Two conducting rails 39.2 cm apart rest on a 4.11° ramp. They are joined at the bottom by a 0.520 Ω resistor, and at the top a copper bar of mass 0.0450 kg is laid across the rails. The whole apparatus is immersed in a vertical 0.580 T field. What is the terminal (steady) velocity of the bar as it slides frictionlessly down the rails?
power into the bar=power in the resistor.
rate of change of GPE=V^2/R
Velocity*sinTheta*mg= (B dA/dt)^2 /R
velocity*sinTheta*mg=( B* dArea/dt )^2/R
Area=2W+2L
dARea/Dt=2 *velocity
so if my algebra is right (check it)
velocity= Resistance*sinTheta*mg/B^2
check that again.
To solve this problem, we need to consider the forces acting on the copper bar and apply the principles of electromagnetism.
First, let's analyze the forces acting on the copper bar:
1. Gravitational Force (Fg): The weight of the copper bar is given by the mass (m) times the acceleration due to gravity (g). The formula for calculating the gravitational force is Fg = m x g.
2. Frictional Force (Ff): The frictional force between the copper bar and the conducting rails is given by the coefficient of friction (μ) multiplied by the normal force (Fn). However, since the problem states that the bar slides frictionlessly down the rails, we can assume that there is no frictional force.
3. Magnetic Force (Fm): The magnetic force on a current-carrying conductor in a magnetic field is given by the formula Fm = ILB, where I is the current flowing through the conductor, L is the length of the conductor, and B is the magnetic field strength.
4. Electrostatic Force (Fe): When there is a potential difference across the resistor (due to the rails being at different potentials), an electric current flows through the circuit, generating an electrostatic force. This force can be calculated using Ohm's Law (V = I x R), where V is the potential difference and R is the resistance.
Since the bar is in a steady state, the sum of all the forces acting on it must be zero (ΣF = 0). Hence, we can equate the forces and calculate the terminal velocity (v) as follows:
Fg + Fm + Fe = 0
Fg = m x g
Fm = ILB
Fe = I x R
Substituting the values given in the problem:
m = 0.0450 kg (mass of copper bar)
g = 9.8 m/s^2 (acceleration due to gravity)
L = 39.2 cm = 0.392 m (length of the rails)
B = 0.580 T (magnetic field strength)
R = 0.520 Ω (resistance)
We can now plug these values into the equations and solve for the current (I) first:
I x 0.520 Ω + 0.0450 kg x 9.8 m/s^2 + I x 0.392 m x 0.580 T = 0
Now, we need to find the value of I that satisfies this equation. Once we have the current, we can calculate the magnetic force (Fm). Finally, using the magnetic force and the other forces, we can find the terminal velocity (v) using the formula:
Fg + Fm + Fe = 0
m x g + ILB + I x R = 0
Solving these equations will give us the terminal velocity of the copper bar as it slides down the rails, given the provided parameters.
To find the terminal velocity of the bar as it slides down the rails, we need to consider the forces acting on the bar.
First, let's find the gravitational force acting on the bar:
Fg = m * g,
where m is the mass of the bar (0.0450 kg) and g is the acceleration due to gravity (9.8 m/s^2).
Fg = 0.0450 kg * 9.8 m/s^2,
Fg = 0.441 N.
Next, let's consider the magnetic force acting on the bar. The magnetic force is given by:
Fm = q * v * B,
where q is the charge, v is the velocity, and B is the magnetic field strength.
The charge q can be calculated using Ohm's law:
V = I * R,
where V is the voltage, I is the current, and R is the resistance.
The voltage across the resistor can be found using the equation:
V = B * d * v,
where d is the distance between the rails (39.2 cm = 0.392 m) and v is the velocity of the bar.
Plugging the values in, we have:
V = 0.580 T * 0.392 m * v,
V = 0.2276 v T.
Using Ohm's law, we can also express the voltage in terms of current and resistance:
V = I * R,
0.2276 v T = I * 0.520 Ω,
I = 0.2276 v T / 0.520 Ω.
The charge q is equal to the current times the time taken to slide down the rails (t):
q = I * t,
q = (0.2276 v T / 0.520 Ω) * t,
q = 0.4377 v T * t Ω.
Now, let's substitute the value of q into the magnetic force equation:
Fm = 0.4377 v T * t Ω * v * B,
Fm = 0.4377 v^2 T * t Ω * B,
Fm = 0.4377 * B * t * v^2 Ω.
Considering the forces acting on the bar, the net force is given by:
Fnet = Fg - Fm,
0.441 N - 0.4377 * B * t * v^2 Ω = 0.
To find the terminal velocity, we need to set Fnet equal to zero:
0.441 N - 0.4377 * B * t * v^2 Ω = 0.
Solving for v^2, we get:
v^2 = 0.441 N / (0.4377 * B * t Ω).
Taking the square root of both sides, we find:
v = sqrt(0.441 N / (0.4377 * B * t Ω)).
Now we can plug in the values to find the terminal velocity of the bar as it slides down the rails.