor = -log(H+) -pH of water = 7 (neutral) -If the hydrogen ion concentration increases, the pH decreases, resulting in a more acidic solution
Determine the final pH and temperature when these two solutions are mixed together in a 3- liter container. Solution A: 2 liters, pH = 3, Temp = 80F Solution B: 1 liter, pH = 5, Temp = 40F
-ph = - log*(m/l) -ph3 = .001 M -ph5 = .00001 M -Find the total moles (mol = M/L) =M of total = mol/l -new pH = -log(M)
Given the figure below, determine the value of x so that when the corners are removed and the flaps folded up, the five-sided box formed will have the maximum volume
-v = lw^2 - 2lx^2 - 2wx^2 + 4x^2 -Find first derivative of v -set to 0 and solve for x -solve for second derivative and plug x in
What is limx->0 sin(x)/x
1 -Use l'hopital's rule -cos(x)/1 lim -> 0
What is the probability of throwing one "7" with two dice?
6/36 = 16%
A. What height h must the car start at to make it around the loop without falling? B. Find the displacement x of the spring when the car impacts the spring. A mass M starts at height h. It goes into a loop with radius r. It then compresses a spring.
A. -mgh = mv^2/r -mg(h-2r) = mv^2/2 -h = 5r/2 -Mgh = 1/2kx^2 B. -mgh = 1/2kx^2 x = sqrt (2Mgh/k)
Take the derivative with respect to x of the following: A. cos4 x sinx B. ae-bx cx2 C. 5x4 D. x(x2 - 4)1/2 E. sin(x), cos(x), tan(x), cot(x), sec(x), csc(x) F. ln(x) and 10x G. x + x3 + sin(x)cos(x) + sin(x) H. x5 + cos(x)(ex ) + sin(x2 /3) I. x1/2 + x2 sin2 x
A. cos^5(x) - 4sin^2(x)cos^3(x) B. CHEGG C. 20x^3
Integrate the following: A. ∫ (x sinx) dx B. ∫ x (x2 - 4) ½ dx C. ∫ e 4 - 3/x dx X 2 D. ∫ (e-x + 3x2 ) dx E. ∫ (x sin2 x + x3 ) dx F. ∫sec(u)tan(u) du G. ∫ xex dx H. ∫(y+3)(y+1) dy R π/2 π/2 I. ∫0-R ∫0-pi/2 ∫0-pi/2 r sin(Ө)dɸ dӨ dr J. ∫ (2x + 1) dx
A. sin(x) - xcos(x) B. 1/3(x^2-4)^3/2 C. 3 - 2e^4 x / (2x^2) D. x^3 - e ^ -x E. 1/8(2x(x^3+x-sin(2x))-cos(2x)) F. sec(u) G. e^x(x-1) H. y^3/3 + 2y^2 + 3y I. piR^2/4 J. x^2 + x
1. What is a solution to the equation (1-y)^2 + 2xy = 0 (1,1) (1,i) (i,1) (1,0)
C. (1,0)
1. Draw the following curves and find the area between them: A. y = 2 + e-x and y = 1 + x2 B. y = x2 and y = x
CHEGG
A baseball initially at a height of 5 feet is hit at an angle of 30 degrees to the horizontal at an initial velocity of 150 feet/sec. Will the baseball clear a fence at a height of 15 feet at a distance of 400 feet?
CHEGG
A spaceship is accelerating at 1000 m/sec2. How much force is required from the backthrusters to completely stop the spaceship?
CHEGG
Analyze the curve y = 1 + e-x by finding the first two derivatives, maxima, minima and inflection points.
CHEGG
Be able to integrate or differentiate by using parts, chain rule or quotient rule
CHEGG
Describe the motion of the block-spring assembly when the block is displaced 4 inches from the equilibrium position. Block is vertical with spring k
CHEGG
Draw the following curves and find the area between them: a. y = x3 and y = x2 b. y = x2 and y = x
CHEGG
Draw the following curves. Plot any maximum, minimum, and points of inflection. A. f(x) = e-x2 B. f(x) = a sinx C. f(x) = e p/2 D. f(x) = 3x2 - 17x -10 E. f(x) = x3 - x2 F. f(x) = xx G. f(x) = x2e(-x2)
CHEGG
Draw the shear and moment diagrams for this beam. There are two fasteners at ends of beam. In the middle of both ends and the center line, there is a force F.
CHEGG
Explain how to solve the following differential equation: A" + A' + A = 0
CHEGG
Find f(x) which best describes the following graph. "A" represents area. Top of function = 3A Bottom of function = 1A
CHEGG
Find the sum of: 100 S n n=1
CHEGG
For the following curve, plot the first and second derivatives (curve then linear)
CHEGG
Given 80 feet of fencing, what is the maximum area that you can enclose along a wall?
CHEGG
Given a closed box, where the length is twice the height, the width is 10 meters less than the length, and the surface area is 10 times the width times the height, what are the dimensions?
CHEGG
Given the figure below with uniform mass, what is the y-coordinate of the center of gravity? Box on another box, each with dimensions a*1*1. Box is off-kilter by a
CHEGG
How does the gravitational force vary between two masses if distance is doubled? How does the electrostatic force vary between two charged particles if the distance is doubled? Explain using both equations and physical applications
CHEGG
Rotate the function y = 1/x around the x axis from x = 1 to x = infinity and find the volume
CHEGG
Show how to solve a differential equation with matrices
CHEGG
Simplify (3+2i)/(3-2i)
CHEGG
Simplify (a^4 + b^4) / (a^2 + b^2)
CHEGG
Solve a system of 3 simultaneous linear equations in three variables. 5x - 4y + 2z = 0 -3x + 4y = 6 x + 4z = 6
CHEGG
Solve the following differential equations: A. y '' + 6y' + 9 = 5 B. dN/dt = -2N C. y' = xy3 at x=0 and y=1
CHEGG
Solve the general and specific homogeneous equation with derivatives: dy/dx + Ky = 10
CHEGG
Solve: dx/dt = x/k
CHEGG
Solve: x" + 5x' + 6x = e-t
CHEGG
Solve: y - 3y' = 0 for y(0) = 3
CHEGG
Solve: y" + 4y' + 3y = sin(x)
CHEGG
The locus of p(x,y), such that the difference of their distance from two fixed points is constant, a(n), is called: ellipse hyperbola parabola circle
CHEGG
The number of square feet in a circle is equal to the number in feet of the circle's circumference. What is the circle's radius?
CHEGG
Two runners start at a distance of 10 miles from each other. They run towards each other at a constant velocity of 5 mph. A fly takes off from runner one's nose at time zero. The fly has a constant velocity of 20 mph and flies between the runners. Find the total distance that the fly has traveled when the runners collide.
CHEGG
Use a first order differential equation to find the function to represent current with respect to time and to find the time constant of the circuit
CHEGG
Use calculus to find the volume of a plastic solo cup.
CHEGG
Using Calculus, derive the formula for the exposed surface area of a ball floating in water
CHEGG
What is the Laplace transform of f(t) = t ?
CHEGG
What is the center of x2 + y2 - 2x - 4y - 17 = 0?
CHEGG
What must the angle Ө be in order for the block of mass M to start sliding? µ = 0.8
CHEGG
You are going to build a rectangular fence alongside a wall so that the wall forms one side of the enclosure. If you have 80 feet of fencing material, what is the largest area you can enclose?
CHEGG
Describe how the following will affect the buoyant force acting on a submarine. a. An increase in depth b. An increase in salinity c. A decrease in water temperature
F = rogV P gage = rogh a. Increasing depth will not affect the buoyant force, but will affect the pressure b. Increasing salinity increases ro, increasing the buoyant force c. A decrease in water temperature increases buoyancy. Molecularly, the molecules tend to go closer together.
Find the area between the curves y = x and y = x2 from x = 0 to x = 3.
Integral from 0 to 3 (x-x^2) dx
A uniform sphere of mass 5 kg and a radius of 0.2 m spins about an axis passing through its center with period t = 0.7 s. What is the angular momentum of the sphere?
L = mr^2 w = v/r L = rmv
What is pH?
The measure of the hydrogen ion concentration in a solution. The greater the H+, the lower the scale. It ranges from 0-14. 7 = neutral < 7 acidic. > 7 basic.
If flow rate is increased by a factor of three, how does the pressure change at points A and B? Assuming point A is in the venturi and point B is after.
Velocity will change by a factor of 3 Pressure will decrease by a factor of nine. Bernoulli's equation states that, in a streamline fluid flow, the greater the speed of the flow, the less the static pressure, and the less the speed of the flow, the greater the static pressure. There exists a simple exchange between the dynamic and static pressures such that their total remains the same. As one increases, the other must decrease. Use the relationship: (1/2)pV^2 + P = P(t)
What happens to the water level with respect to the shore when the sailor throws the lead anchor overboard?
When submerged the anchor displaces a volume of water equal to its own volume. When in the boat, the anchor is associated with displacing a volume of water with a weight equal to its own weight - this is a larger volume than its own volume because the anchor's density is higher than water's. Because the anchor displaces less water when it is submerged, the water level drops when the anchor is thrown overboard. V a = ro lead/ ro water --> 11.34Va
A propeller plane and a jet travel 3000 miles. The velocity of the plane is 1/3 the velocity of the jet. It takes the prop plane 10 hours longer to complete the trip. What is the velocity of the jet?
d = vt = v/3(t+10) t = t+10/3 t= 5 v = d/5
Air is being pumped into a spherical balloon at a rate of 5 cm3 /min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm.
dc/dt = 5 diameter = 20 radius = 10 find dr/dt -volume of sphere = 4/3pir^3 -take derivative : dv/dt = 4/3pi*3*r^2 * dr/dt -solve for dr/dt dr/dt = 0.003978
Plot f(x) = x2 + x - 6. Find the area between the x-axis in the top and the line y = -4 on the bottom and the graph on each side.
integral form -3 to 2 -f(x) dx - integral of -2 to 1 (-4-f(x) dx
Using Calculus, determine the area between two concentric circles of radii 1 and 2 respectively.
integral from 0 to 2pi and r1 to r2 r dr dtheta
Derive the Kinematic Equations using integration.
integral of v(t) = x integral of a(t) = v
A block of mass M1 is attached by string to a support. The block is raised to a height h and released. It then strikes a block of mass M2 on a frictionless surface. Find the velocity of the block M2, assuming a totally elastic collision.
mgh=1/2mv^2 v = sqrt(2gh) V = (2M1/M1+M2)V V = 2M/M1+M2 sqrt(2gh)
