Numerade

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A small sphere with mass m is attached to a massless rod of length L that is pivoted at the top, forming a simple pendulum. The pendulum is pulled to one side so that the rod is at an angle $\theta$ from the vertical, and released from rest. Part A Draw vectors representing the forces acting on the small sphere and the acceleration of the sphere just after it is released. Accuracy counts! Draw the vectors starting at the sphere. The location and orientation of the vectors will be graded. Part B At this point, what is the linear acceleration of the sphere? Express your answer in terms of g, $\theta$. Part C Draw vectors representing the forces acting on the small sphere and the acceleration of the sphere when the pendulum rod is vertical. Draw the vectors starting at the sphere. The location and orientation of the vectors will be graded. Part D At this point, what is the linear speed of the sphere? Express your answer in terms of g, $\theta$, L. >>

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9–5. Locate the center of gravity x¯ of the homogeneous rod. If the rod has a weight per unit length of 0.5 lb/ft, determine the vertical reaction at A and the x and y components of reaction at the pin B. Problem 9–5

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La suma de las tres cifras de un número es 6; y, si se intercambian la primera y la segunda, el número aumenta en 90 unidades. Finalmente, si se intercambian la segunda y la tercera, el número aumenta en 9 unidades. Calcula dicho número.

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Section A The recycling of plastic, with particular reference to the reason why plastic is recycled; which plastics are recyclable; how plastic is collected and how plastic is sorted for recycling. Section B Manufacturing techniques for manufacturing new plastic products from sorted plastic pieces, meaning the processing of the plastic waste into the various plastic products.

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Find v_(O) for the circuit given in the figure, assuming that v_(s)=(6cos(2t)+4sin(4t))V. Take C=350mF. The value of (x) + ") V. c 20 The value of vo= ( 4.65 cos(2t+ 39.62 ) + 3.70 O sin(4t+ 90 ) V.

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Consider the piecewise defined sequence \{c_n\}_{n=0}^\infty given by $c_n = \begin{cases} (-1)^n & 0 \le n \le 1000\\ \frac{n+1}{3n^2 - \cos(n)} & n > 1000 \end{cases}$ Which of the following statements is TRUE? A. The sequence \{c_n\}_{n=0}^\infty converges to $\frac{3}{2}$. B. The sequence \{c_n\}_{n=0}^\infty diverges. C. The series $\sum_{n=0}^\infty c_n$ converges to 0. D. The series $\sum_{n=1001}^\infty c_n$ converges to $\frac{3}{2}$. E. The sequence \{c_n\}_{n=0}^\infty converges to 0

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Exercise 1.14. (i) Define f: T -> R by f(t) = t^2 for all t in T. Find f^Δ. (ii) Define g by g(t) = √t for all t in T with t > 0. Find g^Δ. Some easy and useful relationships concerning the delta derivative are given next. Theorem 1.16. Assume f: T -> R is a function and let t in T. Then we have the following: (i) If f is differentiable at t, then f is continuous at t. (ii) If f is continuous at t and t is right-scattered, then f is differentiable at t with (iii) If t is right-dense, then f is differentiable at t iff the limit lim(s->t) (f(t) - f(s))/(t - s). Exercise 1.14. Define f: T -> R by f(t) = t for all t in T. Find f. (ii) Define g by g(t) = √t for all t in T with t > 0. Find g. Exercise 1.15. Using Definition 1.10 show that if t in T, t min T satisfies (t) = t < o(t), then the jump operator o is not delta differentiable at t. Some easy and useful relationships concerning the delta derivative are given next. Theorem 1.16. Assume f: T -> R is a function and let t in T. Then we have the following: (i) If f is differentiable at t, then f is continuous at t. (ii) If f is continuous at t and t is right-scattered, then f is differentiable at t with (iii) If t is right-dense, then f is differentiable at t iff the limit lim t-s -> 0 (f(t) - f(s))/(t - s).

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(d) Check the linear dependency of the matrices B,C,D in the next problem. (e) Let $v_1 = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$, $v_2 = \begin{bmatrix} 2 & 5 & 6 \end{bmatrix}$, $u = \begin{bmatrix} 2 & 1 & 0 \end{bmatrix}$ and $w = \begin{bmatrix} 1 & 1 & 1 \end{bmatrix}$. Is $u \in \text{span}(v_1, v_2)$? Is $w \in \text{span}(v_1, v_2)$? If yes, write u or w as a linear combination of $v_1$ and $v_2$.

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Problem 2: A paper mill processes rolls of paper having a density of 984 kg/m^3. The paper roll is 1.50 m outside diameter (OD) x 0.22 m inside diameter (ID) x 3.23 m long and is on a simple supported, hollow, steel shaft with Sut = 500 MPa. Find (a) the shaft ID needed to obtain a dynamic safety factor of 2 for a 10-year life if the shaft OD is 22 cm and the roll turns at 55 rpm with 1.5 hp absorbed. (b) Determine the number of stress cycles for a 10-year life if the paper mill runs 3 shifts per day. Assume mean stress is equal to (V3 Mean torque) and the maximum moment is 3.1 x 10^7 N.mm.

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Lab 3: The influence of additional poles and zeros of second order system. 1. (a) Given the transfer function, G(s) = \frac{25}{s^2 + 4s + 25}. Evaluate the percent overshoot, settling time, peak time, and rise time. Also, plot the poles. (b) Add a pole at -200 to the system of (a). Estimate whether the transient response in (a) will be appreciably affected. (c) Repeat (b) with the pole successively placed at -20; -10, and -2. 2. A zero is added to the system of Prelab 1(a) at -200 and then moved to -50; -20; -10; - 5, and -2. List the values of zero location in the order of the most to the least effect upon the pure second-order transient response.

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