Ch 22: Electric Charges and Forces

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo

Knight Calc 5th Edition

Ch 22: Electric Charges and Forces

Problema 24

Capitolo 22, Problema 24

A massless spring is attached to a support at one end and has a 2.0 μC charge glued to the other end. A −4.0 μC charge is slowly brought near. The spring has stretched 1.2 cm when the charges are 2.6 cm apart. What is the spring constant of the spring?

Guida verificata passo dopo passo

Identify the forces acting on the system. The spring is stretched due to the electrostatic force between the two charges. The electrostatic force is given by Coulomb's law:

F = \frac{k e \cdot q_{1} \cdot q_{2}}{r^{2}}

, where

q_{1}

and

q_{2}

are the charges,

r

is the distance between them, and

k_{e}

is Coulomb's constant (

k_{e} =8.99×10

⁹ N·m²/C²).

Relate the electrostatic force to the spring force. The spring force is given by Hooke's law:

F = k \cdot x

, where

k

is the spring constant and

x

is the displacement of the spring. At equilibrium, the spring force equals the electrostatic force:

k_{e} \cdot \frac{q_{1}}{q_{2}} = k \cdot x

Substitute the known values into Coulomb's law. Use

q_{1} =2.0×10

^-6 C,

q_{2} =-4.0×10

^-6 C,

r =2.6×10

^-2 m, and

k_{e} =8.99×10

⁹ N·m²/C² to calculate the electrostatic force:

F = \frac{k_{e} \cdot q_{1} \cdot q_{2}}{r^{2}}

Set the electrostatic force equal to the spring force. Use the displacement of the spring,

x =1.2×10

^-2 m, and solve for the spring constant

k

k = \frac{F}{x}

Simplify the expression to find the spring constant. Substitute the value of the electrostatic force from step 3 and the displacement of the spring into the equation from step 4 to calculate

k

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Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged objects. 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. This law is essential for calculating the force acting on the spring due to the charges.

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Percorso guidato

09:52

Coulomb's Law

Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to its extension or compression from its equilibrium position. Mathematically, it is expressed as F = kx, where F is the force, k is the spring constant, and x is the displacement. This principle is crucial for determining the spring constant in the context of the problem.

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Percorso guidato

05:27

Spring Force (Hooke's Law)

Equilibrium of Forces

In this scenario, the system reaches equilibrium when the electrostatic force exerted by the charges equals the restoring force of the spring. Understanding this balance is key to solving for the spring constant, as it allows us to set the forces equal to each other and solve for the unknown variable.

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Percorso guidato

05:10

Equilibrium

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