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Chapter 14: Problem 38

In deep space, sphere \(A\) of mass \(20 \mathrm{~kg}\) is located at the origin of an \(x\) axis and sphere \(B\) of mass \(10 \mathrm{~kg}\) is located on the axis at \(x=0.80 \mathrm{~m} .\) Sphere \(B\) is released from rest while sphere \(A\) is held at the origin. (a) What is the gravitational potential energy of the two-sphere system as \(B\) is released? (b) What is the kinetic energy of \(B\) when it has moved \(0.20 \mathrm{~m}\) toward \(A\) ?

Short Answer

Expert verified
(a) Initial gravitational potential energy is \( -1.66857 \times 10^{-8} \mathrm{~J} \). (b) Kinetic energy after moving 0.20 m is \( 5.56190 \times 10^{-9} \mathrm{~J} \).

Step by step solution

01

- Calculate the initial gravitational potential energy

The gravitational potential energy between two masses is given by the formula \[ U = -\frac{Gm_1m_2}{r} \]Here, sphere A has mass \(m_1 = 20 \mathrm{~kg}\) and sphere B has mass \(m_2 = 10 \mathrm{~kg}\). The gravitational constant is \(G = 6.67430 \times 10^{-11} \mathrm{~m^3 \cdot kg^{-1} \cdot s^{-2}}\) and the initial separation between the spheres is \(r = 0.80 \mathrm{~m}\). Plug the values into the formula to get the initial potential energy.
02

- Calculate potential energy when B has moved 0.20 m towards A

When sphere B has moved 0.20 m towards sphere A, the new separation is \(r = 0.80 \mathrm{~m} - 0.20 \mathrm{~m} = 0.60 \mathrm{~m}\). Use the same formula \( U = -\frac{Gm_1m_2}{r} \)\ and rearrange it to compute the new potential energy with the updated distance.
03

- Calculate change in potential energy

Calculate the difference between the initial potential energy and the potential energy when sphere B has moved 0.20 m closer to sphere A. This will give us the change in potential energy \( \Delta U = U_{initial} - U_{new} \).
04

- Determine the kinetic energy of sphere B

According to the conservation of energy, the loss in potential energy is converted into kinetic energy. Therefore, the kinetic energy of sphere B when it has moved 0.20 m towards sphere A is equal to the change in potential energy |\( K = \Delta U \).

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Key Concepts

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

Gravitational Force
The gravitational force is fundamental in Newtonian mechanics. It describes the attraction between two masses. Mathematically, the force is expressed as \( F = \frac{Gm_1m_2}{r^2} \) where:
  • \( F \) is the gravitational force between the masses
  • \( G \) is the gravitational constant, approximately \( 6.67430 \times 10^{-11} \mathrm{Nm^2/kg^2} \)
  • \( m_1 \) and \( m_2 \) are the masses of the two objects
  • \( r \) is the distance between the centers of the two masses
The gravitational force keeps celestial bodies in orbit and influences many physical phenomena on Earth. In the exercise, the gravitational force between spheres A and B acts to pull them towards each other.
Kinetic Energy
Kinetic energy is the energy possessed by a body due to its motion. It is given by the formula \( K = \frac{1}{2} mv^2 \). Here,
  • \( K \) represents kinetic energy
  • \( m \) is the mass of the object
  • \( v \) is its velocity
When an object's speed increases, its kinetic energy increases. In our problem, sphere B gains kinetic energy as it moves towards sphere A.
The change in gravitational potential energy transforms into kinetic energy, as seen in the step-by-step solution. When sphere B moves 0.20 m closer, the loss in potential energy equals the gain in kinetic energy.
Conservation of Energy
The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the exercise, the total energy of the system remains constant. Initially, the system has gravitational potential energy. As sphere B moves towards sphere A, this potential energy converts into kinetic energy:
  • Initial total energy = gravitational potential energy
  • Final total energy = leftover potential energy + kinetic energy
By calculating the change in potential energy, we can determine how much kinetic energy the sphere gains. This principle ensures that our calculations are consistent and accurate.
Newtonian Mechanics
Newtonian mechanics is the framework that describes the motion of objects under the influence of forces. Isaac Newton's laws of motion are fundamental to this framework. They state:
  • First Law: An object remains at rest, or in uniform motion, unless acted upon by a force.
  • Second Law: The force acting on an object is equal to its mass times its acceleration \( F = ma \).
  • Third Law: For every action, there is an equal and opposite reaction.
In the given exercise, Newtonian mechanics helps us understand how sphere B accelerates towards sphere A due to gravitational attraction. It also explains why the kinetic energy increases as the potential energy decreases, maintaining the conservation of energy.

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Most popular questions from this chapter

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