# Oxford University Engineering Interview

TLDRThe interview transcript features a session with Ali, an interviewer, and Eddie, the interviewee, focusing on engineering science topics. They delve into mathematical problems, including estimating powers and integrals, followed by a physics problem concerning the center of mass of a hemisphere. The discussion shifts to thermodynamics, where they analyze a system's pressure changes due to heat extraction and mechanical work. The conversation explores energy conservation, equilibrium states, and the relationship between work done and heat extracted in a system. The interview concludes with a focus on the stability of equilibrium positions, providing a comprehensive overview of various scientific concepts.

###### Takeaways

- ๐ Introduction and greetings between Ali, Adam, and Eddie.
- ๐ The interview will cover maths, mechanics, and thermodynamics questions.
- ๐งฎ Estimation of 7 to the power of 6 using approximation methods.
- ๐ Calculation of the integral of 1 over the square root of 1 minus x squared using trigonometric substitution.
- โ๏ธ Determination of the center of mass of a hemisphere of radius r with constant density.
- ๐ Analysis of how varying density affects the center of mass calculation.
- ๐ก๏ธ Thermodynamics problem involving initial and final states of a system, considering pressure changes and equilibrium.
- ๐ง Detailed discussion on work done by the piston and its relationship with heat extraction and internal energy reduction.
- ๐ Equilibrium analysis using energy considerations and differentiation to find stable and unstable positions.
- ๐ Conclusion of the interview with a brief wrap-up and goodbyes.

###### Q & A

### What is the initial topic of the interview?

-The initial topic of the interview is to start with some maths questions and then build up to cover mechanics and thermodynamics.

### How does Eddie attempt to estimate 7 to the power of 6?

-Eddie attempts to estimate 7 to the power of 6 by writing 7 as 6 plus 1, then expanding this expression and focusing on the largest terms, which are 6 to the power of 6 and 6 times 7.

### What substitution does Eddie use to simplify the integral of 1/sqrt(1-x^2) dx?

-Eddie uses the substitution x = sin(theta) to simplify the integral, which changes the denominator to cos^2(theta).

### How does Eddie approach the problem of finding the center of mass of a hemisphere?

-Eddie approaches the problem by considering the symmetry of the hemisphere and focusing on the z coordinate. He then splits the hemisphere into small disks and integrates over these disks to find the center of mass.

### What is the general formula for the center of mass according to Eddie's explanation?

-The general formula for the center of mass, as explained by Eddie, is the sum over small component masses times their distances equals the total mass times the center of mass position.

### How does the density variation affect the calculation of the center of mass in the hemisphere?

-If the density varies linearly, the density can be written as a function of z, and this varying density must be integrated over the volume elements when calculating the center of mass.

### What is the initial state of the thermodynamic system described in the interview?

-The initial state of the thermodynamic system is at pressure p1 and volume v1 with a constant amount of substance m.

### How does Eddie determine the relationship between p1, p2, and the position of the mass m?

-Eddie determines the relationship by considering the torque generated by the mass m at an angle theta and balancing the torques to relate p1 and p2 to the position of the mass.

### What is the assumption made regarding the work done by the piston and the heat extracted from the system?

-The assumption made is that the work done by the piston equals the heat extracted from the system, given that there is no friction in the piston or the pulley and energy is conserved.

### How does Eddie define equilibrium in terms of energy during the interview?

-Eddie defines equilibrium in terms of energy as the condition where the first derivative of energy with respect to the system's parameters (e.g., theta) is zero, and the second derivative is positive for stable equilibrium or negative for unstable equilibrium.

###### Outlines

##### ๐ค Introduction and Estimation of Seven to the Power of Six

The interviewer, Ali, greets the interviewee, Adam, and introduces the interview format, starting with a math question on estimating seven to the power of six. Adam struggles initially but then breaks down the problem by approximating seven squared and cubed to arrive at an estimated answer of 125,000.

##### ๐ Solving an Integral Using Trigonometric Substitution

Adam is asked to solve an integral involving 1 over the square root of 1 minus x squared. He uses a trigonometric substitution, setting x equal to sine theta, which simplifies the integral to be in terms of theta. After changing variables, Adam successfully solves the integral and verifies his result.

##### ๐งฎ Calculating the Center of Mass of a Hemisphere

Adam is tasked with finding the center of mass of a hemisphere with constant density. He splits the hemisphere into thin disks, uses Pythagorean theorem to relate the radius of each disk to the height, and sets up the integral for the mass distribution. He concludes that the center of mass lies on the z-axis.

##### ๐ Adjusting for Linearly Varying Density

The interviewer modifies the problem to account for a linearly varying density from 2 to 1. Adam adjusts his approach by writing the density as a function of z and incorporating it into the integral, leading to a more complex but accurate calculation of the center of mass.

##### ๐ Thermodynamics and Mechanics Problem

Adam tackles a thermodynamics problem involving a piston, pulley, and mass system. He deduces that extracting heat reduces pressure (p2 < p1), relates pressures to mass position, and uses torque balance to find equilibrium conditions. He discusses work done by the piston and heat extraction, noting that some heat reduces the internal energy of the gas.

##### โ๏ธ Energy Balance and Equilibrium

The discussion shifts to writing the work done in mechanical terms. Adam identifies gravitational potential energy changes and sets up an energy balance equation. He differentiates to find equilibrium points and determines the stability of the system, concluding that theta = 0 is a stable equilibrium and theta = pi is unstable.

##### ๐ Conclusion of the Interview

The interview concludes with Ali thanking Adam for his participation. Adam expresses gratitude, and they bid each other goodbye, marking the end of the engineering science interview.

###### Mindmap

###### Keywords

##### ๐กEngineering Science

##### ๐กMathematical Estimation

##### ๐กBinomial Expansion

##### ๐กIntegration

##### ๐กSubstitution

##### ๐กCenter of Mass

##### ๐กDensity

##### ๐กThermodynamics

##### ๐กEquilibrium

##### ๐กWork Done

##### ๐กPotential Energy

###### Highlights

Interview begins with an introduction and an outline of topics including maths, mechanics, and thermodynamics.

Estimation of seven to the power of six using binomial expansion and approximation methods.

Discussion on the simplification of mathematical expressions and the use of different representations.

Application of trigonometric identities in solving integrals, specifically for a function involving square root and cosine.

Use of substitution method in calculus to simplify complex integrals.

Explanation of the center of mass calculation for a hemisphere with constant density.

Introduction of the concept of breaking down a complex shape into smaller, manageable components for calculation.

Integration as a method to find the center of mass in the z-direction due to symmetry.

Consideration of varying density in a hemisphere and its impact on the center of mass calculation.

Thermodynamics problem involving a system with changing pressure and volume due to heat extraction.

Analysis of mechanical equilibrium in a system with a piston connected to a pulley and a mass.

Calculation of work done by the piston and its relation to heat extraction in a thermodynamic process.

Discussion on energy conservation and the role of friction in the system.

Differentiation between stable and unstable equilibrium in terms of energy and potential energy.

Use of calculus to find equilibrium positions by setting derivatives equal to zero.

End of the interview with a summary of the discussed topics and a closing exchange.

###### Transcripts

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