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Ch 22: Electric Charges and Forces
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 22, Problem 71

A 5.0 g ball charged to 1.5 μC is tied to a 25-cm-long string. It swings at 250 rpm in a horizontal circle around a stationary ball charged to −2.5 μC. What is the tension in the string?

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Convert all given quantities to SI units: mass of the ball (m = 5.0 g = 0.005 kg), charge of the moving ball (q₁ = 1.5 μC = 1.5 × 10⁻⁶ C), charge of the stationary ball (q₂ = -2.5 μC = -2.5 × 10⁻⁶ C), string length (r = 25 cm = 0.25 m), and angular velocity (ω = 250 rpm = (250 × 2π) / 60 rad/s).
Determine the centripetal force required to keep the ball moving in a horizontal circle. The centripetal force is provided by the horizontal component of the tension in the string. Use the formula for centripetal force: F_c = m * r * ω².
Calculate the electrostatic force between the two charges using Coulomb's law: F_e = (k * |q₁ * q₂|) / r², where k is Coulomb's constant (k = 8.99 × 10⁹ N·m²/C²). This force acts along the line connecting the two charges.
Resolve the tension in the string into two components: the horizontal component (T_h) provides the centripetal force (T_h = F_c), and the vertical component (T_v) balances the weight of the ball (T_v = m * g, where g = 9.8 m/s²).
Combine the components of tension to find the total tension in the string using the Pythagorean theorem: T = √(T_h² + T_v²). Substitute the expressions for T_h and T_v to calculate the total tension.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. In this scenario, the tension in the string provides the necessary centripetal force to keep the charged ball moving in a horizontal circle. The formula for centripetal force is F_c = m(v^2/r), where m is the mass, v is the tangential speed, and r is the radius of the circle.
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Electric Force

The electric force is the attractive or repulsive force between charged objects, described by Coulomb's Law. In this case, the charged stationary ball exerts an electric force on the swinging ball, which affects the net force acting on it. The magnitude of the electric force can be calculated using F_e = k * |q1 * q2| / r^2, where k is Coulomb's constant, and q1 and q2 are the charges of the two balls.
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Tension in a String

Tension is the force transmitted through a string, rope, or cable when it is pulled tight by forces acting from opposite ends. In this problem, the tension in the string must balance both the centripetal force required for circular motion and the electric force acting on the charged ball. The net force acting on the ball can be expressed as T - F_e = F_c, where T is the tension, F_e is the electric force, and F_c is the centripetal force.
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Related Practice
Textbook Question

In Section 22.3 we claimed that a charged object exerts a net attractive force on an electric dipole. Let's investigate this. FIGURE CP22.80 shows a permanent electric dipole consisting of charges +q and −q separated by the fixed distance s. Charge +Q is distance r from the center of the dipole. We'll assume, as is usually the case in practice, that s≪r. Use the binomial approximation (1+x)n1nx(1+x)^{-n}\(\thickapprox\)1-nx if x≪1 to show that your expression from part a can be written Fnet=2KqQs/r3F_{net}=2KqQs/r^3.

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Textbook Question

A small 1.0 g block charged to 75 nC is placed on a 30° inclined plane. The coefficients of static and kinetic friction are 0.20 and 0.10, respectively. What minimum strength horizontal electric field is needed to keep the block from sliding down the plane?

Textbook Question

Space explorers discover an 8.7×1017 kg asteroid that happens to have a positive charge of 4400 C. They would like to place their 3.3×105 kg spaceship in orbit around the asteroid. Interestingly, the solar wind has given their spaceship a charge of −1.2C. What speed must their spaceship have to achieve a 7500-km-diameter circular orbit?

Textbook Question

An electric field E=200,000i^\(\overrightarrow{E}\)=200,000\(\hat{i}\) N/C causes the point charge in FIGURE P22.68 to hang at an angle. What is θ?

Textbook Question

Three 1.0 nC charges are placed as shown in FIGURE P22.66. Each of these charges creates an electric field E at a point 3.0 cm in front of the middle charge. What are the three fields E₁, E₂, and E₃ created by the three charges? Write your answer for each as a vector in component form.

Textbook Question

Starting from rest, how long does it take an electron to move 1.0 cm in a steady electric field of magnitude 100 N/C?

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