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Question: In the figure, in a circuit, there are two motors, one is the Arm motor of \[12\] watts and the seco...

In the figure, in a circuit, there are two motors, one is the Arm motor of 1212 watts and the second one is the Wheel motor of   18\;18 watts and one 66 volts battery. Motors are in parallel connection. There are three points 11, 22 and 33 shown in the figure. Which of the following lists these three points in order from lowest to highest current?

(A) 1,2,3\left( {\text{A}} \right){\text{ }}1,2,3
(B) 1,3,2\left( {\text{B}} \right){\text{ }}1,3,2
(C) 2,1,3\left( {\text{C}} \right){\text{ }}2,1,3
(D) 2,3,1\left( {\text{D}} \right){\text{ }}2,3,1
(E) 3,2,1\left( {\text{E}} \right){\text{ }}3,2,1

Explanation

Solution

The current is passing through points 11, 22 and 33. There are acting two resistances along with two motors i.e. points 22 and 33.
Also, the equivalent resistance of these two resistances is acting in point 11.
We can calculate the value of the resistances by the given voltage and pressures of the motors.
We can calculate the value of the current of each point using Ohm’s law.
Hence we found the increasing order (lowest to highest) of flowing current along with the given points.

Formula used:
R1,R2,R3{R_1},{R_2},{R_3} are the resistances of the points 11, 22 and 33 respectively.
And, R1{R_1} is the equivalent resistance of R2{R_2}and R3{R_3}.
Since R2R3{R_2}||{R_3}
R1=R2R3R2+R3\therefore {R_1} = \dfrac{{{R_2}{R_3}}}{{{R_2} + {R_3}}},
R2=V2PArmmotor{R_2} = \dfrac{{{V^2}}}{{{P_{Arm - motor}}}} and R3=V2PWheelmotor{R_3} = \dfrac{{{V^2}}}{{{P_{Wheel - motor}}}}
VV is the voltage of the battery.
I1,I2,I3{I_1},{I_2},{I_3} are the Current of the points11, 22 and 33 respectively.
According to Ohm’s law, I1=VR1{I_1} = \dfrac{V}{{{R_1}}},
I2=VR2{I_2} = \dfrac{V}{{{R_2}}}
I3=VR3{I_3} = \dfrac{V}{{{R_3}}}

Complete step by step answer:

In the question stated as the current is passing through points 11, 22 and 33.
There are acting two resistances along with two motors i.e. points 22 and 33.
Also, the equivalent resistance of these two resistances is acting in point 11.
HereR1,R2,R3{R_1},{R_2},{R_3} are the resistances of the points 11, 22 and 33 respectively.
Also, R1{R_1} is the equivalent resistance of R2{R_2} and R3{R_3}.
Here I1,I2,I3{I_1},{I_2},{I_3} are the Current of the points 11, 22 and 33 respectively.
Also, I1{I_1} is the total current of the circuit.
I1=VR1{I_1} = \dfrac{V}{{{R_1}}}
I2=VR2{I_2} = \dfrac{V}{{{R_2}}}and I3=VR3{I_3} = \dfrac{V}{{{R_3}}}
Now, VV is the voltage of the battery which is given 66 Volt.
Since the two motors are in parallel connection the voltage across them is the same as the voltage of the battery i.e. 66Volt.
The power of the Arm motor is PArmmotor=12watt{P_{Arm - motor}} = 12watt.
So, we can write it as
R2=V2PArmmotor{R_2} = \dfrac{{{V^2}}}{{{P_{Arm - motor}}}}
Here V=6V = 6 and we get,
R2=6212=3Ω\Rightarrow {R_2} = \dfrac{{{6^2}}}{{12}} = 3\Omega
And, The power of the Wheel motor is PWheelmotor=18watt{P_{Wheel - motor}} = 18watt.
R3=V2PWheelmotor{R_3} = \dfrac{{{V^2}}}{{{P_{Wheel - motor}}}}
Putting the values and we get,
R3=6218=2Ω\Rightarrow {R_3} = \dfrac{{{6^2}}}{{18}} = 2\Omega
Since,R2R3{R_2}||{R_3}
R1=R2R3R2+R3\therefore {R_1} = \dfrac{{{R_2}{R_3}}}{{{R_2} + {R_3}}}
Putting the finding values in R1{R_1} and we get,
R1=3×23+2=65Ω\therefore {R_1} = \dfrac{{3 \times 2}}{{3 + 2}} = \dfrac{6}{5}\Omega
\therefore The total current of the circuit I1=VR1=665=5A{I_1} = \dfrac{V}{{{R_1}}} = \dfrac{6}{{\dfrac{6}{5}}} = 5A. (Current flowing through point 11 )
Also, the current flowing through the two motors are,
I2=VR2=63=2A{I_2} = \dfrac{V}{{{R_2}}} = \dfrac{6}{3} = 2A (Current flowing through point 22 )
I3=VR3=62=3A{I_3} = \dfrac{V}{{{R_3}}} = \dfrac{6}{2} = 3A (Current flowing through point 33 )
So, the order of the given three points (1,2, and 3) according to the value of the lowest to the highest current flowing through them is 2,3,12,3,1.
Hence the right option is in option (D)\left( {\text{D}} \right).

Note: It has to be noted that we take the voltage drop across the two resistances along the two motors is 6 volt which is the voltage of the battery Since the two motors are in parallel connection.
Similarly, the resistances along them are also in parallel connection as they are acting along with the two motors.