Ch 08: Dynamics II: Motion in a Plane

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 08: Dynamics II: Motion in a Plane

Problem 7

Chapter 8, Problem 7

In the Bohr model of the hydrogen atom, an electron (mass m = 9.1 x 10^-31 kg) orbits a proton at a distance of 5.3 x 10^-11 m. The proton pulls on the electron with an electric force of 8.2 x 10^-8 N. How many revolutions per second does the electron make?

Verified step by step guidance

Step 1: Begin by understanding the relationship between the centripetal force and the electric force acting on the electron. In circular motion, the centripetal force is provided by the electric force. Use the formula for centripetal force:

F = m v^{2} / r

, where

F

is the force,

m

is the mass of the electron,

v

is the velocity, and

r

is the radius of the orbit.

Step 2: Equate the centripetal force to the given electric force:

m v^{2} / r = F

. Rearrange the equation to solve for the velocity

v

v = \sqrt{(F r / m)}

. Substitute the values for

F

r

, and

m

into the equation.

Step 3: Once the velocity

v

is determined, calculate the time it takes for the electron to complete one revolution. The circumference of the orbit is given by

C = 2 π r

. The time for one revolution is

t = C / v

Step 4: To find the number of revolutions per second, calculate the reciprocal of the time for one revolution:

f = 1 / t

. Substitute the values for

t

from the previous step.

Step 5: Perform the calculations step by step, ensuring unit consistency throughout. The final result will give the frequency of revolutions per second, which is the number of revolutions the electron makes in one second.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Centripetal Force

Centripetal force is the net force that acts on an object moving in a circular path, directed towards the center of the circle. In the context of the Bohr model, the electric force between the proton and electron provides the necessary centripetal force to keep the electron in its circular orbit. This force can be calculated using Newton's second law, where the centripetal force is equal to the mass of the electron multiplied by the square of its velocity divided by the radius of the orbit.

Electric Force

Electric force is the attractive or repulsive force between charged particles, described by Coulomb's law. In the hydrogen atom, the proton and electron exert an electric force on each other due to their opposite charges. This force is crucial for determining the electron's motion and can be calculated using the formula F = k * (|q1 * q2| / r^2), where k is Coulomb's constant, q1 and q2 are the charges, and r is the distance between them.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a central point, expressed in radians per second or revolutions per second. For the electron in the Bohr model, the angular velocity can be derived from the relationship between the centripetal force and the radius of the orbit. By equating the centripetal force to the electric force and solving for the velocity, one can find the angular velocity, which can then be converted to revolutions per second.

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In the Bohr model of the hydrogen atom, an electron (mass m = 9.1 x 10-31 kg) orbits a proton at a distance of 5.3 x 10-11 m. The proton pulls on the electron with an electric force of 8.2 x 10-8 N. How many revolutions per second does the electron make?

Verified video answer for a similar problem:

Key Concepts

Centripetal Force

Electric Force

Angular Velocity

In the Bohr model of the hydrogen atom, an electron (mass m = 9.1 x 10^-31 kg) orbits a proton at a distance of 5.3 x 10^-11 m. The proton pulls on the electron with an electric force of 8.2 x 10^-8 N. How many revolutions per second does the electron make?