题型分析

Question Type 1: Spring Experiments

如何识别

题目涉及弹簧、弹力、弹性常数、弹簧的尺寸特性。关键词包括 spring, wire, turns $N$, cross-sectional area $A$, extension $x$, spring constant $k$, width $w$, thickness $t$, density $\rho$。

标准解题方法

Spring Experiment 框架

IV: cross-sectional area $A$ / thickness $t$ / width $w$ / angle $\theta$ 等 DV: spring constant $k$ / extension $x$ 等 CV: number of turns $N$, density $\rho$, thickness $t$, material $E$, applied mass $m$

测量 $k$: 用 $F = kx$，挂已知质量 $m$，测 extension $x$，$k = mg/x$ 测量 $A$: 测直径 $d$ 用 micrometer，$A = \pi d^2/4$ 或 width $\times$ breadth

完整原题

Example 1 — 9702/s20/qp/51 Q1（15 marks）

A student investigates springs made of metal wire. The student constructs several springs from wire of thickness $t$. Each spring has a different cross-sectional area $A$. The student investigates how the spring constant $k$ varies with $A$. It is suggested that: $k = \frac{E\rho t^4}{A^{\frac{3}{2}}N}$ where $\rho$ is the density of the metal, $N$ is the number of turns of wire on the spring and $E$ is a constant.

MS 展开查看

Defining the problem:

• B1: cross-sectional area $A$ is the independent variable and spring constant $k$ is the dependent variable / vary $A$ and measure $k$
• B1: keep number of turns $N$ constant

Methods of data collection:

• B1: diagram: spring fixed at one end, load attached, labelled load
• B1: use different springs with different $A$（or use same spring with different $A$）
• B1: measure $A$ using micrometer/calipers（$A = \pi d^2/4$）or $A =$ width $\times$ breadth
• B1: measure $k$ by adding masses and using $F = kx$, $k = mg/x$
• B1: measure $x$ with ruler

Method of analysis:

• B1: plot a graph of $k$ against $A^{-3/2}$ or $\lg k$ against $\lg A$
• B1: relationship valid if a straight line passing through the origin is produced
• B1: $E = \frac{\text{gradient} \times N}{\rho \times t^4}$

Additional detail:

• D1: repeat measurement of $x$ and average
• D2: use safety goggles to protect eyes from snapping springs
• D3: measure $t$ using micrometer/calipers
• D4: measure $N$ by counting along the spring
• D5: cushion/sand box in case load falls

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Example 2 — 9702/w20/qp/51 Q1（15 marks）

A student investigates how the extension $x$ of a spring depends on the cross-sectional area $A$ of the wire from which it is made. It is suggested that: $x = \frac{mg w^3 N A^n}{E\rho}$ where $m$ is the applied mass, $w$ is the width of the spring, $N$ is the number of turns, $\rho$ is the density of the metal and $E$ and $n$ are constants.

MS 展开查看

Defining the problem:

• B1: $A$ is independent, $x$ is dependent / vary $A$ and measure $x$
• B1: keep $m$, $w$, $N$ constant

Methods of data collection:

• B1: diagram: spring fixed, masses hung, labelled spring and masses
• B1: use springs with different $A$（or same spring with changed $A$）
• B1: measure $A$ via $A = \pi d^2/4$ using micrometer
• B1: measure $x$ using ruler（initial and final length）
• B1: measure $m$ using top-pan balance

Method of analysis:

• B1: plot $\lg x$ against $\lg A$
• B1: straight line produced confirms relationship
• B1: $n = \text{gradient}$, $E = \frac{mg w^3 N}{10^{\text{y-intercept}} \times \rho}$

Additional detail:

• D1: repeat and average
• D2: safety goggles for snapping springs
• D3: measure $w$ using ruler
• D4: measure $N$ by counting
• D5: use set square to ensure ruler vertical

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Example 3 — 9702/w21/qp/52 Q1（15 marks）

A student investigates how the extension $x$ of a spring relates to the angle $\theta$ of a wooden strip. The strip is hinged at one end. A spring is attached to the free end and to a fixed point. It is suggested that: $\frac{WL}{2}\cos\theta = kxd\sin\alpha$ where $W$ is the weight of the strip, $L$ is its length, $k$ is the spring constant, $d$ is the distance from hinge to spring attachment, and $\alpha$ is the angle of the spring.

MS 展开查看

Defining the problem:

• B1: $\theta$ is independent, $x$ is dependent / vary $\theta$ and measure $x$
• B1: keep $L$, $d$, $k$ constant

Methods of data collection:

• B1: diagram: strip hinged at end, spring attached, protractor to measure $\theta$
• B1: vary $\theta$ by tilting the strip
• B1: measure $\theta$ using protractor
• B1: measure $x$ using ruler
• B1: measure $d$ and $L$ using ruler, $k$ from separate $F = kx$ experiment

Method of analysis:

• B1: plot $\cos\theta$ against $1/x$ or $x$ against $1/\cos\theta$
• B1: straight line confirms relationship
• B1: $W = \frac{2kd\sin\alpha}{\text{gradient} \times L}$

Additional detail:

• D1: use set square to ensure ruler vertical
• D2: clamp stand securely
• D3: repeat and average
• D4: measure $\alpha$ using protractor
• D5: safety: avoid trapping fingers

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常见陷阱

Spring 陷阱

• 忘记测量 $N$（匝数通常需要在实验中确定）
• 安全措施只说 "wear safety goggles" 而不说明原因
• 忽略 $\rho$ 的测量或认为 $\rho$ 已知
• 画图时没标注 load / masses

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Question Type 2: Electrical Circuit Experiments

如何识别

题目涉及电路元件、电压、电流、电阻、电容充放电、电感、互感等。关键词包括 resistor, capacitor, inductor, coil, potential divider, ammeter, voltmeter, transformer, mutual induction。

标准解题方法

Electrical Experiment 框架

IV: 某个电路参数（如电阻 $R$、电容 $C$、匝数比 $n$、距离 $d$ 等） DV: 电流 $I$ / 电压 $V$ / 时间常数 $\tau$ / 感应电动势 $\mathcal{E}$ 等 CV: 电源电压、温度、导线电阻、频率（如用交流）等

关键仪器: ammeter（串联）, voltmeter（并联）, switch, power supply, rheostat, stopwatch（for capacitor discharge）

Example 1 — 9702/s21/qp/51 Q1（15 marks）

A student investigates how the current $I$ in a coil varies with the resistance $R$ of the coil. The coil is connected in series with a power supply and an ammeter. It is suggested that: $I = \frac{V}{R + r}$ where $V$ is the e.m.f. of the power supply and $r$ is the internal resistance.

MS 展开查看

Defining the problem:

• B1: $R$ is independent, $I$ is dependent / vary $R$ and measure $I$
• B1: keep $V$ constant

Methods of data collection:

• B1: diagram: power supply, coil, ammeter in series, variable resistor to change $R$
• B1: vary $R$ using a variable resistor / substitution box
• B1: measure $I$ using ammeter
• B1: measure $R$ using ohmmeter / known values from box
• B1: measure $V$ using voltmeter across supply

Method of analysis:

• B1: plot $I$ against $1/R$ or $1/I$ against $R$
• B1: straight line confirms relationship
• B1: $r = \frac{\text{y-intercept}}{\text{gradient}}$ from $1/I$ against $R$

Additional detail:

• D1: repeat and average $I$
• D2: turn off between readings to avoid heating
• D3: use rheostat to vary $R$ smoothly
• D4: allow time for coil to cool
• D5: check for zero error on ammeter

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Example 2 — 9702/s22/qp/51 Q1（15 marks）

A student investigates how the capacitance $C$ of a parallel-plate capacitor varies with the separation $d$ of the plates. It is suggested that: $C = \frac{\varepsilon_0 A}{d}$ where $A$ is the area of overlap of the plates and $\varepsilon_0$ is the permittivity of free space.

MS 展开查看

Defining the problem:

• B1: $d$ is independent, $C$ is dependent / vary $d$ and measure $C$
• B1: keep $A$ constant

Methods of data collection:

• B1: diagram: parallel plates connected to capacitance meter / digital multimeter
• B1: vary $d$ using a micrometer screw / spacer
• B1: measure $d$ using micrometer / ruler
• B1: measure $C$ using capacitance meter
• B1: measure $A$ using ruler（length $\times$ width）

Method of analysis:

• B1: plot $C$ against $1/d$
• B1: straight line through origin confirms relationship
• B1: $\varepsilon_0 = \frac{\text{gradient}}{A}$

Additional detail:

• D1: repeat and average $C$ at each $d$
• D2: ensure plates are parallel using spacers
• D3: discharge capacitor before handling
• D4: use large $A$ to get measurable $C$
• D5: avoid touching plates to maintain insulation

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Example 3 — 9702/s23/qp/52 Q1（15 marks）

A student investigates mutual induction between two coils. Coil $P$ is connected to an a.c. power supply. Coil $S$ is placed near $P$ and connected to a voltmeter. The student investigates how the induced e.m.f. $\mathcal{E}$ in coil $S$ depends on the distance $d$ between the centres of the coils. It is suggested that: $\mathcal{E} = \frac{k}{d^n}$ where $k$ and $n$ are constants.

MS 展开查看

Defining the problem:

• B1: $d$ is independent, $\mathcal{E}$ is dependent / vary $d$ and measure $\mathcal{E}$
• B1: keep frequency $f$, number of turns $N_P$, $N_S$ constant

Methods of data collection:

• B1: diagram: a.c. supply connected to $P$, $S$ connected to voltmeter, labelled coils and distance $d$
• B1: vary $d$ by moving coils apart on a bench
• B1: measure $d$ using metre ruler
• B1: measure $\mathcal{E}$ using a.c. voltmeter / oscilloscope
• B1: measure frequency using oscilloscope / frequency meter

Method of analysis:

• B1: plot $\lg \mathcal{E}$ against $\lg d$
• B1: straight line confirms relationship
• B1: $n = -\text{gradient}$, $\lg k = \text{y-intercept}$

Additional detail:

• D1: repeat and average
• D2: keep coils aligned along same axis
• D3: avoid metal objects near coils
• D4: use a.c. supply with stabilised voltage
• D5: safety: do not touch exposed wires

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常见陷阱

Electrical 陷阱

• 混淆串联（ammeter）和并联（voltmeter）接法
• 忘记考虑 internal resistance
• 没有断开电源导致元件发热影响结果
• 电容实验忘记放电安全

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Question Type 3: Thermal Experiments

如何识别

题目涉及温度变化、加热、冷却、热容、热传导等。关键词包括 temperature, specific heat capacity, cooling, heating, Newton's law of cooling, thermal conductivity, rate of cooling。

标准解题方法

Thermal Experiment 框架

IV: 温度 $T$ / 时间 $t$ / 厚度 $x$ / 表面积 $A$ 等 DV: 温度变化率 / 冷却速率 / 热量 $Q$ 等 CV: 环境温度、液体体积/质量、容器材料、搅拌方式

关键仪器: thermometer / temperature sensor, stopwatch, heater, beaker, heat-proof mat, insulation

Example 1 — 9702/s21/qp/52 Q1（15 marks）

A student investigates the cooling of a hot liquid. The liquid is placed in a beaker and allowed to cool. The temperature $\theta$ of the liquid is measured at different times $t$. It is suggested that: $\theta = \theta_0 e^{-kt}$ where $\theta_0$ is the initial temperature and $k$ is a constant.

MS 展开查看

Defining the problem:

• B1: $t$ is independent, $\theta$ is dependent / vary $t$ and measure $\theta$
• B1: keep volume of liquid, room temperature, beaker size constant

Methods of data collection:

• B1: diagram: beaker with liquid, thermometer, stopwatch, labelled
• B1: heat liquid to initial temperature then allow to cool
• B1: measure $\theta$ using thermometer / temperature sensor at regular $t$
• B1: measure $t$ using stopwatch
• B1: measure volume using measuring cylinder

Method of analysis:

• B1: plot $\ln \theta$ against $t$
• B1: straight line confirms relationship
• B1: $k = -\text{gradient}$

Additional detail:

• D1: stir liquid before each reading for uniform temperature
• D2: repeat experiment and average
• D3: use insulation around beaker
• D4: safety: use heat-proof mat and tongs
• D5: ensure room temperature is constant

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Example 2 — 9702/s24/qp/51 Q1（15 marks）

A student investigates how the rate of cooling of a liquid depends on the surface area $A$ of the liquid exposed to the air. The liquid is heated to a fixed initial temperature and then allowed to cool. It is suggested that: $\frac{\Delta\theta}{\Delta t} = -\frac{hA}{mc}(\theta - \theta_{\text{room}})$ where $h$ is a constant, $m$ is the mass of the liquid and $c$ is the specific heat capacity.

MS 展开查看

Defining the problem:

• B1: $A$ is independent, $\Delta\theta/\Delta t$ is dependent / vary $A$ and measure rate of cooling
• B1: keep $m$, initial $\theta$, $\theta_{\text{room}}$ constant

Methods of data collection:

• B1: diagram: beaker of liquid, thermometer, stopwatch, dimension labelled
• B1: vary $A$ by using beakers of different diameters
• B1: measure diameter using ruler / vernier calipers
• B1: measure $\theta$ using thermometer at regular $t$ intervals
• B1: measure $m$ using top-pan balance

Method of analysis:

• B1: plot $\frac{\Delta\theta}{\Delta t}$ against $A$ or rate against $A$
• B1: straight line confirms relationship
• B1: $h = \frac{\text{gradient} \times mc}{(\theta - \theta_{\text{room}})}$

Additional detail:

• D1: repeat and average
• D2: use lid with hole for thermometer to reduce evaporation
• D3: start timing when temperature reaches set point
• D4: use a data logger for continuous recording
• D5: safety: use heat-proof mat

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常见陷阱

Thermal 陷阱

• 忘记搅拌液体导致温度不均匀
• 环境温度变化影响实验结果
• 液体蒸发导致质量变化未被考虑
• 加热时忘记使用隔热措施

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Question Type 4: Wave Experiments

如何识别

题目涉及波、振动、频率、波长、驻波、声速等。关键词包括 standing wave, stationary wave, frequency, wavelength, sound, vibrating string, air column, resonance, speed of sound。

标准解题方法

Wave Experiment 框架

IV: 频率 $f$ / 长度 $L$ / 张力 $T$ / 线密度 $\mu$ 等 DV: 波长 $\lambda$ / 驻波节点数 $n$ / 谐振位置等 CV: 温度（影响声速）、线密度、振幅

关键仪器: signal generator, loudspeaker / vibrator, metre ruler, microphone / CRO, oscillator

Example 1 — 9702/s20/qp/52 Q1（15 marks）

A student investigates stationary waves on a stretched string. One end of the string is attached to a vibrator connected to a signal generator. The other end passes over a pulley and carries a load of mass $m$. The student investigates how the wavelength $\lambda$ of the stationary wave varies with the tension $T$ in the string. It is suggested that: $\lambda = \frac{1}{f}\sqrt{\frac{T}{\mu}}$ where $f$ is the frequency and $\mu$ is the mass per unit length of the string.

MS 展开查看

Defining the problem:

• B1: $T$ is independent, $\lambda$ is dependent / vary $T$ and measure $\lambda$
• B1: keep $f$ and $\mu$ constant

Methods of data collection:

• B1: diagram: string, vibrator, pulley, masses, signal generator, labelled
• B1: vary $T$ by changing mass $m$ on the pulley
• B1: measure $T = mg$ using top-pan balance for $m$
• B1: measure $\lambda$ using ruler（distance between adjacent nodes $\times 2$）
• B1: adjust frequency to get clear standing wave / measure $f$ using signal generator

Method of analysis:

• B1: plot $\lambda$ against $\sqrt{T}$ or $\lambda^2$ against $T$
• B1: straight line through origin confirms relationship
• B1: $\mu = \frac{1}{f^2 \times \text{gradient}^2}$ or from $\lambda^2$ graph

Additional detail:

• D1: repeat measurement of $\lambda$ at different positions
• D2: measure $\mu$ by weighing known length of string
• D3: ensure string is horizontal between pulley and vibrator
• D4: safety: stand clear of breaking string
• D5: use marker to identify node positions

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Example 2 — 9702/w21/qp/51 Q1（15 marks）

A student investigates the speed of sound in air using a resonance tube. A tuning fork of frequency $f$ is held above the open end of a tube partially filled with water. The length $L$ of the air column is adjusted until resonance is heard. It is suggested that: $f = \frac{v}{4(L + c)}$ where $v$ is the speed of sound and $c$ is the end correction.

MS 展开查看

Defining the problem:

• B1: $L$ is independent, $f$ is dependent / vary $L$ and measure $f$
• B1: keep temperature constant（affects $v$）

Methods of data collection:

• B1: diagram: resonance tube, water reservoir, tuning fork, ruler, labelled
• B1: vary $L$ by adjusting water level
• B1: measure $L$ using metre ruler
• B1: measure $f$ using frequency of tuning fork（known / calibrated）
• B1: use set of tuning forks of different frequencies

Method of analysis:

• B1: plot $1/f$ against $L$ or $L$ against $1/f$
• B1: straight line confirms relationship
• B1: $v = 4 \times \text{gradient}$, $c = \text{y-intercept} / (-\text{gradient})$

Additional detail:

• D1: repeat at different $L$ for same $f$
• D2: measure temperature using thermometer
• D3: strike tuning fork on rubber pad（not hard surface）
• D4: hold tuning fork at same height
• D5: ensure tube is vertical using set square

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Example 3 — 9702/s23/qp/51 Q1（15 marks）

A student investigates how the speed $v$ of a wave on a stretched string depends on the tension $T$. It is suggested that: $v = \sqrt{\frac{T}{\mu}}$ where $\mu$ is the mass per unit length. The student measures the frequency $f$ and wavelength $\lambda$ to determine $v$. It is suggested that: $f = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$ where $n$ is the number of loops in the standing wave and $L$ is the length of the string.

MS 展开查看

Defining the problem:

• B1: $T$ is independent, $f$ is dependent / vary $T$ and measure $f$
• B1: keep $L$, $\mu$, $n$ constant

Methods of data collection:

• B1: diagram: string, vibrator, pulley, masses, signal generator, labelled with $L$
• B1: vary $T$ by adding masses to the hanger
• B1: measure $T = mg$ using balance
• B1: measure $f$ from signal generator / stroboscope / CRO
• B1: measure $L$ using ruler

Method of analysis:

• B1: plot $f$ against $\sqrt{T}$
• B1: straight line through origin confirms relationship
• B1: $\mu = \left(\frac{n}{2L \times \text{gradient}}\right)^2$

Additional detail:

• D1: repeat and average
• D2: measure $\mu$ by weighing a known length of string
• D3: ensure pulley is frictionless
• D4: adjust frequency to get stable $n$ loops
• D5: use marker to identify node positions

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常见陷阱

Wave 陷阱

• 驻波中 node 和 antinode 混淆
• 忘记 end correction $c$（resonance tube）
• 线密度 $\mu$ 的测量方法不具体
• 频率计/信号发生器的使用没说明

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Question Type 5: Damped Oscillations — Copper Sheet in Magnetic Field

如何识别

题目涉及金属片在磁场中摆动、振幅衰减、涡流阻尼。关键词包括 damped oscillations, copper sheet, magnetic field, eddy current, amplitude decay, electromagnetic braking。

标准解题方法

Damped Oscillations 框架

IV: 金属片面积 $A$ / 厚度 $z$ DV: 5 次摆动后的距离 $s$ / 停止时间 $t$ CV: 磁场强度 $B$、初始位移 $s_0$、材料类型

关键关系: $s = s_0 e^{-ABKt}$，振幅随面积指数衰减

测量 $t$: 用 stopwatch 测量从释放到停止的时间 测量 $A$: 用 ruler 测长宽计算面积

Example 1 — 9702/w22/qp/51 Q1（15 marks）

A student investigates the damping of oscillations of a copper sheet placed between the poles of a strong magnet. The sheet is suspended by a thread and set into oscillation. The amplitude of the oscillations decreases due to eddy currents induced in the sheet. It is suggested that the distance $s$ moved by the sheet after 5 complete oscillations is given by: $s = s_0 e^{-ABKt}$ where $s_0$ is the initial amplitude, $A$ is the area of the sheet, $B$ is the magnetic flux density, $K$ is a constant and $t$ is the thickness of the sheet.

MS 展开查看

Defining the problem:

• B1: cross-sectional area $A$ is the independent variable, distance $s$ after 5 oscillations is the dependent variable / vary $A$ and measure $s$
• B1: keep $s_0$, $B$, $t$ constant

Methods of data collection:

• B1: diagram: copper sheet suspended by thread between magnet poles, ruler beside to measure amplitude
• B1: use sheets of different area $A$（vary length/width）
• B1: measure $A$ using ruler / vernier calipers（$A = \text{length} \times \text{width}$）
• B1: measure $s$ using ruler adjacent to sheet
• B1: measure $t$ using micrometer

Method of analysis:

• B1: plot $\ln s$ against $A$
• B1: straight line confirms relationship $s = s_0 e^{-ABKt}$
• B1: $K = -\frac{\text{gradient}}{B t}$

Additional detail:

• D1: repeat and average $s$ at each $A$
• D2: ensure same initial displacement $s_0$ using marker / release mechanism
• D3: use marker on sheet for consistent reading point
• D4: avoid air currents / draughts
• D5: safety: cushion in case sheet falls

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Example 2 — 9702/w22/qp/52 Q1（15 marks）

A student investigates how the time $t$ taken for a copper sheet to stop oscillating depends on the thickness $z$ of the sheet. The sheet is suspended between the poles of a magnet and set into oscillation. It is suggested that: $t = \frac{K z^q}{A B \rho}$ where $A$ is the area of the sheet, $B$ is the magnetic flux density, $\rho$ is the resistivity of copper and $K$ and $q$ are constants.

MS 展开查看

Defining the problem:

• B1: thickness $z$ is the independent variable, time $t$ is the dependent variable / vary $z$ and measure $t$
• B1: keep $A$, $B$ constant

Methods of data collection:

• B1: diagram: copper sheet suspended, magnet poles, stopwatch, labelled
• B1: use sheets of different thickness $z$（same area $A$）
• B1: measure $z$ using micrometer
• B1: measure $t$ using stopwatch（from release to rest）
• B1: measure $A$ using ruler

Method of analysis:

• B1: plot $\lg t$ against $\lg z$
• B1: straight line confirms relationship
• B1: $q = \text{gradient}$, $\lg K = \text{y-intercept} + \lg(A B \rho)$

Additional detail:

• D1: repeat and average $t$
• D2: use same initial amplitude for each trial
• D3: ensure magnet poles are aligned
• D4: clamp magnet securely
• D5: safety: gloves for sharp edges of sheet

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常见陷阱

Damped Oscillations 陷阱

• 忘记说明如何确保每次初始振幅相同
• 测量 $s$ 时视差问题未处理
• 忽略空气流动对阻尼的影响
• 磁场强度 $B$ 的恒定性未考虑

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Question Type 6: Ballistic Pendulum / Projectile Motion

如何识别

题目涉及抛射体撞击摆锤、摆动高度与质量关系。关键词包括 ballistic pendulum, projectile, clay block, pendulum, conservation of momentum, height of swing。

标准解题方法

Ballistic Pendulum 框架

IV: 抛射体质量 $Z$ DV: 摆锤摆动高度 $h$ CV: 摆锤质量 $M$、初速度 $u$、重力加速度 $g$

关键关系: $h = \frac{u^2}{2g}\left(\frac{Z}{M+Z}\right)^2$

测量 $h$: 用 ruler / protractor 测最大摆动高度 测量 $Z$: 用 top-pan balance 称质量

Example 1 — 9702/w23/qp/51 Q1（15 marks）

A student investigates the behaviour of a ballistic pendulum. A projectile of mass $Z$ is fired horizontally into a stationary block of clay of mass $M$ suspended as a pendulum. The projectile embeds itself in the clay and the pendulum swings to a maximum height $h$. It is suggested that: $h = \frac{u^2}{2g}\left(\frac{Z}{M+Z}\right)^2$ where $u$ is the speed of the projectile and $g$ is the acceleration of free fall.

MS 展开查看

Defining the problem:

• B1: mass of projectile $Z$ is the independent variable, height $h$ is the dependent variable / vary $Z$ and measure $h$
• B1: keep $M$, $u$ constant

Methods of data collection:

• B1: diagram: pendulum with clay block, projectile gun, ruler to measure $h$, labelled
• B1: use projectiles of different mass $Z$
• B1: measure $Z$ using top-pan balance
• B1: measure $h$ using ruler / marker on the scale behind pendulum
• B1: measure $M$ using top-pan balance

Method of analysis:

• B1: plot $\sqrt{h}$ against $\frac{Z}{M+Z}$ or $h$ against $\left(\frac{Z}{M+Z}\right)^2$
• B1: straight line through origin confirms relationship
• B1: $u = \sqrt{2g \times \text{gradient}}$ for $h$ vs $\left(\frac{Z}{M+Z}\right)^2$

Additional detail:

• D1: repeat and average $h$ for each $Z$
• D2: ensure pendulum swings freely without friction
• D3: use plumb line to check vertical alignment
• D4: use safety screen around apparatus
• D5: safety: stand clamped securely, wear goggles

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Example 2 — 9702/w23/qp/53 Q1（15 marks）

A student investigates how the angle $\theta$ of swing of a ballistic pendulum depends on the mass $m$ of the projectile. A projectile of mass $m$ is fired into a stationary pendulum of mass $M$. The pendulum swings to a maximum angle $\theta$. It is suggested that: $\cos\theta = 1 - \frac{u^2}{2gL}\left(\frac{m}{M+m}\right)^2$ where $L$ is the length of the pendulum and $u$ is the speed of the projectile.

MS 展开查看

Defining the problem:

• B1: $m$ is independent, $\theta$ is dependent / vary $m$ and measure $\theta$
• B1: keep $M$, $L$, $u$ constant

Methods of data collection:

• B1: diagram: pendulum, protractor to measure $\theta$, projectile launcher, labelled
• B1: vary $m$ using projectiles of different masses
• B1: measure $m$ using top-pan balance
• B1: measure $\theta$ using protractor at maximum swing
• B1: measure $L$ using ruler

Method of analysis:

• B1: plot $\cos\theta$ against $\left(\frac{m}{M+m}\right)^2$ or $\theta$ against $m$
• B1: straight line confirms relationship
• B1: $u = \sqrt{-2gL \times \text{gradient}}$

Additional detail:

• D1: repeat and average $\theta$
• D2: use marker on the scale to read $\theta$ consistently
• D3: ensure projectile is fired horizontally
• D4: clamp stand securely
• D5: safety: stand behind safety screen

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常见陷阱

Ballistic Pendulum 陷阱

• 忘记保持 $u$ 恒定（需要相同的发射装置/弹簧压缩量）
• 没说明如何测量 $h$（从起始位置到最高点）
• 空气阻力和摩擦力忽略导致系统误差
• 摆长 $L$ 未测量

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Question Type 7: Capillary Rise / Surface Tension

如何识别

题目涉及毛细管中液面上升高度与管径关系、表面张力测量。关键词包括 capillary, surface tension, meniscus, tube diameter, rise height, adhesion。

标准解题方法

Capillary Rise 框架

IV: 毛细管直径 $d$ DV: 液面上升高度 $h$ CV: 液体种类（$\rho$、$\gamma$）、温度

关键关系: $h = \frac{4\gamma}{d\rho g}$

测量 $h$: 用 travelling microscope 或 ruler 配合 set square 读取弯月面底部 测量 $d$: 用 travelling microscope / micrometer

Example 1 — 9702/w20/qp/52 Q1（15 marks）

A student investigates the rise of water in capillary tubes of different diameters. The student places a capillary tube vertically into a container of water and measures the height $h$ to which the water rises above the surface of the water in the container. It is suggested that the height $h$ is related to the diameter $d$ of the tube by: $h = \frac{4\gamma}{d\rho g}$ where $\gamma$ is the surface tension of the water, $\rho$ is the density of water and $g$ is the acceleration of free fall.

MS 展开查看

Defining the problem:

• B1: diameter $d$ is the independent variable, height $h$ is the dependent variable / vary $d$ and measure $h$
• B1: keep temperature, liquid type constant

Methods of data collection:

• B1: diagram: capillary tube in water container, ruler / travelling microscope beside tube, labelled
• B1: use capillary tubes of different internal diameters $d$
• B1: measure $d$ using travelling microscope / micrometer
• B1: measure $h$ using travelling microscope / ruler with set square
• B1: measure temperature using thermometer

Method of analysis:

• B1: plot $h$ against $1/d$
• B1: straight line through origin confirms relationship
• B1: $\gamma = \frac{\rho g \times \text{gradient}}{4}$

Additional detail:

• D1: repeat and average $h$ for each tube
• D2: ensure tube is vertical using set square
• D3: measure from water surface to bottom of meniscus
• D4: clean tubes thoroughly before use
• D5: use dye in water to improve visibility

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常见陷阱

Capillary Rise 陷阱

• 弯月面读数位置不明确（应读底部）
• 忘记清洁毛细管（污染影响表面张力）
• 温度变化导致 $\gamma$ 和 $\rho$ 变化
• 毛细管未完全垂直放置

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Question Type 8: LR Circuit / Time Constant

如何识别

题目涉及电感线圈、电阻、时间常数、电流上升/衰减。关键词包括 inductor, LR circuit, time constant, current growth, decay, coil, resistance。

标准解题方法

LR Circuit 框架

IV: 电阻 $R$ DV: 电流达到最大值的时间 $t$ CV: 线圈匝数 $N$、线圈面积 $A$、线圈长度 $L$

关键关系: $t = \frac{K N^2 A}{L R}$

测量 $t$: stopwatch 测量从闭合开关到电流达到某值的时间 测量 $R$: ohmmeter / 从电阻箱读取

Example 1 — 9702/s21/qp/51 Q1（15 marks）

A student investigates how the time $t$ for the current in an LR circuit to reach a maximum value depends on the resistance $R$ in the circuit. The circuit consists of a coil connected in series with a variable resistor, a switch, and a power supply. It is suggested that: $t = \frac{K N^2 A}{L R}$ where $N$ is the number of turns on the coil, $A$ is the cross-sectional area of the coil, $L$ is the length of the coil and $K$ is a constant.

MS 展开查看

Defining the problem:

• B1: resistance $R$ is the independent variable, time $t$ is the dependent variable / vary $R$ and measure $t$
• B1: keep $N$, $A$, $L$ constant

Methods of data collection:

• B1: diagram: power supply, coil, variable resistor, ammeter, switch in series, stopwatch
• B1: vary $R$ using a variable resistor / resistance box
• B1: measure $R$ using ohmmeter / read from resistance box
• B1: measure $t$ using stopwatch（from switch closing to current reaching maximum）
• B1: measure $I$ using ammeter to identify when maximum is reached

Method of analysis:

• B1: plot $t$ against $1/R$
• B1: straight line through origin confirms relationship
• B1: $K = \frac{L \times \text{gradient}}{N^2 A}$

Additional detail:

• D1: repeat and average $t$
• D2: switch off between readings to avoid heating
• D3: measure $N$ by counting turns
• D4: measure $A$ using ruler（$A = \pi d^2/4$）
• D5: safety: switch off before changing resistor

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Example 2 — 9702/s21/qp/53 Q1（15 marks）

A student investigates the growth of current in an LR circuit. A coil of inductance $L_c$ and resistance $R_c$ is connected in series with a resistor of resistance $R$ and a battery. The student measures the time $T$ taken for the current to reach half its final value. It is suggested that: $T = \frac{L_c \ln 2}{R + R_c}$ where $L_c$ is the inductance of the coil.

MS 展开查看

Defining the problem:

• B1: $R$ is independent, $T$ is dependent / vary $R$ and measure $T$
• B1: keep $L_c$, $R_c$ constant

Methods of data collection:

• B1: diagram: coil, resistor, ammeter, switch, battery in series, stopwatch
• B1: vary $R$ using a variable resistor
• B1: measure $R$ using ohmmeter
• B1: measure $T$ using stopwatch（switch on, stop when $I = I_{\text{max}}/2$）
• B1: measure $I_{\text{max}}$ using ammeter

Method of analysis:

• B1: plot $T$ against $1/(R + R_c)$
• B1: straight line through origin confirms relationship
• B1: $L_c = \frac{\text{gradient}}{\ln 2}$

Additional detail:

• D1: repeat and average $T$
• D2: allow coil to cool between readings
• D3: use a data logger for precise timing
• D4: ensure switch makes good contact
• D5: safety: disconnect supply when not in use

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常见陷阱

LR Circuit 陷阱

• 忽略线圈自身电阻 $R_c$
• 电流最大值判断不准确
• 开关接触电阻未考虑
• 线圈发热导致电感变化

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Question Type 9: Doppler Effect

如何识别

题目涉及运动声源、观察频率变化、多普勒效应。关键词包括 Doppler effect, moving source, observed frequency, speed of source, signal generator。

标准解题方法

Doppler Effect 框架

IV: 声源速度 $v$ DV: 观察频率 $f$ CV: 声源频率 $f_s$、声速 $k$

关键关系: $f = \frac{f_s k}{k - v}$

测量 $f$: 用 microphone 连接 data logger / CRO 测量频率 测量 $v$: 用 light gates / motion sensor 测声源速度

Example 1 — 9702/s24/qp/51 Q1（15 marks）

A student investigates the Doppler effect using a loudspeaker attached to a moving trolley. The loudspeaker emits a sound of constant frequency $f_s$ from a signal generator. The trolley moves towards a stationary microphone connected to a data logger. It is suggested that the frequency $f$ measured by the microphone is given by: $f = \frac{f_s k}{k - v}$ where $k$ is the speed of sound and $v$ is the speed of the trolley.

MS 展开查看

Defining the problem:

• B1: speed of trolley $v$ is the independent variable, measured frequency $f$ is the dependent variable / vary $v$ and measure $f$
• B1: keep $f_s$, $k$ constant（$k$ depends on temperature）

Methods of data collection:

• B1: diagram: trolley with loudspeaker, signal generator, microphone, light gates, data logger, labelled
• B1: vary $v$ by adjusting the slope / pushing force on trolley
• B1: measure $v$ using light gates connected to timer
• B1: measure $f$ using microphone connected to data logger / CRO
• B1: measure $f_s$ using signal generator reading / CRO

Method of analysis:

• B1: plot $1/f$ against $1/v$ or $f$ against $1/(k - v)$
• B1: straight line confirms relationship
• B1: $k = \frac{f_s \times \text{gradient}}{\text{gradient} - 1}$（depending on linearised form）

Additional detail:

• D1: repeat and average $f$ at each $v$
• D2: measure temperature using thermometer to estimate $k$
• D3: ensure trolley moves directly towards microphone
• D4: use a long run-up for steady speed
• D5: safety: ensure trolley path is clear

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Example 2 — 9702/s24/qp/53 Q1（15 marks）

A student investigates how the observed wavelength $\lambda$ of sound from a moving source depends on the speed $v$ of the source. The source emits sound of frequency $f_s$ and moves away from a stationary observer. It is suggested that: $\lambda = \frac{k + v}{f_s}$ where $k$ is the speed of sound.

MS 展开查看

Defining the problem:

• B1: $v$ is independent, $\lambda$ is dependent / vary $v$ and measure $\lambda$
• B1: keep $f_s$, $k$ constant

Methods of data collection:

• B1: diagram: moving loudspeaker, stationary microphone, ruler, signal generator, labelled
• B1: vary $v$ using trolley at different speeds
• B1: measure $v$ using light gates
• B1: measure $\lambda$ using microphone and CRO（distance between successive wavefronts）
• B1: measure $f_s$ using signal generator

Method of analysis:

• B1: plot $\lambda$ against $v$
• B1: straight line confirms relationship
• B1: gradient $= 1/f_s$, y-intercept $= k/f_s$ → find $k$

Additional detail:

• D1: repeat and average $\lambda$
• D2: use data logger for accurate frequency measurement
• D3: minimise background noise
• D4: ensure microphone is aligned with source path
• D5: safety: secure all cables to avoid tripping

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常见陷阱

Doppler Effect 陷阱

• 声速 $k$ 受温度影响但未测量温度
• 声源是否做匀速运动未确认
• microphone 和 data logger 的采样率不足
• 公式中 $v$ 的符号（接近/远离）混淆

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Question Type 10: Resistivity / Wheatstone Bridge

如何识别

题目涉及电阻率、金属丝电阻与尺寸关系、Wheatstone 桥电路。关键词包括 resistivity, Wheatstone bridge, wire diameter, length, resistance ratio。

标准解题方法

Resistivity / Wheatstone Bridge 框架

IV: 金属丝长度 $L$ / 直径 $d$ DV: 电阻比 $Z/R$（从 bridge 平衡获得） CV: 材料电阻率 $\rho$、温度

关键关系: $\frac{Z}{R} = \frac{4\rho L}{\pi Y d^2}$

测量 $d$: 用 micrometer 在不同位置测量取平均 测量 $L$: 用 ruler

Example 1 — 9702/w22/qp/51 Q1（15 marks）

A student uses a Wheatstone bridge to determine how the ratio $Z/R$ of resistances depends on the length $L$ of a metal wire of diameter $d$. The wire forms part of one arm of the bridge. It is suggested that: $\frac{Z}{R} = \frac{4\rho L}{\pi Y d^2}$ where $\rho$ is the resistivity of the metal and $Y$ is a constant.

MS 展开查看

Defining the problem:

• B1: length $L$ is the independent variable, ratio $Z/R$ is the dependent variable / vary $L$ and measure $Z/R$
• B1: keep $d$, $\rho$, temperature constant

Methods of data collection:

• B1: diagram: Wheatstone bridge circuit with wire of length $L$, galvanometer, labelled
• B1: vary $L$ by using wires of different lengths
• B1: measure $L$ using ruler
• B1: measure $Z/R$ from balance point on bridge / known resistor values
• B1: measure $d$ using micrometer（at several points, average）

Method of analysis:

• B1: plot $Z/R$ against $L$
• B1: straight line through origin confirms relationship
• B1: $\rho = \frac{\pi Y d^2 \times \text{gradient}}{4}$

Additional detail:

• D1: repeat balance measurement and average
• D2: avoid heating by switching off between readings
• D3: clean wire ends for good electrical contact
• D4: use short pulses of current to minimise heating
• D5: safety: disconnect power when adjusting circuit

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Example 2 — 9702/w22/qp/53 Q1（15 marks）

A student investigates how the resistance $X$ of a metal wire varies with its diameter $d$. The wire is connected in one arm of a Wheatstone bridge. It is suggested that: $X = \frac{4\rho L}{\pi d^2}$ where $L$ is the length of the wire and $\rho$ is the resistivity of the metal.

MS 展开查看

Defining the problem:

• B1: $d$ is independent, $X$ is dependent / vary $d$ and measure $X$
• B1: keep $L$, $\rho$, temperature constant

Methods of data collection:

• B1: diagram: Wheatstone bridge circuit, wire under test, labelled
• B1: vary $d$ by using wires of different diameters
• B1: measure $d$ using micrometer
• B1: measure $X$ using Wheatstone bridge balance / ohmmeter
• B1: measure $L$ using ruler

Method of analysis:

• B1: plot $X$ against $1/d^2$ or $\lg X$ against $\lg d$
• B1: straight line through origin（or straight line for log-log）confirms relationship
• B1: $\rho = \frac{\pi \times \text{gradient}}{4L}$ from $X$ vs $1/d^2$

Additional detail:

• D1: repeat and average $X$
• D2: measure $d$ at several points along wire
• D3: stretch wire to remove kinks before measuring
• D4: use a constant current to prevent heating
• D5: safety: avoid touching bare wires

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常见陷阱

Resistivity / Wheatstone Bridge 陷阱

• 直径 $d$ 只测一端不取平均
• 忘记温度对电阻率的影响
• 接触电阻未消除
• 桥路平衡判断不准确（galvanometer 灵敏度）

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Question Type 11: Stellar Physics (lg-lg Plots)

如何识别

题目涉及恒星物理数据、光度与质量关系、对数坐标图。关键词包括 luminosity, mass, star, stellar, lg-lg plot, data analysis, published data。

标准解题方法

Stellar Physics 框架

IV: 恒星质量 $M$ DV: 恒星光度 $L$ CV: 数据来源可靠性

关键关系: $L = S Z M^n$，线性化为 $\lg L = \lg(SZ) + n \lg M$

注意: 此题型通常为 Q2（数据分析），但可作为 Q1（实验规划）出现，需要学生从文献/数据库获取数据

数据处理: 用 $\lg L$ 对 $\lg M$ 作图求 $n$ 和常数

Example 1 — 9702/s22/qp/51 Q1（15 marks）

A student investigates the relationship between the luminosity $L$ of a main sequence star and its mass $M$. The student uses published data from a star catalogue. It is suggested that: $L = S Z M^n$ where $S$ and $Z$ are known constants for the star and $n$ is a constant to be determined.

MS 展开查看

Defining the problem:

• B1: mass $M$ is the independent variable, luminosity $L$ is the dependent variable
• B1: use stars of the same type（main sequence）to keep composition constant

Methods of data collection:

• B1: obtain data from published star catalogue / database
• B1: select stars with known mass $M$ and luminosity $L$
• B1: record $M$ in solar masses $M_\odot$ and $L$ in solar luminosities $L_\odot$
• B1: use a wide range of $M$ values for reliable graph
• B1: ensure data is from reliable sources

Method of analysis:

• B1: plot $\lg L$ against $\lg M$
• B1: straight line confirms relationship
• B1: $n = \text{gradient}$, $\lg(SZ) = \text{y-intercept}$

Additional detail:

• D1: include error bars on graph if uncertainty given
• D2: use at least 6 data points
• D3: check for outliers in published data
• D4: state assumptions（stars are main sequence）
• D5: reference data sources in report

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Example 2 — 9702/s22/qp/52 Q1（15 marks）

A student investigates how the temperature $T$ of a star relates to its colour index $C$. The student collects data from an astronomical database. It is suggested that: $T = \frac{a}{C + b}$ where $a$ and $b$ are constants.

MS 展开查看

Defining the problem:

• B1: colour index $C$ is the independent variable, temperature $T$ is the dependent variable
• B1: select stars of similar spectral type to minimise variation

Methods of data collection:

• B1: obtain data from astronomical database / star catalogue
• B1: record $T$ and $C$ for a range of stars
• B1: ensure data covers a wide range of $C$
• B1: use data processed by standard methods

Method of analysis:

• B1: plot $1/T$ against $C$
• B1: straight line confirms relationship
• B1: $a = 1/\text{gradient}$, $b = \text{y-intercept} / \text{gradient}$

Additional detail:

• D1: consider uncertainties in published data
• D2: use data from multiple sources for verification
• D3: check that temperature scale is consistent
• D4: note any selection bias in data
• D5: calibrate using known standard stars

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常见陷阱

Stellar Physics 陷阱

• 混淆 $\lg$ 和 $\ln$
• 数据的 uncertainty 未考虑
• 选择的恒星不是同一类型
• 对数图的坐标轴标签不完整

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Question Type 12: Gas Laws (pV vs T / V vs θ)

如何识别

题目涉及理想气体、压强-体积-温度关系、恒温/恒压实验。关键词包括 gas law, Boyle's law, Charles' law, pressure, volume, temperature, gas syringe, water bath。

标准解题方法

Gas Laws 框架

IV: 温度 $\theta$ DV: $pV$ 乘积 / $V$ 体积 CV: 气体质量 $n$（物质的量）、体积（如定容）

关键关系: $pV = Y k (\theta + Z)$

测量 $p$: 用 pressure gauge / manometer 测量 $V$: 用 gas syringe / ruler（测柱体长度） 控制 $\theta$: 用 water bath + thermometer

Example 1 — 9702/s23/qp/52 Q1（15 marks）

A student investigates how the product $pV$ of pressure and volume of a fixed mass of gas varies with temperature $\theta$. The gas is contained in a syringe placed in a water bath. The temperature is changed by heating or cooling the water. It is suggested that: $pV = Y k (\theta + Z)$ where $Y$, $k$ and $Z$ are constants.

MS 展开查看

Defining the problem:

• B1: temperature $\theta$ is the independent variable, $pV$ product is the dependent variable / vary $\theta$ and measure $p$ and $V$
• B1: keep mass of gas constant（sealed system）

Methods of data collection:

• B1: diagram: gas syringe in water bath, thermometer, pressure gauge, labelled
• B1: vary $\theta$ by using water at different temperatures
• B1: measure $\theta$ using thermometer / temperature sensor
• B1: measure $p$ using pressure gauge connected to syringe
• B1: measure $V$ using scale on syringe / ruler

Method of analysis:

• B1: plot $pV$ against $\theta$
• B1: straight line confirms relationship
• B1: $Z = -\text{intercept}/\text{gradient}$, $Yk = \text{gradient}$

Additional detail:

• D1: repeat and average $p$ and $V$ at each $\theta$
• D2: stir water bath for even temperature
• D3: allow time for gas to reach thermal equilibrium
• D4: clamp syringe securely
• D5: safety: use heat-proof gloves for hot water

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常见陷阱

Gas Laws 陷阱

• 忘记气体必须达到热平衡就读数
• 水浴温度不均匀
• 气体泄漏导致质量变化
• 压强单位与体积单位不匹配导致计算错误

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题型对比总结（完整版）

题型	常见 IV	常见 DV	典型 CV	分析图
Spring	$A$, $t$, $w$, $\theta$	$k$, $x$	$N$, $\rho$, $m$	$\lg k$ vs $\lg A$ / $k$ vs $A^{-3/2}$
Electrical	$R$, $d$, $n$	$I$, $V$, $C$, $\mathcal{E}$	$V$, $f$, $A$	$C$ vs $1/d$ / $\lg \mathcal{E}$ vs $\lg d$
Thermal	$t$, $A$	$\theta$, rate	volume, $\theta_{\text{room}}$	$\ln \theta$ vs $t$ / rate vs $A$
Wave	$T$, $L$, $f$	$\lambda$, $f$, $v$	$\mu$, temperature	$\lambda$ vs $\sqrt{T}$ / $1/f$ vs $L$
Damped Oscillations	$A$, $z$	$s$, $t$	$B$, $s_0$, material	$\ln s$ vs $A$ / $\lg t$ vs $\lg z$
Ballistic Pendulum	$Z$, $m$	$h$, $\theta$	$M$, $u$, $L$	$\sqrt{h}$ vs $Z/(M+Z)$ / $\cos\theta$ vs $(m/(M+m))^2$
Capillary Rise	$d$	$h$	liquid, temperature	$h$ vs $1/d$
LR Circuit	$R$	$t$, $T$	$N$, $A$, $L$	$t$ vs $1/R$ / $T$ vs $1/(R+R_c)$
Doppler Effect	$v$	$f$, $\lambda$	$f_s$, $k$	$1/f$ vs $1/v$ / $\lambda$ vs $v$
Resistivity / Bridge	$L$, $d$	$Z/R$, $X$	$\rho$, temperature	$Z/R$ vs $L$ / $X$ vs $1/d^2$
Stellar Physics	$M$, $C$	$L$, $T$	star type	$\lg L$ vs $\lg M$ / $1/T$ vs $C$
Gas Laws	$\theta$	$pV$, $V$	$n$, mass of gas	$pV$ vs $\theta$