PLANCKS是国际物理学生协会为全球本科生和硕士生举办的物理竞赛。比赛为期三天，参赛队伍需解答高难度的物理问题。本文选取了PLANCKS英国和爱尔兰预赛及2024年都柏林决赛中的部分难题，包括四维宇宙中的太阳颜色、引力常数的计算、摩天大楼顶层摆钟的计时误差以及量子棒的倾倒时间。这些问题涵盖了经典物理学和量子力学等多个领域，旨在挑战参赛者的物理知识和解决问题的能力。如果你对物理充满热情，不妨尝试解答这些问题，检验自己的实力。

☀️ 假设存在一个四维宇宙，太阳的表面温度与我们宇宙相同，那么太阳的颜色会是什么？这需要我们运用物理学知识，对四维空间中的光线传播和能量分布进行推导和计算。

⚖️ 在一个平行的宇宙中，两个质量均为1千克的物体，在相距1米的情况下，仅受彼此的引力作用，经过26小时42分钟相遇。我们需要计算出这个宇宙中的引力常数G，这考验我们对万有引力定律的理解和应用。

⏱️ 一个在地球表面运行精确的摆钟，如果被移到摩天大楼的百层楼顶，每天会慢多少？这需要我们理解重力加速度随高度的变化，并计算其对摆钟周期的影响。

⚛️ 一根质量为1千克，长度为1米的无限细棒，在量子力学中不可能完全平衡。即使在理想的暗真空和冷却条件下，不确定性原理也会导致其倾倒。我们需要计算出该棒在不确定性原理下倾倒的最大时间，这需要我们理解量子力学的不确定性原理，并将其应用于实际问题中。

Each year, the International Association of Physics Students organizes a physics competition for bachelor’s and master’s students from across the world. Known as the Physics League Across Numerous Countries for Kick-ass Students (PLANCKS), it’s a three-day event where teams of three to four students compete to answer challenging physics questions.

In the UK and Ireland, teams compete in a preliminary competition to be sent to the final. Here are some fiendish questions from past PLANCKS UK and Ireland preliminaries and the 2024 final in Dublin, written by Anthony Quinlan and Sam Carr, for you to try this holiday season.

Question 1: 4D Sun

Imagine you have been transported to another universe with four spatial dimensions. What would the colour of the Sun be in this four-dimensional universe? You may assume that the surface temperature of the Sun is the same as in our universe and is approximately T = 6 × 10³ K. [10 marks]

Boltzmann constant, k_B = 1.38 × 10⁻²³ J K⁻¹

Speed of light, c = 3 × 10⁸ m s⁻¹

Question 2: Heavy stuff

In a parallel universe, two point masses, each of 1 kg, start at rest a distance of 1 m apart. The only force on them is their mutual gravitational attraction, F = –Gm₁m₂/r². If it takes 26 hours and 42 minutes for the two masses to meet in the middle, calculate the value of the gravitational constant G in this universe. [10 marks]

Question 3: Just like clockwork

Consider a pendulum clock that is accurate on the Earth’s surface. Figure 1 shows a simplified view of this mechanism.

A pendulum clock runs on the gravitational potential energy from a hanging mass (1). The other components of the clock mechanism regulate the speed at which the mass falls so that it releases its gravitational potential energy over the course of a day. This is achieved using a swinging pendulum of length l (2), whose period is given by

$T=2\pi \sqrt{\frac{l}{g}}$

where g is the acceleration due to gravity.

Each time the pendulum swings, it rocks a mechanism called an “escapement” (3). When the escapement moves, the gear attached to the mass (4) is released. The mass falls freely until the pendulum swings back and the escapement catches the gear again. The motion of the falling mass transfers energy to the escapement, which gives a “kick” to the pendulum that keeps it moving throughout the day.

Radius of the Earth, R = 6.3781 × 10⁶ m

Period of one Earth day, τ₀ = 8.64 × 10⁴ s

How slow will the clock be over the course of a day if it is lifted to the hundredth floor of a skyscraper? Assume the height of each storey is 3 m. [4 marks]

Question 4: Quantum stick

Imagine an infinitely thin stick of length 1 m and mass 1 kg that is balanced on its end. Classically this is an unstable equilibrium, although the stick will stay there forever if it is perfectly balanced. However, in quantum mechanics there is no such thing as perfectly balanced due to the uncertainty principle – you cannot have the stick perfectly upright and not moving at the same time. One could argue that the quantum mechanical effects of the uncertainty principle on the system are overpowered by others, such as air molecules and photons hitting it or the thermal excitation of the stick. Therefore, to investigate we would need ideal conditions such as a dark vacuum, and cooling to a few milli­kelvins, so the stick is in its ground state.

Moment of inertia for a rod,

$I=\frac{1}{3}m{l}^2$

where m is the mass and l is the length.

Uncertainty principle,

$\Delta x\Delta p\ge \frac{\hslash }{2}$

There are several possible approximations and simplifications you could make in solving this problem, including:

sinθ ≈ θ for small θ

${\cosh }^{-1}x=ln \left(x+\sqrt{x^2-1}\right)$

and

${\sinh }^{-1}x=ln \left(x+\sqrt{x^2+1}\right)$

Calculate the maximum time it would take such a stick to fall over and hit the ground if it is placed in a state compatible with the uncertainty principle. Assume that you are on the Earth’s surface. [10 marks]

Hint: Consider the two possible initial conditions that arise from the uncertainty principle.

• Answers will be posted here on the Physics World website next month. There are no prizes.
• If you’re a student who wants to sign up for the 2025 edition of PLANCKS UK and Ireland, entries are now open at plancks.uk

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