A metal rod of mass m slides without friction along two parallel horizontal rails separated by a distance L and connected by a resistor R. A uniform magnetic field of magnitude B is applied perpendicular to the plane of the paper. A force is applied to give the bar an initial speed v. In terms of m, L, R, B, and v, find the distance the rod will slide as it coasts to a stop.
The distance the rod will slide as it coasts to a stop is given by:
d = (mv^2)/(2BLR)
To find the distance the rod will slide as it coasts to a stop, we need to consider the forces acting on the rod.
First, let's determine the magnetic force acting on the rod. The magnetic force is given by the formula F = BIL, where B is the magnetic field strength, I is the current flowing through the rod, and L is the length of the rod.
The current flowing through the rod is determined by Ohm's law, which states that the current (I) is equal to the voltage (V) divided by the resistance (R). In this case, the voltage across the resistor is V = IR, where I is the current and R is the resistance.
Using Newton's second law, F = ma, we can equate the magnetic force with the force required to accelerate or decelerate the rod. Since there is no friction, the only force acting against the rod's motion is the magnetic force.
Therefore, we have BIL = ma.
Rearranging the equation to solve for a, we get a = BIL/m.
The acceleration of the rod is the rate at which its speed decreases. We can use the kinematic formula v^2 = u^2 - 2as, where v is the final speed, u is the initial speed, a is the acceleration, and s is the distance traveled.
Since the final speed is zero (as the rod comes to a stop), we get 0 = u^2 - 2as.
Rearranging this equation to solve for s, we have s = u^2 / (2a).
Now we can substitute the expression for acceleration (a = BIL/m) into the equation for distance traveled (s = u^2 / (2a)).
s = u^2 / (2(BIL/m)).
Since u^2 is proportional to v^2, we can rewrite the equation as:
s = v^2 / (2(BIL/m)).
Finally, simplifying the equation further, we get:
s = (v^2 * m) / (2BIL).
Therefore, the distance the rod will slide as it coasts to a stop is given by (v^2 * m) / (2BIL).
To find the distance the rod will slide as it coasts to a stop, we need to analyze the forces acting on the rod.
1. Magnetic Force:
The magnetic force acting on the rod is given by the equation: Fm = BIL, where B is the magnetic field strength, I is the current, and L is the length of the rod.
2. Electric Force:
The electric force acting on the rod is given by the equation: Fe = I * V, where I is the current and V is the voltage drop across the resistor (which is equal to the product of the current and the resistance, V = IR).
3. Net Force:
The net force acting on the rod is the difference between the magnetic force and the electric force, so we have: Fnet = Fm - Fe.
4. Acceleration:
Using Newton's second law (F = ma), we can relate the net force to acceleration: Fnet = ma. Rearranging the equation, we have: a = Fnet / m.
5. Time:
The time it takes for the rod to come to a stop can be calculated using the equation: v = u + at, where v is the final velocity (which is 0 when the rod comes to a stop), u is the initial velocity (given as v), a is the acceleration, and t is the time.
6. Distance:
The distance the rod will slide can be calculated using the equation: s = ut + (1/2)at², where s is the distance, u is the initial velocity (given as v), a is the acceleration (calculated in step 4), and t is the time (calculated in step 5).
So, the distance the rod will slide can be calculated as follows:
1. Calculate the electric force: Fe = I * V = I * (IR) = I²R.
2. Calculate the net force: Fnet = Fm - Fe.
3. Calculate the acceleration: a = Fnet / m.
4. Calculate the time: t = -v / a.
5. Calculate the distance: s = v * t + (1/2) * a * t².
Substituting the given values for m, L, R, B, and v into these equations will give you the final answer, which is the distance the rod will slide as it coasts to a stop.