Numerade

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Let X be a random variable equal to the outcome of a 6 sided die roll squared. (For exam- ple, if you roll a 2 the outcome of X is 22 = 4.) (a) What is the state space of X ? (b) What is the probability mass function associated with X ? (c) What is the expected value of X : E [X ]? (d) What is the variance of X : V (X ) = E [X 2] − E [X ]2?

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Problem 1: Deriving translational and rotational equations of motion (Rolling Machine) Consider the system shown in Figure 1. A cylinder with a mass $m$, an inertia $J$ and a radius $r$ spins about an axle that can translate horizontally and rolls on the ground without slipping (No slip means that $\dot{x} = r\omega = r\dot{\theta}$). Attached to the axle housing are a force source $f$ (input), a linear spring that obeys $F_s = kx$ and a linear damper that behaves according to $F_c = c\dot{x}$. (a) (5 points) Show the a free body diagram of the mass/inertia, and determine the equation of motion for $x(t)$. Hint: Consider the rotational motion about the contact point B. Moment of inertia of the machine about B is given by $J + mr^2$ (by the parallel axis theorem). Note also that the damping force $F_c$ opposes motion and the spring is stretched with increasing displacement $x$, while the force source $f$ drives the movement of the rolling machine. (b) (3 points) Define the state variables of the system. The input is $u = f$, the applied force on the machine. Convert the 2nd-order ODE above into two 1st-order ODE's that are functions of the state variables and input. (c) (2 points) Write the two 1st-order ODE's from part (b) in the linear state-space form $\dot{x} = Ax + Bu$, where $x$ is the state vector and $u$ is the input vector (in this case, a scalar $u$).

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Explain the concept of wave-particle duality and discuss its significance in modern physics.

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Consider the following code segment. A 9-line code segment reads as follows. Line 1: int x equals 1, semicolon. Line 2: while, open parenthesis, forward slash, asterisk, condition asterisk, forward slash, close parenthesis. Line 3: open brace. Line 4: if, open parenthesis, x, modulo, 2, equals, equals 0, close parenthesis. Line 5: open brace. Line 6: System, dot, out, dot, print, open parenthesis, x plus, open double quote, space, close double quote, close parenthesis, semicolon. Line 7: close brace. Line 8: x equals x plus 2, semicolon. Line 9: close brace. The following conditions have been proposed to replace /* condition */ in the code segment. x < 0 x <= 1 x < 10 For which of the conditions will nothing be printed?

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An indentured servant in Virginia who completed his term of service would most likely do which of the following? A. Set up a small farm. B. Return to England to buy land. C. Purchase slaves and establish a plantation. D. Become a merchant exporting cash crops. E. Return to Europe to fight wars.

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New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $435 per night. Assume that room rates have a standard deviation of $73. What is the cost for the upper 15% per night?

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If the domain for both variables $x, y$ consists of all integers $Z$. Which of the following is false : A) $\forall x \exists y : x^2 \ge y^2$ B) $\exists x \exists y : x^2 = -y^2$ C) $\forall x \forall y : x^2 \ge -y^2$ D) $\exists x \forall y : x^2 \ge y^2$

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4. Consider this circuit of fully-charged, identical 3.00-µF capacitors. a. What would be the magnitude of the reading on an accurate and properly operated voltmeter whose probes were touched to points A and B? b. By what maximum factor could you increase the energy storage by re-arranging these capacitors, then connecting the modified circuit to the same battery (and allowing full charging)? r 600 V A B

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Q2. You are familiar with the graphical method of finding the take-off and landing performances of an Aircraft. Give the advantages and disadvantages of the analytical and graphical methods of finding the Landing and Take-off performances of an Aircraft. Give your answer in a tabular form. It should be summarized to a maximum of one (1) A4 page.

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How long can the turbines produce 210.0 MW of power from the release of the mass of stored water calculated above? Assume the water sustains a friction loss of 46.2 m²/s² on the way down to the turbine. Number min What is the efficiency of this process? (Assume that the pump and tubines operate at 100% efficiency, and the only loses are due to frictional loses within the pipes.) Efficiency is defined as the total energy recovered/ total energy input. Number %

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jerry s.