A small sphere of charge q1=0.800 μC hangs from the end of a...

A small sphere of charge $q_{1}=0.800 \mu \mathrm{C}$ hangs from the end of a spring as in Figure P22.32a. When another small sphere of charge $q_{2}=$ $-0.600 \mu \mathrm{C}$ is held beneath the first sphere as in Figure P22.32b, the spring stretches by $d=3.50 \mathrm{~cm}$ from its original length and reaches a new equilibrium position with a separation between the charges of $r=5.00 \mathrm{~cm}$. What is the force constant of the spring?

A small sphere of charge $q_{1}=0.800 \mu \mathrm{C}$ hangs from
the end of a spring as in Figure P22.32a. When another small sphere of charge $q_{2}=$ $-0.600 \mu \mathrm{C}$ is held beneath the first sphere as in Figure P22.32b, the spring stretches by $d=3.50 \mathrm{~cm}$ from its original length and reaches a new equilibrium position with a separation between the charges of $r=5.00 \mathrm{~cm}$. What is the force constant of the spring?

Key Concepts

Gravitational Force

Gravitational force is the force with which the Earth attracts objects towards its center and is calculated using F = mg, where m is the mass of the object and g is the acceleration due to gravity. This force is a key part of many equilibrium problems because it must be balanced by other forces (like spring force and, in this context, additional forces like the Coulomb force) to achieve a state of rest.

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from its equilibrium length. In other words, F = kx, where k is the spring constant and x is the displacement. This concept is essential for determining how much restoring force the spring exerts in response to being stretched or compressed, which is critical when analyzing systems in which a spring interacts with external forces.

Coulomb’s Law

Coulomb’s Law describes the force between two point charges. The magnitude of the electric force is given by F = k_e |q1 · q2| / r^2, where k_e is the Coulomb constant, q1 and q2 are the charges, and r is the separation between them. This law is fundamental in understanding how charged objects interact with each other, particularly when calculating the additional force produced by electrostatic attraction or repulsion in equilibrium systems.

Static Equilibrium

Static equilibrium occurs when all the forces acting on an object balance out so that the net force is zero, resulting in no acceleration. In problems involving a mass on a spring subject to gravitational, elastic, and electrostatic forces, setting up the equilibrium condition (sum of forces equals zero) allows for solving unknown quantities, such as the spring constant, by equating the restoring force to the sum of other external forces.

Transcript

00:01 A small sphere with a charge of q1 .8 .00 microculems hanging from a spring and another sphere q2 of negative 0 .6 .0 microcolum is held beneath it.

00:13 The string stretches by 3 .5 centimeters and reaches a new equilibrium with a separation between the charges of 5 .00 centimeters and we're asked to find the force constant of the spring.

00:31 So first of all, we know that the sum of our forces is going to equal zero.

00:37 Hook's law tells us that our spring constant is equal to kx, k is the spring constant, and the force is needed to extend our spring, and our force is equal to k -e times q1, q2, times r -squared.

01:07 Where k is equal to 8 .98 times 10 to the 9th newton meters squared per coulum squared.

01:21 And our given data will be q1 and let's do this in coolums will be 8 .00 times 10 to the minus 7th coolums.

01:37 Q2 will be 6 .00 negative times 10 to the minus 7th coulamps.

01:49 Our d, which is 3 .5 centimeters, will 0 .0 3 .50 meters.

01:59 And our separation will be r, which will be 0 .050 meters.

02:12 Okay.

02:19 So let's take a look at this.

02:22 So i've got a spring hanging from a spring and i've got q1.

02:34 And then i have a spring hanging and i've got q1 and i've got q1 and down here i have q2.

02:51 So here i'll have my spring constant.

02:59 F my charge.

03:02 This will be down...