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Chapter 8: Problem 2

Three particles each having a mass of \(100 \mathrm{gm}\) are placed on the vertices of an equilateral triangle of side \(20 \mathrm{~cm}\). The work done in increasing the side of this triangle to \(40 \mathrm{~cm}\) is \(\left(G=6.67 \times 10^{\prime \prime} \mathrm{Nm}^{2} \mathrm{~kg}^{2}\right)\) (a) \(5.0 \times 10^{-12} \mathrm{~J}\) (b) \(2.25 \times 10^{-10} \mathrm{~J}\) (c) \(4.0 \times 10^{-11} \mathrm{~J}\) (d) \(6.0 \times 10^{-15} \mathrm{~J}\)

Short Answer

Expert verified
(a) \(5.0 \times 10^{-12} \text{ J}\).

Step by step solution

01

Calculate Initial Potential Energy

The gravitational potential energy of the system with side length 20 cm is given by the formula for potential energy between two masses, repeated for all three sides of the triangle:\[U_i = -G\frac{m^2}{r} + -G\frac{m^2}{r} + -G\frac{m^2}{r}\]Substitute the given values, \(m = 0.1 \text{ kg}\), \(r = 0.2 \text{ m}\) (20 cm), and \(G = 6.67 \times 10^{-11} \text{ Nm}^2\text{/kg}^2\):\[U_i = -3 \times \frac{6.67 \times 10^{-11} \times (0.1)^2}{0.2}\]Simplifying gives:\[U_i = -1 \times 10^{-11} \text{ J}\]
02

Calculate Final Potential Energy

Repeat the calculation for the new side length of 40 cm (0.4 m):\[U_f = -G\frac{m^2}{r} + -G\frac{m^2}{r} + -G\frac{m^2}{r}\]Substitute the new radius:\[U_f = -3 \times \frac{6.67 \times 10^{-11} \times (0.1)^2}{0.4}\]Simplifying gives:\[U_f = -0.5 \times 10^{-11} \text{ J}\]
03

Calculate the Work Done

The work done in changing the configuration is the difference in gravitational potential energy:\[W = U_f - U_i\]Replacing \(U_i\) and \(U_f\):\[W = (-0.5 \times 10^{-11}) - (-1 \times 10^{-11})\]This further simplifies to:\[W = 0.5 \times 10^{-11} \text{ J} = 5.0 \times 10^{-12} \text{ J}\]
04

Conclusion: Identify the Correct Answer

The calculated work done is \(5.0 \times 10^{-12} \text{ J}\), which corresponds to option (a). Therefore, the correct answer is (a) \(5.0 \times 10^{-12} \text{ J}\).

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Key Concepts

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

Work Done
Work done refers to the amount of energy transferred when a force is applied to move an object a certain distance. In this exercise, work done involves changing the configuration of a system of particles by altering the side length of the equilateral triangle they form. Here, gravitational potential energy changes due to the change in distance between the masses, and this energy change represents the work done on the system.

In practical terms, to find the work done in altering the triangle from a side length of 20 cm to 40 cm, we calculate the difference between the initial and final gravitational potential energies. Work done can be mathematically expressed as:
  • \[ W = U_f - U_i \]
  • Where \(W\) is the work done, \(U_f\) is the final potential energy, and \(U_i\) is the initial potential energy.
Calculating work done in such contexts helps us understand how energy is conserved and transformed in physical systems.
Equilateral Triangle
An equilateral triangle is a special type of triangle where all three sides are of equal length, and all angles are equal, typically each being 60 degrees. This symmetry is crucial in solving problems involving forces and energies evenly distributed across the vertices.

In this exercise, the masses are placed at the vertices of an equilateral triangle, and we're interested in understanding how the configuration's gravitational potential energy changes when the triangle's side is doubled. This type of symmetry ensures that any changes in the side length affect all particle-pair interactions equally.

Utilizing the equilateral triangle properties simplifies calculations and provides insight into how geometric configurations influence the system's energy.
Distance Effect on Energy
The distance between masses plays a critical role in determining the gravitational potential energy of the system. Gravitational potential energy between any two masses is inversely proportional to the distance between them. This means as the distance increases, the potential energy becomes less negative (or less bound), indicating reduced attractive force.

When the side of the triangle increases from 20 cm to 40 cm, the potential energy of the system becomes less negative. Initially, the potential energy was more negative due to closer proximity of the masses. As distance doubles, gravitational influences decrease, and the potential energy lessens in magnitude.
  • Mathematically, this relationship is shown as:\[ U = -G \frac{m^2}{r} \],where \(U\) is the gravitational potential energy, \(G\) is the gravitational constant, \(m\) is the mass of each particle, and \(r\) is the separation distance.
  • Understanding this relation helps visualize how distance manipulation affects system energy, informing engineering, astronomy, and physics applications.

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