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Ch. 31 - Maxwell's Equations and Electromagnetic Waves
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 31 - Maxwell's Equations and Electromagnetic WavesProblem 65
Chapter 30, Problem 65

A powerful laser portrayed in a movie provides a 3-mm-diameter beam of green light with a power of 3 W. A good agent inside a spacecraft aims the laser beam at an enemy astronaut hovering outside. The mass of the enemy astronaut is 120 kg and the spacecraft 185,000 kg. (a) Determine the “radiation-pressure” force exerted on the enemy by the laser beam assuming her suit is perfectly reflecting. (b) If the enemy is 30 m from the spacecraft’s center of mass, estimate the gravitational force the spacecraft exerts on the enemy. (c) Which of the two forces is larger, and by what factor?

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Step 1: Calculate the radiation pressure force exerted by the laser beam. For a perfectly reflecting surface, the radiation pressure is given by the formula: \( P = \frac{2I}{c} \), where \( I \) is the intensity of the laser beam and \( c \) is the speed of light. The intensity \( I \) can be calculated as \( I = \frac{P_{laser}}{A} \), where \( P_{laser} \) is the power of the laser and \( A \) is the cross-sectional area of the beam. The area \( A \) is \( \pi r^2 \), where \( r \) is the radius of the beam (half the diameter).
Step 2: Use the formula for gravitational force to calculate the force exerted by the spacecraft on the enemy astronaut. The gravitational force is given by \( F_g = \frac{G m_1 m_2}{r^2} \), where \( G \) is the gravitational constant, \( m_1 \) is the mass of the spacecraft, \( m_2 \) is the mass of the astronaut, and \( r \) is the distance between the two masses.
Step 3: Compare the two forces calculated in steps 1 and 2. To determine which force is larger, compare the magnitudes of the radiation pressure force and the gravitational force. To find the factor by which one force is larger than the other, divide the larger force by the smaller force.
Step 4: Analyze the results conceptually. Radiation pressure is typically very small compared to gravitational forces in most scenarios. However, the specific values in this problem will determine the outcome. Reflect on the implications of the results for the astronaut's motion.
Step 5: Summarize the findings. State which force is larger and by what factor, based on the calculations. Discuss the physical significance of the comparison in the context of the problem, such as the practicality of the laser beam affecting the astronaut's motion compared to gravitational attraction.

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Key Concepts

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

Radiation Pressure

Radiation pressure is the force exerted by electromagnetic radiation on a surface. When light strikes a perfectly reflecting surface, it transfers momentum, resulting in a pressure that can be calculated using the formula P = I/c, where P is the pressure, I is the intensity of the light, and c is the speed of light. In this scenario, the laser beam's power and area determine the intensity, which is crucial for calculating the force on the enemy astronaut.
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Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. The force can be calculated using the formula F = G(m1*m2)/r^2, where G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between their centers of mass. In this case, the gravitational force between the spacecraft and the enemy astronaut is essential for understanding the interaction between them.
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Comparative Force Analysis

Comparative force analysis involves evaluating two or more forces to determine their relative magnitudes. In this problem, we need to compare the radiation pressure force exerted by the laser beam on the enemy astronaut with the gravitational force from the spacecraft. By calculating both forces, we can identify which one is larger and by what factor, providing insight into the dynamics of the situation.
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