A 100 gram resistive slider (thick line) moves without friction along a vertical U-shaped conducting track (thin line) that is 50 centimeters wide in a magnetic field of 5 Tesla as shown.
The slider's resistance is 20 Ohms. It falls under the influence of gravity. What is the terminal velocity of the slider in meters/second? Assume it can reach terminal velocity before it falls off.
Well, if the slider falls off, then it's not really terminal... it's more like "flight mode." But let's not let semantics get in the way!
To calculate the terminal velocity of the slider, we need to consider the balance between the gravitational force pulling it down and the electromagnetic force pushing it up due to the magnetic field.
First, let's calculate the gravitational force. The weight of the slider can be calculated using the equation:
Weight = mass * gravity
Since the mass of the slider is given as 100 grams, we need to convert it to kilograms before plugging it into the equation. So, 100 grams is equal to 0.1 kilograms. And gravity is approximately 9.8 meters per second squared. Therefore:
Weight = 0.1 kg * 9.8 m/s^2 = 0.98 N
Now, let's calculate the electromagnetic force. The electromagnetic force can be calculated using the equation:
Force = magnetic field strength * length of conductor * current
The length of the conductor is given as 50 centimeters, so we need to convert it to meters. Therefore, 50 centimeters is equal to 0.5 meters. And the current can be calculated using Ohm's Law:
Current = voltage / resistance
Since the voltage is not provided, let's go with a fun estimate and say it's 42 volts. Now, let's calculate the current:
Current = 42 V / 20 Ohms = 2.1 A
Now that we have the current, let's calculate the electromagnetic force:
Force = 5 T * 0.5 m * 2.1 A = 5.25 N
Finally, let's calculate the terminal velocity using the equation:
Terminal velocity = force / drag coefficient
The drag coefficient is not provided, so let's just say it's a nice round number, like 0.5. Therefore:
Terminal velocity = 5.25 N / 0.5 = 10.5 m/s
So, according to my calculations, the terminal velocity of the slider would be approximately 10.5 meters per second. Please note that these calculations are just an estimate for the sake of humor, and should not be considered scientifically accurate.
To find the terminal velocity of the resistive slider, we need to consider the forces acting on it. There are two main forces at play: the gravitational force (mg) and the magnetic force (BIL), where B is the magnetic field strength, I is the current, and L is the length of the conductor moving in the field.
1. Calculate the weight of the slider:
Weight (W) = mass (m) x acceleration due to gravity (g)
Weight (W) = 0.1 kg x 9.8 m/s^2
Weight (W) = 0.98 N
2. Calculate the magnetic force:
Magnetic Force (F) = BIL
The length of the track moving in the magnetic field is equal to the width of the track (50 cm) because the slider is moving along the U-shaped conducting track.
Convert the width to meters: 50 cm = 0.5 m
F = 5 T x I x 0.5 m
F = 2.5I N (since the magnetic field applies force per unit current)
3. At terminal velocity, the gravitational force and the magnetic force are equal:
W = F
0.98 N = 2.5I N
I = 0.98 N / 2.5 N
I = 0.392 A
4. The voltage across the slider is given by Ohm's Law:
Voltage (V) = Current (I) x Resistance (R)
V = 0.392 A x 20 Ω
V = 7.84 V
5. Now, we can calculate the power dissipated in the slider:
Power (P) = Voltage (V) x Current (I)
P = 7.84 V x 0.392 A
P = 3.06 W
6. The power dissipated is equal to the product of power and velocity:
Power (P) = Force (F) x Velocity (v)
3.06 W = (0.98 N - 2.5I N) x v
2.52 W = 2.5I N x v
v = 2.52 W / (2.5I N)
v = 2.52/ 2.5 m/s
v = 1.008 m/s
The terminal velocity of the resistive slider is approximately 1.008 meters/second.
To find the terminal velocity of the slider, we need to consider the forces acting on it. In this case, the two main forces are the force of gravity and the electromagnetic force.
The force of gravity acting on the slider can be calculated using the formula: F_gravity = mass x acceleration due to gravity.
Given that the mass of the slider is 100 grams, which is equal to 0.1 kg, and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the gravitational force:
F_gravity = 0.1 kg x 9.8 m/s² = 0.98 N
Next, we need to consider the electromagnetic force acting on the slider. The electromagnetic force experienced by a moving charged object in a magnetic field can be calculated using the formula: F_em = qvb.
In this case, the charge (q) of the slider is not given. However, we can relate it to the resistance (R) of the slider using Ohm's Law: V = IR, where V is the voltage and I is the current.
Since the slider is moving at a constant velocity (terminal velocity), the current (I) in the circuit must be constant. Therefore, we can use the formula: V = I(R + r), where r represents the internal resistance of the circuit (which we can assume to be negligible).
Rearranging the formula, we have: I = V / (R + r).
Given that the resistance (R) of the slider is 20 Ohms, we can calculate the current (I):
I = 5V / (20Ω) = 0.25 A
The electromagnetic force (F_em) can now be calculated using the formula F_em = qvb.
Since the slider is moving vertically in a U-shaped track, the velocity (v) can be related to the width (w) of the track and the time (t) it takes to cross the width: v = w / t.
The time (t) can be related to the current (I) and the charge (q) using the formula: t = q / I.
Combining the above formulas, we obtain: v = w / (q / I) = Iw / q.
We can now determine the electromagnetic force F_em:
F_em = qvb = (q)(0.25A)(5T).
To reach terminal velocity, the electromagnetic force must balance the force of gravity. Therefore, we have:
F_em = F_gravity
(q)(0.25A)(5T) = 0.98N
Now, we can solve for the charge (q):
q = (0.98N) / [(0.25A)(5T)]
Finally, we can determine the velocity v by substituting the value of q into the equation:
v = (0.25A)(50cm) / q
Since the track is 50 cm wide, we have v in cm/s. To convert it to m/s, we divide by 100:
v = [(0.25A)(50cm) / q] / 100
Now you can substitute the known values into the equation above to calculate the terminal velocity of the slider in meters per second.