A small sphere with a charge of +2.44 μC is attached to a relaxed horizontal spring whose force

A small sphere with a charge of +2.44 μC is attached to a...

A small sphere with a charge of $+2.44 \mu C$ is attached to a relaxed horizontal spring whose force constant is 89.2 $\mathrm{N} / \mathrm{m} .$ The spring extends along the $x$ axis, and the sphere rests on a frictionless surface with its center at the origin. A point charge $Q=-8.55 \mu C$ is now moved slowly from infinity to a point $x=d>0$ on the $x$ axis. This causes the small sphere to move to the position $x=0.124 \mathrm{m} .$ Find $d$ .

A small sphere with a charge of $+2.44 \mu C$ is attached to a relaxed
horizontal spring whose force constant is 89.2 $\mathrm{N} / \mathrm{m} .$ The spring
extends along the $x$ axis, and the sphere rests on a frictionless surface
with its center at the origin. A point charge $Q=-8.55 \mu C$ is now
moved slowly from infinity to a point $x=d>0$ on the $x$ axis. This
causes the small sphere to move to the position $x=0.124 \mathrm{m} .$ Find $d$ .

Key Concepts

Static Equilibrium

Static equilibrium occurs when the net force acting on a system is zero, meaning there is no acceleration. In problems involving a charged particle attached to a spring and subject to external forces, setting the elastic force equal to the electrostatic force allows one to solve for unknown parameters, such as displacements or distances, ensuring the system remains at rest.

Hooke's Law

This principle describes the behavior of springs, specifying that the force exerted by a spring is proportional to its displacement from its natural (unstressed) length, with the constant of proportionality being the spring constant. This concept is crucial for determining the restoring force in a spring in mechanical equilibrium problems.

Coulomb's Law

This law quantifies the electrostatic force between two point charges. It states 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. Its formulation helps determine both the magnitude and direction (attraction or repulsion) of the force based on the signs of the charges.

Transcript

00:01 Question 87.

00:03 States that a small sphere with a charge of positive 2 .44 microculems is attached to a relaxed horizontal spring, whose force constant is 89 .2 newton's per meter.

00:14 The spring extends along the x -axis, and the sphere rests on a frictionless surface with the center at the origin.

00:20 A point charge capital q equals negative 0 .55 microculems is now slowly moved from infinity to point x equals d, which is greater than 0, on the x -axis.

00:30 This causes the small sphere to move to position x equals 0 .124 meters.

00:37 Find d.

00:38 So a very interesting situation that i think would benefit very much with a diagram.

00:47 So we have this scenario.

00:49 We have a sphere attached to a spring.

00:54 So say it's attached to some wall.

00:57 We have our spring here.

00:59 Excuse me.

01:01 We have charge q here represented by this.

01:04 Sphere, charge q.

01:06 Oh, sorry, that's not charged q.

01:07 My diagram i'm calling that from my given, i say that's a little q.

01:12 And we say that this is fixed at the origin.

01:14 This is zero on the x -axis.

01:17 Find this to be the x -axis.

01:24 So the spring extends and it rests there.

01:27 Once we place capital q here, it is negative.

01:31 So this will draw this smaller sphere to position x, just given as 0 .124 meters.

01:43 Oh, no, sorry, that's a it was to position b, i meant to say.

01:47 X is the extension of the spring.

01:50 So the sphere of our little charge q moves to this position, given my d, and our goal is to solve for how much more does this spring extend, essentially, to provide this new position.

02:05 Okay, so in both situations, the spring is not moving, so we know this a static type of problem...

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