In lecture today the day was started with a teaser problem involving lights and circuits. Then charge and current were reviewed and a problem was solved about the relationship between the two. Then, voltage, power and energy was also reviewed from physics 4B and a problem relating all of these concepts together was solved. The Solderless Breadboards lab was then performed, and its review is found below under the lab section. After the lab power and energy was dived into more detail, and a problem solving for energy and power from voltage and current was solved. An overview of dependent and independent power sources, what they are and how they are indicated and used was reviewed after the problem, and then a problem concerning conservation of energy in a circuit was solved to find the voltage across an object in the circuit.
Found above is the teaser problem. It consisted of a circuit with two voltage sources and three light bulbs. As seen above, there is a light bulb found between the sources with a switch next to it, Initially the center light bulb is off because the switch is off (there is no connection to the sources). The upper and lower light bulb were one at a certain brightness. A hypothesis had to be performed about all three light bulbs after the switch was turned on. It was found that the two light bulbs previously turned on stayed at the same brightness whereas the center bulb stayed off after the switch was turned on. The two light bulbs stayed the same because the current running through them and the voltages across them was the same even after turning on the switch. The center bulb stayed off because there was no potential difference across it, and therefore no current was running through it.
In this problem, a graph of charge as a function of time was given, and current as a function of time was asked to be found. By knowing that i(t) = dq/dt, the current was easily found by first finding the formula for charge and then taking the derivative. The graph is a sinusoidal graph that begins at 0 at the origin, so it is know that charge is a sin function. The period was found to be pi s, so therefore the angular frequency was found to be 2 rad/s. Using the general form q = q(max) sin (wt), the formula for charge was found, and taking the derivative the current as a function of time was obtained.
In this problem i(t) was given and total charge in 2 seconds was wanted to be found. Knowing that q is the integral of i times dt, the charge function was found. With the limits from 0 < t < 2, the charge obtained in 2 sec was found.
It this problem voltage and a graph of current as a function of time was given. The problem asked for power and energy as a function of time. Multiplying the graph of i(t) by V, the graph of power as a function of time was found. Then, integrating P(t) allowed for finding the graph of energy as a function of time. From 0 < t < 2, the integral of a constant is a linear graph. Same goes from 2 < t < 4. From 4 < t < 6, the integral of 0 is just 0.
Graphs of current and voltage as functions of time was provided in this problem. The problem asked for finding the graph of power and energy, as well as finding the total energy from 0 < t < 4 seconds, the period. Piece-wise functions of i(t) and v(t) were found, and multiplying them together gave power as a function of time in a piece-wise function. Using the function the graph of P(t) was drawn. Then integrating P using dt intervals was done to find the piece-wise function of energy, and then E(t) was graphed. Then, the total energy in a period was found was integrating P * dt from 0 < t < 4 seconds.
The previous circuit was given, and the voltage Vo of the part of the circuit in the middle of the diamond shape.Using the conservation of energy principle, it was found that the power provided is equal to the power used in a circuit. Using these principles, the voltage Vo was found to be 20 volts.
LAB:
Purpose:
The purpose of this experiment was to experimentally determine the circuit layout of a breadboard, and how the holes in rows and columns are connected and whether the connections form open or closed circuits. Another purpose of this experiment was to learn how to use a digital multimeter to measure the resistance of a circuit.
Apparatus:
The materials used in this experiment include a digital multimeter (as an ohmmeter), alligator clips, jumper wires and a breadboard. The digital multimeter was used to measure the resistance of different wires and circuits. The jumper wires and alligator clips were used to attach the ohmmeter to the breadboard to measure its resistance. The breadboard was used to study its internal circuits and how the holes in the rows and columns are connected to form open and short circuits.
Procedure:
First, the leads of the ohmmeter were attached to holes in the breadboard that were in the same row on one side. The leads were attached to the holes in columns f and j on row 57. The ohmmeter measured a small resistance of 1.6 Ω.
Next, the leads of the ohmmeter were connected to holes in the breadboard that were in the same row but on different sides of the breadboard. The leads were connected to holes in columns a and j of row 57. An immeasurable resistance was picked up by the ohmmeter.
Then, one of the leads of the ohmmeter was disconnected and reattached to a hole in the breadboard that was on a different row and column on different sides. The leads were placed on rows 50 and 57 in columns j and a, respectively. Again, the ohmmeter measured a very large resistance, as shown in the picture above.
Lastly, the leads were left in the same position as before (one in row 57 column j and the other in row 50 column a). However, a jumper wire was used to connect the two rows (nodes) together. The resistance turned to be small again, measured by the ohmmeter as 2.7 Ω.
Data Analysis:
The resistances measured by the ohmmeter for the first and last procedures were low, whereas the resistances for the middle two procedures were immeasurable by the ohmmeter. Because of this, it shows that the first and last procedures (situations 1 and 4) consisted of connections between the holes. These connections correspond to "short circuits" or "closed circuits", where all of the wires are connected to form a circuit loop. On the other hand, the middle two procedures (situations 2 and 3) had no connections for the circuits, indicated by the infinite resistance. These disconnections show that the circuits in situations 2 and 3 were "open circuits", where there is a disconnection in the circuit, not allowing the electrons to flow and cause a current if the circuit were attached to a voltage supply.
Conclusion:
The connections layout of a breadboard and the use of a digital multimeter as an ohmmeter were determined in this experiment. The connections of a breadboard consist of all the holes in each row on one side connected to each other, as shown in the top figure. Also, the holes of two columns on each side of the breadboard are also connected vertically, as seen in the above picture. This layout of the breadboard was determined by connecting the leads of the digital multimeter to different holes in either same rows and/or columns or different rows and/or columns, as explained in the four steps of the procedure above. The resistances read by the ohmmeter was also used in order to determine whether the way the ohmmeter was connected resulted in an open or closed circuit. An infinite resistance was found to imply an open circuit, and a very small resistance was found to imply a closed circuit.
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