Two small spheres of mass m are suspended from strings of length ℓthat are connected at a common

step 1 Okay, so Theta 1 should be equal to Theta 2. Because according to Newton's third law, for every

Two small spheres of mass m are suspended from strings of...

Two small spheres of mass $m$ are suspended from strings of length $\ell$ that are connected at a common point. One sphere has charge $Q$ and the other has charge 2$Q$ The strings make angles $\theta_{1}$ and $\theta_{2}$ with the vertical. (a) Explain how $\theta_{1}$ and $\theta_{2}$ are related. (b) Assume $\theta_{1}$ and $\theta_{2}$ are small. Show that the distance $r$ between the spheres is approximately $$ r \approx\left(\frac{4 k_{e} Q^{2} \ell}{m g}\right)^{1 / 3} $$

Two small spheres of mass $m$ are suspended from strings of length $\ell$ that are connected at a common point. One sphere has charge $Q$ and the other has charge 2$Q$ The strings make angles $\theta_{1}$ and $\theta_{2}$ with the vertical. (a) Explain how $\theta_{1}$ and $\theta_{2}$ are related. (b) Assume $\theta_{1}$ and $\theta_{2}$ are small. Show that the distance $r$ between the spheres is approximately
$$
r \approx\left(\frac{4 k_{e} Q^{2} \ell}{m g}\right)^{1 / 3}
$$

Key Concepts

Small Angle Approximation

When angles are sufficiently small, trigonometric functions like sine and tangent can be approximated by the angle itself in radians. This simplifies the mathematical analysis in systems where angles are small, making it easier to derive relationships between distances and forces.

Geometric Relations in Mechanical Systems

These relations involve using trigonometry to relate physical lengths and angles in a system, which is important for expressing distances (such as the separation between objects) in terms of other known quantities like string length and angles. This concept bridges the gap between the physical geometry of the setup and the forces acting within it.

Coulomb’s Law

This principle describes the force between two point charges, stating that the magnitude of the repulsive or attractive force is proportional to the product of the charges and inversely proportional to the square of the distance between them. It is essential for calculating electrostatic forces in systems with charged objects.

Force Equilibrium

In a static system, all forces acting on an object must balance so that there is no net motion. In such problems, understanding how gravitational, tension, and electrostatic forces combine to produce equilibrium is crucial for setting up and solving the appropriate equations.

Transcript

00:01 Okay, so theta 1 should be equal to theta 2.

00:03 Because according to newton's third law, for every action in the nature, there should be an equal and opposite reaction, which means that the electric force on each sphere should have in the same magnitude by opposite direction.

00:17 So the deflection should be equal, which is the angle here.

00:20 So theta 1 is equal to theta 2.

00:22 For question b, when the electric force is equal to k -eq times 2 k2 over r squared, and r is the distance between two small spheres.

00:29 And this is a 3 bi diagram i drew from 1st2.

00:32 Sphere and so this sphere should experience three different forces which is electric force from another sphere and a tension from string and the gravity from itself.

00:39 If we decompose the tension we have xy components of tension and the y components of tension is t cosine theta 1.

00:46 You can use theta 2 as well because theta 1 is equal to theta 2.

00:50 And the x component of tension is t sine theta 1.

00:55 And according to graph we have t sine theta 1 is equal to electric force which is equal to ke q times 2 q over r square which we will we have r square is equal to 2 k eq squared over t sine theta 1.

01:07 And according to the graph, we have t cosine theta 1 should have the same magnitude as a gravity.

01:14 So we have t cosine equal to mg.

01:16 So therefore tension t is equal to mg divided by cosine theta...

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