Here is a HARD Physics Question from the Oxford PAT exam

ZPhysics
26 Jun 202208:01
EducationalLearning
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TLDRIn this educational video, the presenter tackles a complex physics problem from the 2018 Oxford Physics Admission Test involving masses connected by strings and deflected by gravity. The focus is on proving the approximate formula for small deflections using the small angle approximation. The solution involves resolving tensions, applying gravitational force equations, and simplifying the resulting quadratic equation to demonstrate the relationship between mass, gravitational constant, and the deflection angle. The presenter also highlights the importance of dimensional analysis and the physical intuition behind the solution.

Takeaways
  • 📚 The problem is from the 2018 Oxford Physics Admission Test and involves masses connected by strings and deflected by gravitational force.
  • 🔍 The goal is to prove an approximate expression for small deflection delta x using the small angle approximation.
  • 📐 The small angle approximation is introduced, where sine theta is approximately equal to tangent theta, which is approximately equal to theta itself.
  • ⚖️ Tension in the strings is resolved into vertical and horizontal components, with the vertical balancing the weight and the horizontal causing the deflection.
  • 📉 The equations are rearranged and combined to relate the tension, gravitational force, and the deflection angle.
  • 🔢 The small angle approximation simplifies the equation, allowing for the substitution of theta with delta x/l.
  • 🔄 The equation is manipulated to form a quadratic in terms of delta x, with the higher power terms of delta x considered negligible.
  • 🧩 The quadratic formula is applied to find the roots of the equation, leading to an expression for delta x.
  • 🔑 The solution involves a simplification using the square root of 64 x squared to further simplify the expression for delta x.
  • 📉 The final expression for delta x is derived, which is the main result the script aims to prove.
  • 💡 The script suggests that if the mass is zero, delta x would also be zero, which is a physically sensible result.
  • 🔍 The script mentions dimensional analysis as an important technique in physics competitions, hinting at the importance of understanding units in problem-solving.
Q & A

  • What is the main problem being discussed in the script?

    -The main problem is to prove a specific expression for the small deflection 'delta x' of masses connected by strings under the influence of gravitational force.

  • What is the small angle approximation used in the script?

    -The small angle approximation used is that sine of theta is approximately equal to the tangent of theta, which is also approximately equal to theta itself, especially when dealing with small quantities like 'delta x'.

  • How does the script resolve the tension in the strings?

    -The script resolves the tension by considering the vertical component balancing the weight (t cos theta = mg) and the horizontal component causing the deflection (t sine theta = gmm/r^2).

  • What is the significance of the term 'delta x to the power of 3' in the script?

    -The term 'delta x to the power of 3' is considered negligible in the approximation because it represents a very small quantity when 'delta x' is already small.

  • How does the script transform the equation involving 'sine theta over cos theta'?

    -The script transforms 'sine theta over cos theta' into 'tan theta' using the small angle approximation, and then further approximates it as 'delta x/l'.

  • What is the quadratic form of the equation derived in the script?

    -The quadratic form derived is -4x * delta x squared + delta x * x squared - g * m * l / g, which is simplified to a standard quadratic equation form.

  • How is the quadratic formula applied in the script?

    -The quadratic formula is applied to find the roots of the derived quadratic equation, with the first root being calculated as -b +/- sqrt(b^2 - 4ac).

  • What is the physical interpretation of the mass being zero in the script?

    -If the mass is zero, the deflection 'delta x' would also be zero, which makes physical sense as it's the presence of mass that causes the deflection.

  • Why is dimensional analysis important in physics problems like the one in the script?

    -Dimensional analysis is important to ensure that the units on both sides of an equation are consistent, which is crucial for the validity of the solution.

  • What is the final expression the script aims to prove?

    -The final expression to be proven is delta x ≈ x / (8 * sqrt(g * m * l / g)) simplified to delta x ≈ 1/(4x) * sqrt(g * m * l).

  • How does the script ensure the approximation is valid?

    -The script ensures the approximation is valid by considering the negligible terms and checking the physical interpretation, such as the deflection being zero when the mass is zero.

Outlines
00:00
🔍 Solving a Physics Problem with Small Angle Approximation

This paragraph discusses a complex physics problem from the 2018 Oxford Physics admission test involving masses connected by strings of length 'l', deflected by gravitational force. The goal is to prove an expression for small deflection 'delta x'. The speaker uses the small angle approximation, simplifying trigonometric functions to linearize the problem. They resolve the tension forces and equate vertical and horizontal components to the weight and gravitational force, respectively. By substituting and rearranging equations, the problem is transformed into a quadratic form, which is then solved using the quadratic formula, neglecting higher order terms due to their insignificance in the approximation.

05:04
📚 Applying Quadratic Formula and Dimensional Analysis

The second paragraph continues the solution of the physics problem, focusing on the application of the quadratic formula to find the roots of the derived equation. The speaker simplifies the equation further by using algebraic manipulations and introduces a trick involving the square root of a term to simplify the expression. They then square the entire expression to eliminate the square root and arrive at the final simplified form of the equation. The speaker also validates the solution by considering a hypothetical scenario where the mass is zero, which logically results in no deflection. Lastly, the paragraph touches on the importance of dimensional analysis in physics competitions and refers to a related video for further study.

Mindmap
Keywords
💡Small Angle Approximation
The small angle approximation is a mathematical technique where sin(θ), tan(θ), and θ are approximated as nearly equal when θ is small, usually in radians. In the video, this concept is used to simplify the trigonometric expressions involved in solving the physics problem, specifically for the angle θ formed by the deflected masses. This approximation allows the equation to be more manageable, particularly when dealing with small deflections (Δx).
💡Tension
Tension refers to the force exerted by a string or rope when it is pulled tight by forces acting from opposite ends. In the video, tension is a key factor in balancing the vertical force due to gravity (mg) and in contributing to the horizontal deflection (Δx) caused by gravitational attraction between the masses. Understanding tension is essential for analyzing the forces at play in the system described.
💡Deflection (Δx)
Deflection (Δx) represents the small displacement or deviation of the masses from their equilibrium position due to external forces, in this case, gravitational force. The video seeks to prove that Δx can be approximated by a specific expression using the small angle approximation. The deflection is central to the problem as it is the primary variable being analyzed.
💡Gravitational Force
Gravitational force is the attractive force that acts between any two masses. In this context, it causes the deflection of the masses connected by strings. The video uses Newton's law of gravitation to express the force between the two masses as GMm/R², which is crucial for deriving the expression for Δx. Gravitational force is what initiates the deflection and is balanced by the tension in the string.
💡Quadratic Equation
A quadratic equation is a second-degree polynomial equation in one variable, typically in the form ax² + bx + c = 0. In the video, the problem is reduced to solving a quadratic equation after applying the small angle approximation and simplifying the terms involving Δx. The quadratic equation is crucial for finding the roots, which represent possible values for the deflection Δx.
💡Dimensional Analysis
Dimensional analysis is a technique used in physics to check the consistency of equations by analyzing the dimensions (units) of the physical quantities involved. The video briefly mentions this as a helpful tool in physics problems, particularly in exams. Although not deeply explored in the provided script, dimensional analysis would ensure that the derived equation for Δx has consistent units on both sides.
💡Newton's Law of Universal Gravitation
Newton's law of universal gravitation states that every point mass attracts every other point mass with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them (F = GMm/R²). This law is fundamental to the problem in the video as it describes the gravitational force causing the deflection Δx, and it is used to set up the equations that are solved using the small angle approximation.
💡Tangent (tanθ)
Tangent of an angle (tanθ) in a right triangle is the ratio of the opposite side to the adjacent side. In the video, tanθ is used to describe the horizontal deflection caused by gravitational forces. By using the small angle approximation, tanθ is approximated to θ, simplifying the mathematical expressions and making the problem easier to solve.
💡Approximation
Approximation refers to the process of finding a value or expression that is close enough to the actual value for a particular purpose. The video relies on approximation techniques, such as the small angle approximation, to simplify complex trigonometric and algebraic expressions into more manageable forms. Approximation is key to solving the problem efficiently without losing significant accuracy.
💡Gravitational Constant (G)
The gravitational constant (G) is a fundamental physical constant involved in the calculation of gravitational forces between two masses. It appears in Newton's law of universal gravitation and is crucial for determining the magnitude of the force causing the deflection Δx. In the video, G is used in the equation to relate the gravitational force to the deflection of the masses.
Highlights

Introduction to the small deflection problem from the 2018 Oxford Physics Admission Test.

Explanation of the small angle approximation: sine of theta is approximately equal to tangent of theta and to theta itself.

Resolution of the tension forces: the vertical component balances the weight, while the horizontal component causes deflection.

Substitution of tension into the equation using the small angle approximation.

Simplification of the equation by canceling out the masses and applying small angle approximation to delta x.

Formulation of a quadratic equation by ignoring higher-order terms of delta x.

Use of the quadratic formula to find the roots, leading to an expression for delta x.

Simplification of the expression by representing terms in a more manageable form.

Explanation of the physical meaning of the solution, showing that delta x equals zero when mass is zero.

Dimensional analysis as a powerful technique in solving physics problems and ensuring correct results.

Reinforcement of the importance of using small angle approximations in physics to simplify complex problems.

Demonstration of the quadratic solution process, which is a common technique in physics problem-solving.

Importance of identifying and eliminating negligible terms in approximations to simplify calculations.

Highlighting the physical intuition behind why certain roots of the equation make physical sense.

Encouragement to use dimensional analysis and approximation techniques in physics exams and competitions.

Transcripts

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