A tiny sphere with a charge of 7.0 μC is attached to a spring. Two other tiny charged spheres, e

A tiny sphere with a charge of 7.0 μC is attached to a...

A tiny sphere with a charge of $7.0 \mu \mathrm{C}$ is attached to a spring. Two other tiny charged spheres, each with a charge of $-4.0 \mu \mathrm{C},$ are placed in the positions shown in the figure and the spring stretches $5.0 \mathrm{cm}$ from its previous equilibrium position toward the two spheres. Calculate the spring constant.

Key Concepts

Static Equilibrium

This concept refers to a state where the net force acting on an object is zero, meaning that all the forces balance each other out. In problems involving a spring and electrostatic forces, static equilibrium is used to set the restoring force equal to the net electrostatic force to solve for unknowns such as the spring constant.

Coulomb's Law

This principle quantifies the electrostatic force between two point charges, asserting that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. It is essential for determining the magnitude of the force acting between charged objects.

Hooke's Law

This law states that the force exerted by a spring is directly proportional to its displacement from its equilibrium position, with the proportionality constant being the spring constant (k). It is crucial for relating the displacement of the spring to the restoring force it generates.

Transcript

00:01 Now in this problem we are given that a tiny sphere with a charge of 7 microcoulomb, q is 7 .0 microcoulomb, it is attached at the end of the spring that you can see here in this image.

00:14 Now two other tiny charged sphere each with a charge of minus 4 microcoulomb, so these two charges are there with a charge minus 4 .0 microcoulomb are placed in this position that you can see here.

00:33 Now due to this the spring is stretched by 5 centimeter, this is the x value it is stretched by 5 centimeter from its previous equilibrium position.

00:45 Now we need to find the spring constant k.

00:50 So to solve this problem we're going to use the coulomb slope and the hook slope.

00:55 Now by symmetry we know that the horizontal component of the force cancel each other only the vertical component we're going to consider.

01:02 So first let me draw the figure here.

01:06 So here you can see these are the three charges q and this is small q with the charge minus 4 microcoulomb.

01:16 Now this distance let this be b which we are given to be 4 centimeter and let this distance be a and here also this will be a which we are given to be 2 centimeter.

01:30 Okay now we'll be considering the forces in the y direction.

01:33 So this is the y direction i'm taking.

01:35 Now let's calculate the magnitude of the force in the y direction due to this charge.

01:40 So this is along the y direction let this force be f.

01:44 Now by coulomb slope i can write that f this would be equals to k.

01:49 I'm just taking the magnitude so q times q small q divided by r square times cosine theta.

01:59 Correct? here this is the theta here i'm taking this theta and this is the r.

02:06 Now r from the pythagoras theorem i can write that r square would be a square plus b square and cosine theta this will be equals to b over square root a square plus b square...

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