(Electrical - Medium) Node Voltage Analysis

Node voltage analysis is one the most commonly used methods for analyzing circuits. With this method, given the value of a circuit's resistors and voltage/current sources, you can find the voltage at each node and the currents through each branch.

Review

A node is where a circuit path splits into two or more paths, like a copper wire being split in two. The copper wire in the picture has a current $i_1$ entering a node from which two wires with currents $i_2$ and $i_3$ emerge.

Kirchoff's Current Law (KCL) says that the current entering a node, is equal to the sum of currents leaving the node. So in the copper wire,  $i_1=i_2+i_3$

The next picture shows two equivalent circuits to illustrate the way to think of nodes in a rectangular circuit. 

The node voltage analysis method starts by selecting a reference node, usually ground, which we'll set to 0. Then you label the current in each branch and the voltage at each node. Using Ohm's law to write an equation for every current and voltage, we end up with a system of equations we can use to solve for all unknowns. For example:
$i_S=i_1+i_2$
$i_2=i_3$
$i_1=\frac {v_a-v_c}{R_1}$
$i_3=\frac {v_b-v_c}{R_3}$

Question

Given:

$i_1=10mA$; $i_2=50mA$; $R_1=1k$; $R_2=2k$; $R_3=10k$; $R_4=2k$


Find all the unknown currents and voltages in the circuit above.

(Physics - Medium) Car Rounding a Banked Curve

Highways are sometimes built with banked (inclined) curves. This makes it possible for a car traveling at velocity v to bank a curve and maintain the turning radius, even if the road surface has low friction (rainy/icy surfaces). Assuming a worst-case scenario of no friction, what incline angle θ would a curve have to be so that a car moving at velocity v can safely make the turn?

                A) θ = asin( gr / v2 )
                B) θ = atan( gr / v2 )
                C) θ = asin( v2 / gr )
                D) θ = atan( v2 / gr )
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(Mechanics - Easy) Two Dogs On A Diving Board

Statics 1 (2-2-2014) - Google Drive1.pdf-page-001.jpg

A dog owner has hired you to build a custom diving board for his award winning diving dogs. You've determined the diving board will weigh 150 lbs. In the most extreme loading case, the owner will have two dogs at D & E. The dog at D will weigh 43 lbs, and the dog at E will weigh 88 lbs. You want to make sure the joints are strong enough to support that loading case.

Assume that the convention for the direction of gravity is negative.

So what do you get when you calculate out the reaction forces of A and B?

A) A = 230 lbs, B = 511 lbs

B) A = -230 lbs, B = 511 lbs

C) A = 230 lbs, B = -511 lbs

D) A = -230 lbs, B = -511 lbs

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(Physics - Medium) Car Rounding a Flat Curve

A car is rounding a flat curve with radius R. If the coefficient of friction between the tires and road is µs, what is the maximum speed vmax at which the driver can take the curve without sliding?

(Logic - Medium) Rope Burning Puzzle

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You have two ropes, a lighter, and nothing else. You know that for each of the two ropes when you light one end of the rope, it will take exactly one hour to burn all the way to the other end. However, neither rope burns uniformly, and they burn at different rates from each other. For instance, half of the first rope may burn in the first 10 minutes, and then the other half would finish burning in 50 minutes, while the second rope may burn evenly throughout. Who knows? You don't!

How do you measure 45 minutes?