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Jacob S.
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Physics (Newtonian Mechanics)
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Question:

If an object of mass 5kg is at rest on a ramp with a slope of 15 degrees, what is the magnitude of the normal force acting on it from the ramp? $($)

Jacob S.
Answer:

We are given that the object has a mass of 5kg, the slope is 15 degrees, and that we are looking for the normal force. This implies that Newton's First Law will be helpful to us in this problem. This Law states that the sum of all forces acting on the object is equal to zero. For this problem, we are not concerned with friction for it is not a factor in determining the normal force. After visualizing and drawing a Free-Body-Diagram of the problem, we can sum the forces in the direction of the normal force and see that: $(N_F = mgcos(15)$) $(N_F = (5kg)(9.81m/s^2)cos(15)$) $(N_F = 47N$) From Newton's First Law, we can determine that the normal force acting on the object is 47N

Trigonometry
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Question:

Using the Law of Cosines, find the side length, C, if the following are known: Side A = 14cm with a corresponding angle of 34 degrees. Side B = 20cm with a corresponding angle of 78 degrees. $($)

Jacob S.
Answer:

To be able to use Law of Cosines we must find the remaining angle corresponding to side C. A triangle must have all internal angles add to 180 degrees so we can easily find the third angle. $(180 = 34 + 78 + A_C$) This implies that the angle corresponding to C is 68 degrees. Now we can use Law of Cosines easily... $(C^2 = A^2+B^2-2ABcos(c)$) $(C^2 = 14^2+20^2-2*14*20*cos(68) $) $(C^2 = 386 \Longrightarrow C = 19.65cm$)

Electrical Engineering
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Question:

Given the following transfer function for a circuit, determine: A) If the system is stable, unstable, or marginally stable. B) Without finding the time domain expression for the output, what is the magnitude of the impusle response as time approaches infinity? $(H(s) = \frac{s+7}{s(s+10)}$)

Jacob S.
Answer:

A) To begin, we check for stability. First we check if the system is unstable. For a system to be unstable, it must be true that there are poles in the right half plane or repeated poles on the imaginary axis. Put simply, this means that the denominator has to have positive or any repeated purely imaginary roots. If the system is not unstable, then next we check if it is stable. For a system to be stable, it must be true that all poles lie in the left half plane on the real/imaginary axis. This simply means that all terms in the denominator have negative roots. Lastly, if the system is neither unstable or stable then we check the marginally stable case. For a system to be marginally stable, it must be true that all poles are either in the left half plane or on the imaginary axis, but not repeated. This simply means that the denominator has to have either negative roots or non-repeated purely imaginary roots. For our case, the denominator roots are 0 and -10. The system has no positive roots or repeated imaginary roots so it is not unstable. It does not have purely negative roots so it is not stable. It does however satisfy the condition for marginally stable with a root in the left half plane, -10, and a non-repeated root on the imaginary axis, 0, so the system is marginally stable. B) Without taking the inverse Laplace transform of the step response, we can still determine the beginning or end behavior of a system from the initial and final value theorems. In this case we are interested in the final value of the system in the time domain so we choose the final value theorem. $(\lim_{t\to\infty} f(t) = \lim_{s\to0} sF(s)$) The impulse response is simply the response of the transfer function itself so... $(\lim_{s\to0} \frac{s(s+7)}{s(s+10)} = \frac{7}{10} $) This agrees with the previous answer because we have a bounded signal meaning the system is stable for the impulse response.$($)

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