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Chapter 18: Problem 24

An unstrained horizontal spring has a length of \(0.32 \mathrm{m}\) and a spring constant of \(220 \mathrm{N} / \mathrm{m}\). Two small charged objects are attached to this spring, one at each end. The charges on the objects have equal magnitudes. Because of these charges, the spring stretches by \(0.020 \mathrm{m}\) relative to its unstrained length. Determine (a) the possible algebraic signs and (b) the magnitude of the charges.

Short Answer

Expert verified
Both charges can be positive or negative, with a magnitude of approximately \(7.5 \times 10^{-6} \ \text{C}\).

Step by step solution

01

Understanding Spring Force and Hooke's Law

The spring force can be described using Hooke's Law: \( F_{spring} = k \cdot x \), where \( k = 220 \ \text{N/m} \) is the spring constant, and \( x = 0.020 \ \text{m} \) is the elongation of the spring. Thus, the spring force is \( F_{spring} = 220 \cdot 0.020 \).
02

Calculate the Spring Force

Plug the values into Hooke's Law: \( F_{spring} = 220 \times 0.020 = 4.4 \ \text{N} \). This is the force exerted by the spring due to its elongation.
03

Relate Spring Force to Electrostatic Force

Since the force exerted by the spring is balanced by the electrostatic force, according to Coulomb's Law, we equate them: \( F_{spring} = F_{electric} = \frac{k_e \cdot q^2}{r^2} \), where \( k_e = 8.99 \times 10^9 \ \mathrm{N \, m^2 / C^2} \), and \( r \) is the stretched length of the spring.
04

Calculate the Stretched Length of the Spring

The stretched length is the original length plus the elongation: \( r = 0.32 + 0.020 = 0.34 \ \text{m} \).
05

Solve for Charge Magnitude

Set up the equation from Step 3: \( 4.4 = \frac{8.99 \times 10^9 \cdot q^2}{(0.34)^2} \). Rearrange to find \( q^2 \), then solve for \( q \). \( q^2 = \frac{4.4 \times (0.34)^2}{8.99 \times 10^9} \approx 0.561 \times 10^{-9} \). Taking the square root gives: \( q \approx 7.5 \times 10^{-6} \ \text{C} \).
06

Determine Possible Algebraic Signs of Charge

Since the charges cause repulsion, they must both be of the same sign. The possible signs for both charges are either both positive (both \(+q\)) or both negative (both \(-q\)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hooke's Law
Hooke's Law is a fundamental principle in physics that describes the behavior of springs when they are compressed or elongated. It states that the force exerted by a spring is directly proportional to the distance it is stretched or compressed away from its natural (rest) length. This relationship can be expressed with the formula: \( F_{spring} = k \cdot x \), where
\( F_{spring} \) is the force exerted by the spring,
\( k \) is the spring constant, a measure of the spring's stiffness, and
\( x \) is the displacement from the spring's natural length.
A higher spring constant means a stiffer spring, which requires more force to produce the same amount of stretch as a less stiff spring. In our exercise, with an unstrained length of 0.32 meters and a spring constant of 220 N/m, the spring stretches by 0.020 meters due to electrostatic forces exerted on the charges at its ends.
Coulomb's Law
Coulomb's Law explains how electrostatic forces work between charged objects. It provides a formula to calculate the force of interaction between two point charges. According to Coulomb's Law, this force is directly proportional to the product of the magnitudes of the charges, and inversely proportional to the square of the distance between them: \( F_{electric} = \frac{k_e \cdot q_1 \cdot q_2}{r^2} \).Here,
\( k_e = 8.99 \times 10^9 \ \mathrm{N \ m^2 / C^2} \) is a constant known as Coulomb's constant,
\( q_1 \) and \( q_2 \) are the magnitudes of the two charges, and
\( r \) is the distance between the charges.
In this problem, the tension from the spring and the electrostatic force balance each other, making it possible to relate the spring's stretch to the forces between the charges.
Spring Force
The spring force is the force exerted by a spring when it is stretched or compressed from its rest position. It is a restoring force, meaning it acts to return the spring to its original length. According to Hooke's Law, this force is given by: \( F_{spring} = k \cdot x \).
In our scenario, with the values provided (spring constant \( k = 220 \ ext{N/m} \) and elongation \( x = 0.020 \ ext{m} \)), the spring force calculates to \( 4.4 \ ext{N} \).
This force is what keeps the charged objects in equilibrium and is counterbalanced by the electrostatic repelling force between two identically charged objects. The equilibrium shows us the beauty of how these two forces manage to balance out when equal magnitudes of opposing forces are at play.
Charge Magnitude
Charge magnitude refers to the size of an electric charge, a crucial concept in understanding forces in electrostatic fields. To determine the magnitude of charges in this exercise, we use the balance of spring and electrostatic forces. By rearranging Coulomb's equation (equated to spring force since they balance each other), and substituting the values: \( 4.4 = \frac{8.99 \times 10^9 \cdot q^2}{(0.34)^2} \),
we solve for \( q^2 \), and find \( q \approx 7.5 \times 10^{-6} \ ext{C} \).Understanding charge magnitude helps us predict how charges interact under certain conditions. In this problem, the equal magnitude of the charges ensures that they either both attract or repel with equal force, resulting in either both being positive or both negative.

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Most popular questions from this chapter

The membrane surrounding a living cell consists of an inner and an outer wall that are separated by a small space. Assume that the membrane acts like a parallel plate capacitor in which the effective charge density on the inner and outer walls has a magnitude of \(7.1 \times 10^{-6} \mathrm{C} / \mathrm{m}^{2} .\) (a) What is the magnitude of the electric field within the cell membrane? (b) Find the magnitude of the electric force that would be exerted on a potassium ion \(\left(\mathrm{K}^{+} ;\right.\) charge \(\left.=+e\right)\) placed inside the membrane.

A proton and an electron are moving due east in a constant electric field that also points due east. The electric field has a magnitude of \(8.0 \times 10^{4} \mathrm{N} / \mathrm{C} .\) Determine the magnitude of the acceleration of the proton and the electron.

ssm Consider three identical metal spheres, A, B, and C. Sphere A carries a charge of \(+5 q .\) Sphere \(\mathrm{B}\) carries a charge of \(-q .\) Sphere \(\mathrm{C}\) carries no net charge. Spheres \(A\) and \(B\) are touched together and then separated. Sphere \(C\) is then touched to sphere \(A\) and separated from it. Last, sphere \(C\) is touched to sphere \(\mathrm{B}\) and separated from it. (a) How much charge ends up on sphere \(\mathrm{C} ?\) What is the total charge on the three spheres (b) before they are allowed to touch each other and (c) after they have touched?

A charge \(+q\) is located at the origin, while an identical charge is located on the \(x\) axis at \(x=+0.50 \mathrm{m}\). A third charge of \(+2 q\) is located on the \(x\) axis at such a place that the net electrostatic force on the charge at the origin doubles, its direction remaining unchanged. Where should the third charge be located?

Two point charges are fixed on the \(y\) axis: a negative point charge \(q_{1}=-25 \mu \mathrm{Cat} y_{1}=+0.22 \mathrm{m}\) and a positive point charge \(q_{2}\) at \(y_{2}=+0.34 \mathrm{m}.\) A third point charge \(q=+8.4 \mu \mathrm{C}\) is fixed at the origin. The net electrostatic force exerted on the charge \(q\) by the other two charges has a magnitude of \(27 \mathrm{N}\) and points in the \(+y\) direction. Determine the magnitude of \(q_{2}.\)
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