Question

Two balls have the same mass of \(0.681 \mathrm{~kg}\) and identical charges of \(18.0 \mu \mathrm{C} .\) They hang from the ceiling on strings of identical length as shown in the figure. If the angle with respect to the vertical of the strings is \(20.0^{\circ}\), what is the length of the strings?

Short answer

(where the mass of the balls is \(0.681\mathrm{~kg}\) and the charge of both balls is \(18.0\mu \mathrm{C}\)) Answer: The length of the strings is approximately \(1.01\) meters.

Step-by-step solution

Determine the electric repulsion force

We will use Coulomb's Law to find the force of repulsion between the two charged balls: \(F_e = \cfrac{k|q_1 q_2|}{r^2}\) Where \(F_e\) is the electric force, \(k\) is the electrostatic constant (\(\approx 8.99\times10^9\,\mathrm{N\cdot m^2/C^2}\)), \(q_1\) and \(q_2\) are the charges of the balls, and \(r\) is the distance between them. Since both charges are identical and positive, we have: \(F_e = \cfrac{k(q)^2}{(2L\sin{\theta})^2}\) Where \(q=18.0\mu \mathrm{C}\), \(L\) is the length of the strings, and \(\theta=20.0^{\circ}\).

Determine the gravitational force

The gravitational force acting on each ball is given by: \(F_g = mg\) Where \(F_g\) is the gravitational force, \(m=0.681\mathrm{~kg}\) is the mass of the balls, and \(g\approx9.81\mathrm{~m/s^2}\) is the acceleration due to gravity.

Find the tension in the strings

The tension in the strings can be found by examining the horizontal and vertical components separately. The horizontal component of the tension (\(T_x\)) equals the electric repulsion force, and the vertical component (\(T_y\)) equals the gravitational force: \(T_x=F_e\) \(T_y=F_g\) Using trigonometry, we can find the total tension in the string as: \(T=\frac{T_y}{\cos\theta}\)

Set up the equation to solve for the length of strings (L)

We have the following equation from using the trigonometric and force equations: \(F_e = T\sin\theta\) \(F_g = T\cos\theta\) Divide the two equations to eliminate tension (T): \(\cfrac{F_e}{F_g}=\cfrac{T\sin\theta}{T\cos\theta} \) \(\cfrac{F_e}{F_g}=\tan\theta \) Now we will substitute \(F_e\) and \(F_g\) into the above equation: \(\cfrac{k(q)^2}{(2L\sin{\theta})^2g}=\tan\theta\) Our goal is to solve for \(L\).

Solve for the length of the strings (L)

Rearrange the equation to isolate L: \(L=\cfrac{k(q)^2}{4g\sin^2\theta\tan\theta}\) Substitute the given values and constants: \(L=\cfrac{8.99\times10^9 \cdot (18\times10^{-6})^2}{4\times 9.81\times\sin^2(20)\times\tan(20)}\) Now, compute the length: \(L\approx1.01\,\mathrm{m}\) Therefore, the length of the strings is approximately \(1.01\) meters.

Key concepts

Coulomb's Law

Coulomb's Law is fundamental when discussing electric forces between charges. It provides a way to calculate the magnitude of the electric force between two stationary charged particles. According to Coulomb's Law, the electric force (\(F_e\)) between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This is represented by the equation:
\[ F_e = \cfrac{k|q_1 q_2|}{r^2} \]
where:

• \(k\) is the electrostatic constant, with a value of approximately \(8.99 \times 10^9\,\mathrm{N\cdot m^2/C^2}\).
• \(q_1\) and \(q_2\) are the charges involved.
• \(r\) is the distance between the centers of the two charges.

Coulomb's Law is useful in determining how strong or weak electric forces are between charges, which in turn helps in understanding how these charges would interact with each other in a physical setup like balls hanging from strings.

Gravitational Force

Gravitational force is the attractive force acting between two masses. For most everyday calculations, such as determining the force on an object at or near the surface of the Earth, gravity can be simplified by using the formula:
\[ F_g = mg \]
where:

• \(F_g\) is the gravitational force acting on an object.
• \(m\) is the mass of the object.
• \(g\) is the acceleration due to gravity, approximately \(9.81\, \mathrm{m/s^2}\) on the surface of the Earth.

Gravitational force is key in calculating the vertical forces involved when determining the tension in the strings from which the balls hang. It helps balance the forces so that they can form a coherent system of equilibrium.

Trigonometry

Trigonometry is crucial when analyzing forces that act at an angle, such as tension in a string. It involves using the properties of triangles to solve problems. In the context of hanging balls, trigonometry helps resolve the tension in the strings into its horizontal and vertical components.
The tension in the string is divided into:

• Horizontal component (\(T_x\)): This is equivalent to the electric force between the balls. Calculated as \(T\sin{\theta}\).
• Vertical component (\(T_y\)): This is balanced by the gravitational force. It can be calculated as \(T\cos{\theta}\).

By understanding these components, we can use trigonometric identities, such as \(\tan{\theta} = \cfrac{T_x}{T_y}\), to find relationships between the forces involved.

String Tension

String tension is a key factor when understanding how the forces interact in a scenario with hanging objects. It is the force exerted by the string to keep the charged balls in equilibrium.
Tension plays a dual role here:

• Vertical role: It balances out the gravitational force, which is pulling the balls straight down.
• Horizontal role: It resists the electric repulsion force that is pushing the balls apart.

The entire purpose of determining string tension is to ensure that the forces are in equilibrium, meaning they cancel each other out perfectly. When balanced, you can then reliably calculate other unknowns in the system, such as the length of the strings. String tension is found using the relationship between its horizontal and vertical components, bringing into play trigonometric identities for a comprehensive solution.