Review problem. Two identical metallic blocks resting on a...

Review problem. Two identical metallic blocks resting on a frictionless horizontal surface are connected by a light metallic spring having a spring constant $k$ as shown in Figure $\mathrm{P} 23.59 \mathrm{a}$ and an unstretched length $L_{i} .$ A total charge $Q$ is slowly placed on the system, causing the spring to stretch to an equilibrium length $L,$ as shown in Figure $\mathrm{P} 23.59 \mathrm{b}$ . Determine the value of $Q,$ assuming that all the charge resides on the blocks and modeling the blocks as point charges.

Key Concepts

Coulomb's Law

Coulomb's Law describes the force between two electric charges. It states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This concept is crucial for calculating the repulsive force when charges are distributed on objects, as is the case in systems where charged blocks interact.

Hooke's Law

Hooke's Law characterizes the behavior of elastic materials, stating that the force required to stretch or compress a spring by a certain distance is proportional to that displacement, with the proportionality constant being the spring constant. This principle is essential in problems where a spring’s extension or compression leads to a restoring force that balances other forces in the system.

Static Equilibrium

Static equilibrium occurs when all forces acting on a system are balanced, meaning there is no net force and the system remains at rest. In problems that involve forces such as those from springs and electrostatic repulsion, understanding equilibrium is key to setting up the equations where the forces cancel out, allowing for the determination of unknown quantities like the amount of charge.

System Modeling

System modeling involves approximating real-world objects with simplified models to facilitate analysis. In this case, treating the metallic blocks as point charges and assuming an ideal spring and frictionless surface allows for the use of fundamental laws like Coulomb’s and Hooke’s without additional complicating factors, thereby simplifying the analysis of the system's behavior.

Transcript

00:01 So in this problem, we are dealing with static electric fields and the relationship between the electric force and some other forces involved with the electrostatic force, like in this problem where the force of the elastic spring is involved in the relationship with the electrostatic repulsive force.

00:24 So we have two blocks and they are basically repulsing each other, repelling each other one from another because they have the same charge q and the same sign of the charge.

00:35 Now the total force between the two blocks can be modeled as the force between two point charges that are pulling each other one from another because we are assuming that the distance between two blocks is large enough so that you can treat them like two point charges.

00:54 And since we're given that l, being l is a separation between two blocks, the electrostatic force is static constant, q squared over l squared.

01:09 Now this force must balance another force, which is the elastic force of the spring, which will be stretched by the acceleration and the motion resulting from the acceleration of the two blocks coming from the electrostatic force repulsion.

01:29 So this force of corresponding to the spring stretching is x.

01:37 Now according to the k is the spring constant and x is the horizontal stretch because we are placing the two blocks to be on the horizontal axis in our coordinate system.

01:52 So from this condition that fe must equal fs in order for this problem to satisfy the static condition, we can directly substitute and say, okay, this is kx equal this expression right here.

02:15 And directly we can say that q squared equals and basically kx times l squared over electrostatic.

02:28 Constant.

02:30 From here we know that q is equal to the square root of k x times cell squared over electrostatic constant...

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