An electrically neutral model airplane is flying in a horizontal circle on a 2.0-m guideline, which is nearly parallel to the ground. The line breaks when the kinetic energy of the plane is 50 J. Reconsider the same situation, except that now there is a point charge of +q on the plane and a point charge of -q at the other end of the guideline. In this case, the line breaks when the kinetic energy of the plane is 52.0 J. Find the magnitude of the charges.
To find the magnitude of the charges, we need to make use of the principle of conservation of energy.
In the first scenario where the model airplane is electrically neutral, the only form of energy involved is kinetic energy. The breaking of the line is caused by the tension in the guideline, which is equal to the centripetal force required for circular motion.
The centripetal force can be calculated using the formula:
F = (mv^2) / r
Where F is the centripetal force, m is the mass of the airplane, v is its velocity, and r is the radius of the circular path (2.0 m in this case).
The kinetic energy can also be expressed in terms of mass and velocity:
KE = (1/2)mv^2
Since the question states that the kinetic energy in the first scenario is 50 J, we can equate this to the centripetal force expression:
50 J = (1/2)mv^2
Simplifying the equation, we get:
100 = mv^2
Now let's consider the second scenario where the airplane has a charge of +q and there is an equal charge of -q at the other end of the guideline. In this case, we need to take into account the additional force due to the electrical interaction between the charges.
The electrostatic force between the charges can be calculated using Coulomb's Law:
F = k(q1q2) / r^2
Where F is the electrostatic force, k is the electrostatic constant, q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
Since the line breaks when the kinetic energy of the plane is 52 J in this scenario, we can equate this to the sum of the electrostatic force and the centripetal force:
52 J = (1/2)mv^2 + k(q1q2) / r^2
Substituting the value of 100 for mv^2 from the first scenario, we have:
52 J = 100 + k(q1q2) / r^2
Now we need to substitute the values we have:
k = 9 x 10^9 N m^2/C^2 (electrostatic constant)
q1 = +q (magnitude of the charge on the airplane)
q2 = -q (magnitude of the charge at the other end of the guideline)
r = 2.0 m (length of the guideline)
After substitution, the equation becomes:
52 J = 100 + (9 x 10^9)(q^2) / (2.0^2)
Now we can solve the equation for q, the magnitude of the charges, by rearranging and solving for q:
(9 x 10^9)(q^2) = (52 J - 100) x (2.0^2)
q^2 = [(52 J - 100) x (2.0^2)] / (9 x 10^9)
Finally, taking the square root of both sides, we can find the magnitude of the charges:
q = sqrt([(52 J - 100) x (2.0^2)] / (9 x 10^9))