Guided Study: Newtonian Gravity and Special Relativity Concepts

Q1a. Who first proposed the Wave in Ether model of light? The Particle in Vacuum model? Clearly articulate the “physical picture” of each model. Discuss why, in the absence of experimental evidence to the contrary, they are both perfectly reasonable hypotheses. Could both be true?

Background

Topic: Historical Models of Light

This question explores the early scientific models for the nature of light, focusing on the wave (ether) and particle (vacuum) theories. Understanding these models is foundational for grasping the development of modern physics and the context for Einstein's work.

Key Terms:

• Wave in Ether Model: The idea that light is a wave that requires a medium (the 'ether') to propagate, similar to how sound waves need air.

• Particle in Vacuum Model: The concept that light consists of particles (corpuscles) that can travel through empty space without a medium.

Step-by-Step Guidance

1. Identify the scientists historically associated with each model. Consider the timeline of scientific discoveries about light.

2. Describe the 'physical picture' for each model: For the wave model, think about how waves generally behave in a medium; for the particle model, consider how particles move through space.

3. Discuss why, before experiments like Young's double-slit, both models were reasonable based on the evidence available at the time.

4. Reflect on whether both models could be true simultaneously, or if they are mutually exclusive, and why.

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Q1b. Assume light is a Particle in Vacuum. Discuss the two “speed of light” statements. Are either reasonable? Why or why not? Mark each as reasonable (√) or not reasonable (XX).

Background

Topic: Speed of Light in Particle Model

This question asks you to analyze how the speed of light would behave if light were made of particles traveling in a vacuum, and whether the speed would depend on the motion of the source or observer.

Key Concepts:

• Relative velocity in classical mechanics: How do velocities add when considering moving sources and observers?

• Galilean relativity: The classical (pre-relativity) way of adding velocities.

Step-by-Step Guidance

1. Recall how, in Newtonian mechanics, the velocity of a particle as measured by different observers depends on their relative motion.

2. Apply this reasoning to the two statements: (1) Speed of light independent of source motion, (2) Speed of light independent of observer motion.

3. For each, decide if it matches what you would expect for particles in a vacuum, and explain your reasoning.

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Q1c. Assume light is a Wave in Ether. Discuss the two “speed of light” statements:

Background

Topic: Speed of Light in Ether Model

This question examines how the speed of light would be observed if light were a wave propagating through a stationary ether, analogous to sound in air.

Key Concepts:

• Reference frames: The importance of being at rest or moving relative to the medium (ether).

• Wave propagation: How the speed of a wave depends on the observer's motion relative to the medium.

Step-by-Step Guidance

1. For (i): If you move toward or away from the source in the ether, use the analogy of sound waves in air to predict the observed speed of light.

2. For (ii): If the source moves but you are at rest in the ether, consider whether the speed of the wavefronts changes for you.

3. For each case, decide if the statement is reasonable under the wave in ether model, and explain why.

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Q1d. What famous experiment did Thomas Young perform in 1802? Did it prove the Wave in Ether model? Did it disprove the Particle in Vacuum model? Name two influential 1800s physicists who supported the Ether model.

Background

Topic: Experimental Evidence for Light Models

This question focuses on the impact of Young's experiment on the debate between wave and particle models of light, and the philosophy of scientific proof.

Key Terms:

• Young's Double-Slit Experiment: Demonstrated interference patterns, a hallmark of wave behavior.

• Proof vs. Disproof in Science: The difference between supporting evidence and absolute proof/disproof.

Step-by-Step Guidance

1. Describe what Young's experiment involved and what was observed.

2. Discuss whether the results conclusively proved the wave model or merely provided strong evidence.

3. Consider whether the experiment ruled out the particle model, and why or why not.

4. Identify two major 19th-century physicists who advocated for the ether model.

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Q1e. Einstein’s 1905 speed of light principle: “Any ray of light moves in the 'stationary' system of coordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body.”

Background

Topic: Einstein's Postulate and Historical Context

This question asks you to consider the prevailing model of light in 1905 and how Einstein's principle would have been received.

Key Concepts:

• Historical context: What physicists believed about light in 1905.

• Interpretation of 'stationary' system in Einstein's writing.

Step-by-Step Guidance

1. Identify which model (wave or particle) was dominant among physicists in 1905.

2. Discuss whether Einstein's principle would have seemed reasonable or surprising to his contemporaries.

3. Interpret what Einstein might have meant by 'stationary' system of coordinates, considering the context of Maxwell's equations and the ether.

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Q2a. Einstein’s 1905 relativity principle: “No physical experiment performed in a closed room moving with constant velocity can detect the motion of the room.”

Background

Topic: Principle of Relativity

This question explores the plausibility of the relativity principle for different types of phenomena: mechanical (particles), waves in a medium, and light in ether.

Key Concepts:

• Relativity principle: The idea that the laws of physics are the same in all inertial frames.

• Medium dependence: How the presence of a medium (like air or ether) affects the relativity principle.

Step-by-Step Guidance

1. For (i): Consider whether mechanical experiments (like billiards) can detect absolute motion.

2. For (ii): Analyze whether wave phenomena in a medium (like sound in air) can reveal the motion of the room relative to the medium.

3. For (iii): Apply this reasoning to the wave in ether model of light, and suggest an experiment that could test the relativity principle for light.

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Q2b. Discuss possible interpretations of the null result of the Michelson–Morley experiment.

Background

Topic: Michelson–Morley Experiment and Its Implications

This question asks you to consider different explanations for why the Michelson–Morley experiment failed to detect Earth's motion through the ether.

Key Concepts:

• Ether drag hypothesis

• Lorentz-FitzGerald contraction

• Rejection of ether

• Re-examination of space and time concepts

Step-by-Step Guidance

1. For each interpretation, briefly explain what it proposes.

2. For (i): Explain how ether drag would affect observations and why stellar aberration is a problem for this idea.

3. For (ii): Describe the Lorentz-FitzGerald contraction and how it attempts to explain the null result.

4. For (iii) and (iv): Discuss the implications of rejecting the ether or modifying our understanding of space and time.

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Q2c. Discuss how Einstein’s “universal” relativity principle rules out the concept of “absolute” velocity, space, and time.

Background

Topic: Relativity and the Nature of Space and Time

This question explores the philosophical shift introduced by Einstein, moving away from Newtonian absolutes to relative quantities.

Key Concepts:

• Relative vs. absolute velocity

• Relativity of space and time

Step-by-Step Guidance

1. Explain why, under Einstein's principle, only relative velocities are meaningful in physics.

2. Discuss how this principle challenges Newton's concepts of absolute space and time.

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Q2d. As budding physicists, which hypothesis feels more likely to be a universal principle of nature: Newton’s absolute time or Einstein’s universal relativity? Was there definitive experimental evidence against either in 1905? Could both be true?

Background

Topic: Competing Worldviews in Physics

This question asks you to reflect on the plausibility and experimental support for Newtonian and Einsteinian views in the early 20th century.

Key Concepts:

• Philosophy of science: How intuition and evidence guide acceptance of physical principles.

• Experimental status in 1905

Step-by-Step Guidance

1. Consider which principle aligns better with your understanding of nature and why.

2. Review whether experiments at the time definitively ruled out either hypothesis.

3. Discuss whether both could be true, or if one must be false, based on logical consistency and experimental evidence.

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Q3a. Calculate: Based on first principles (Newton’s law of universal gravitation and Newton’s law of dynamics), derive the relationship between $r_0$ and $v_0$, Halley’s orbital speed.

Background

Topic: Circular Orbits and Newtonian Gravity

This question tests your ability to derive the speed of an object in a circular orbit using Newton's laws.

Key Formulas:

• Newton’s law of universal gravitation: $F = \frac{GMm}{r^2}$

• Centripetal force for circular motion: $F = \frac{mv^2}{r}$

Step-by-Step Guidance

1. Set the gravitational force equal to the required centripetal force for circular motion.

2. Write out the equations: $\frac{GMm}{r_0^2} = \frac{mv_0^2}{r_0}$

3. Solve for $v_0$ in terms of $G$, $M$, and $r_0$.

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Q3b. Suppose Neptune passes behind Halley and reduces Halley’s speed from $v_0$ to $\alpha v_0$, where $\alpha < 1$.

Background

Topic: Orbital Changes and Elliptical Orbits

This question explores how a change in speed affects the shape and parameters of an orbit, using conservation laws.

Key Concepts:

• Elliptical orbits: Properties and definitions (aphelion, perihelion)

• Conservation of angular momentum and energy

Step-by-Step Guidance

1. Explain why reducing the speed at a given distance from the Sun leads to an elliptical orbit.

2. Identify which point in the new orbit corresponds to aphelion and perihelion.

3. Predict whether the perihelion distance $r_p$ will be less than or greater than $r_0$ based on the energy change.

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Q3c. Using conservation of angular momentum, show that $r_p = (\alpha/\beta) r_0$.

Background

Topic: Conservation of Angular Momentum in Orbits

This question asks you to apply angular momentum conservation to relate perihelion and aphelion distances and speeds.

Key Formula:

• Angular momentum: $L = mvr$ (for motion perpendicular to $r$)

Step-by-Step Guidance

1. Write the angular momentum at aphelion: $L_a = m v_a r_a$

2. Write the angular momentum at perihelion: $L_p = m v_p r_p$

3. Set $L_a = L_p$ and substitute $v_a = \alpha v_0$, $r_a = r_0$, $v_p = \beta v_0$, and solve for $r_p$ in terms of $\alpha$, $\beta$, and $r_0$.

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Q3d. Using conservation of energy, show that $\beta = (2 - \alpha^2)/\alpha$ and derive $r_p$ and $v_p$ in terms of $\alpha$ and $v_0$.

Background

Topic: Conservation of Energy in Orbits

This question requires you to use the total mechanical energy of an orbiting body to relate the speeds and distances at aphelion and perihelion.

Key Formula:

• Total energy: $E = \frac{1}{2}mv^2 - \frac{GMm}{r}$

Step-by-Step Guidance

1. Write the total energy at aphelion and perihelion and set them equal (since energy is conserved).

2. Substitute $v_a = \alpha v_0$, $r_a = r_0$, $v_p = \beta v_0$, and $r_p$ from the previous result.

3. Algebraically solve for $\beta$ in terms of $\alpha$.

4. Express $r_p$ and $v_p$ in terms of $\alpha$, $v_0$, and $r_0$ using your previous results.

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Q3e. Discuss three special cases: (i) $\alpha = 1$, (ii) $\alpha \to 0$, (iii) $\alpha \to \sqrt{2}$.

Background

Topic: Limits and Physical Interpretation of Orbital Parameters

This question asks you to interpret the formulas for perihelion distance and speed in three limiting cases, connecting them to physical scenarios like escape velocity.

Key Concepts:

• Special cases in orbital mechanics

• Escape velocity

Step-by-Step Guidance

1. For each value of $\alpha$, substitute into your formulas for $r_p$ and $v_p$ and analyze the physical meaning.

2. Discuss what happens to the orbit in each case (circular, highly elliptical, or parabolic/escape).

3. Explain why the results make sense physically, especially in the context of escape speed.

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Q3f. Calculate the semimajor axis of the elliptical orbit, $a = (r_a + r_p)/2$. Does your formula make sense for the three special cases in part e)?

Background

Topic: Ellipse Geometry in Orbital Mechanics

This question connects the geometric properties of ellipses to the physical parameters of the orbit.

Key Formula:

• Semimajor axis: $a = \frac{r_a + r_p}{2}$

Step-by-Step Guidance

1. Substitute your expressions for $r_a$ and $r_p$ in terms of $\alpha$ and $r_0$ into the formula for $a$.

2. Check the result for each special case from part e) to see if it matches your physical expectations.

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Q3g. Using the definition of eccentricity, show that $e = |1 - \alpha^2|$. Does this formula make sense for the three special cases in part e)?

Background

Topic: Eccentricity of Elliptical Orbits

This question asks you to relate the eccentricity of the orbit to the parameter $\alpha$ and interpret the result in special cases.

Key Formula:

• Eccentricity: $r_a = (1 + e)a$, $r_p = (1 - e)a$

Step-by-Step Guidance

1. Express $r_a$ and $r_p$ in terms of $a$ and $e$.

2. Use your previous results for $r_a$, $r_p$, and $a$ to solve for $e$ in terms of $\alpha$.

3. Check if the formula $e = |1 - \alpha^2|$ makes sense for the special cases in part e).

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