Static Electricity Legacy Problem #11 Guided Solution

This is a very difficult problem which involves combining principles of circular motion with principles of electrostatic force. It was in the early 1900s that physicist Niels Bohr generated a model of the atom in which he had the electron orbiting about the nucleus almost as though planets orbit about the sun. In this model, it was the electrical force that pulled upon the electrons inward in order to provide the centripetal force in order for the electron to move in this circular orbit. In the Bohr model, we had the electrical force resulting from a positive nucleus with protons, in the case of the hydrogen atom, just one proton, and the negative electron that was in its orbit. Here, we are asked to use circular motion and electrostatic principles to determine the speed at which the electron moved in its orbit about that single proton in the hydrogen atom. What we need to recognize is that the centripetal force is being supplied by the electrical force. In other words, k times q of the electron times q of the proton divided by the distance squared between them is equal to the net force. And for objects moving in a circle, the net force is equal to the mass times the acceleration, where acceleration is simply v squared divided by r. So I can write this as my equation, F electrical equals F net in expanded form, that would be k times q of electron times q of proton over distance squared equals m times v squared over r. The r in the denominator on the right side of this equation is the same distance value as the d squared, or at least the d, in the denominator on the left side of the equation. So I'm going to replace the r by the variable d. Now I notice that in this equation, I know all of the quantities except for one, and that quantity is the v. Theoretically, I should be able to solve for v. In order to do so, I need to begin by doing some good algebra. I'm going to get the v by itself. So I'm going to multiply both sides of the equation by d. That cancels the d in the denominator on the right, and it also cancels one of the d's in the denominator on the left. And then I'm going to divide through both sides of the equation by m, where m is the mass of the electron. Doing so yields the equation that v squared is equal to k times q of electron times q of the proton, divided by the distance between them, divided by the mass of the electron. Now it's time to substitute in values for all of these quantities. For the value of k, I use 8.99 times 10 to the 9th newtons times meters squared per coulomb squared. For the value of q1 and q2, or q electron and q proton, I simply use the quantity 1.60 times 10 to the negative 19th coulombs. I need not worry about the fact that there's a positive charge for the proton and a negative for the electron, because these have nothing to do with the quantity of charge. They have to do with the type of charge. For the mass of the electron, I use the given value of 9.11 times 10 to the negative 31st. And for the distance between the proton and the electron, I use the given value of 5.29 times 10 to the negative 11th meters. When I substitute and solve for the quantity v squared, I get 4.7756 times 10 to the 12th meters squared per second squared. Now what I can do is take the square root of both sides, and I would get v is equal to 2.19 times 10 to the 7th meters per second. That's the speed of the electron orbiting about the nucleus in the Bohr model of the hydrogen atom.