IB Physics HL | Unit A2: Forces and Momentum

Show all working in the space below each question. Marks are awarded for correct method as well as final answers. Draw a free-body diagram and state your sign convention before applying Newton's laws. Give numerical answers to an appropriate number of significant figures.在每题下方空白处写出全部解题过程。方法分（method marks）与最终答案同等重要。应用牛顿定律前先画受力图并写明正方向约定（sign convention）。数值答案保留适当的有效数字。

Q1MEDIUM Paper 1 equilibrium: cables + incline平衡：绳与斜面 [6 marks]

A $8.0\ \mathrm{kg}$ lamp hangs in equilibrium from two cables that each make $25^{\circ}$ with the horizontal ceiling.一盏 $8.0\ \mathrm{kg}$ 的灯由两条与水平天花板各成 $25^{\circ}$ 的绳悬挂，处于平衡。

(a) Draw a free-body diagram of the lamp and write the two equilibrium conditions $\Sigma F_{x} = 0$ and $\Sigma F_{y} = 0$ for the knot.画出灯的受力图，并写出结点的两个平衡条件 $\Sigma F_{x} = 0$ 与 $\Sigma F_{y} = 0$。 [2]

(b) Calculate the tension in each cable.计算每条绳的张力。 [2]

(c) State and explain how the tension changes if the cables are lowered to make a smaller angle with the ceiling.说明并解释：若把绳放低使其与天花板的夹角更小，张力将如何变化。 [2]

Q2MEDIUM Paper 1 mass vs weight, lift N2L质量与重量、电梯牛二 [6 marks]

A $65\ \mathrm{kg}$ student stands on a bathroom scale inside a lift. The scale reads the normal force in newtons.一名 $65\ \mathrm{kg}$ 的学生站在电梯内的体重计上。体重计以牛顿为单位读出法向力。

(a) State the student's weight and explain why the mass would be unchanged on the Moon while the weight would not.写出该学生的重量，并解释为何在月球上质量不变而重量会变。 [2]

(b) The lift accelerates upward at $1.8\ \mathrm{m\,s^{-2}}$. Calculate the scale reading (the apparent weight).电梯以 $1.8\ \mathrm{m\,s^{-2}}$ 向上加速。计算体重计读数（视重）。 [2]

(c) The lift instead accelerates downward at $1.8\ \mathrm{m\,s^{-2}}$. Calculate the new reading, and state what the reading would be if the lift were in free fall.改为电梯以 $1.8\ \mathrm{m\,s^{-2}}$ 向下加速。计算新读数，并写出若电梯处于自由落体时的读数。 [2]

Q3HARD Paper 1 Newton's third law + Atwood牛顿第三定律 + 阿特伍德机 [6 marks]

Two blocks $m_{1} = 2.0\ \mathrm{kg}$ and $m_{2} = 3.0\ \mathrm{kg}$ hang over a light, frictionless pulley by a light inextensible string and are released from rest.两物块 $m_{1} = 2.0\ \mathrm{kg}$ 与 $m_{2} = 3.0\ \mathrm{kg}$ 由轻质不可伸长绳跨过轻质无摩擦滑轮悬挂，从静止释放。

(a) State Newton's third law, and identify the third-law partner of the pull of the string on $m_{2}$.陈述牛顿第三定律，并指出绳对 $m_{2}$ 拉力的第三定律配对力。 [2]

(b) Calculate the acceleration of the system.计算系统的加速度。 [2]

(c) Calculate the tension in the string and explain why it is not simply equal to the weight of either block.计算绳中张力，并解释它为何不简单等于任一物块的重量。 [2]

Q4HARD Paper 1 static vs kinetic friction静摩擦与动摩擦 [6 marks]

A $15\ \mathrm{kg}$ crate sits on a level floor with coefficients $\mu_{s} = 0.45$ and $\mu_{k} = 0.30$.一个 $15\ \mathrm{kg}$ 的板条箱置于水平地面，摩擦系数 $\mu_{s} = 0.45$、$\mu_{k} = 0.30$。

(a) A horizontal force of $60\ \mathrm{N}$ is applied. Determine, with a calculation, whether the crate moves, and state the magnitude of the friction force acting.施加 $60\ \mathrm{N}$ 的水平力。通过计算判断箱子是否移动，并写出此时摩擦力的大小。 [3]

(b) The applied force is increased to $90\ \mathrm{N}$. Calculate the acceleration of the crate.所加力增至 $90\ \mathrm{N}$。计算箱子的加速度。 [2]

Q5HARD Paper 1 vertical circular motion竖直圆周运动 [6 marks]

A ball of mass $0.25\ \mathrm{kg}$ is whirled on a light string in a vertical circle of radius $0.90\ \mathrm{m}$.一个 $0.25\ \mathrm{kg}$ 的小球用轻绳在半径 $0.90\ \mathrm{m}$ 的竖直圆中旋转。

(a) Explain why the term "centripetal force" does not denote a new, separate force, and state the direction of the net force.解释"向心力"为何并非一种新的、独立的力，并写出合力的方向。 [2]

(b) Calculate the minimum speed at the top of the circle for the string to remain taut.计算使绳在圆顶保持张紧的最小速率。 [2]

(c) When the ball moves at $4.0\ \mathrm{m\,s^{-1}}$ at the lowest point, calculate the tension in the string there.当小球在最低点以 $4.0\ \mathrm{m\,s^{-1}}$ 运动时，计算此处绳中张力。 [2]

Graph and Data Questions图像与数据题

These items reward correct reading of areas and gradients, and careful handling of uncertainties. Quote uncertainties to one significant figure and round the value to match.这些题考查对面积与斜率的正确读取以及对不确定度的细致处理。不确定度保留 1 位有效数字，并使数值的末位与之对齐。

Q6HARD Paper 1B HL ONLY impulse as area under $F$-$t$冲量即 $F$-$t$ 图面积 [10 marks]

A $0.60\ \mathrm{kg}$ trolley, initially at rest on a frictionless track, is pushed by a horizontal force that varies with time as follows: it rises linearly from $0$ to $18\ \mathrm{N}$ over $0$ to $0.20\ \mathrm{s}$, stays constant at $18\ \mathrm{N}$ from $0.20\ \mathrm{s}$ to $0.50\ \mathrm{s}$, then falls linearly to zero from $0.50\ \mathrm{s}$ to $0.60\ \mathrm{s}$.一辆 $0.60\ \mathrm{kg}$ 的小车初始静止于无摩擦轨道上，受一随时间变化的水平力作用：$0$ 至 $0.20\ \mathrm{s}$ 内由 $0$ 线性升到 $18\ \mathrm{N}$，$0.20\ \mathrm{s}$ 至 $0.50\ \mathrm{s}$ 内保持 $18\ \mathrm{N}$，$0.50\ \mathrm{s}$ 至 $0.60\ \mathrm{s}$ 内线性降为零。

(a) State what the area under a force-time graph represents, and write the impulse-momentum theorem.说明力-时间图下的面积代表什么，并写出动量定理。 [2]

(b) By splitting the graph into a triangle, a rectangle and a triangle, calculate the total impulse delivered to the trolley.将图分为三角形、矩形、三角形，计算传递给小车的总冲量。 [3]

(d) Calculate the average force over the whole $0.60\ \mathrm{s}$, and explain why this average produces the same final speed as the varying force.计算整个 $0.60\ \mathrm{s}$ 内的平均力，并解释为何该平均力产生与变力相同的末速度。 [3]

Q7HARD Paper 1B drag data + terminal velocity阻力数据与收尾速度 [12 marks]

A small sphere of mass $0.030\ \mathrm{kg}$ falls through a liquid. The drag force $F_{D}$ is measured at several steady speeds $v$ and is believed to obey $F_{D} = b v^{2}$:一个 $0.030\ \mathrm{kg}$ 的小球在液体中下落。在若干稳定速率 $v$ 下测得阻力 $F_{D}$，认为其满足 $F_{D} = b v^{2}$：

$v\ /\ \mathrm{m\,s^{-1}}$	$2.0$	$4.0$	$6.0$	$8.0$
$v^{2}\ /\ \mathrm{m^{2}\,s^{-2}}$	$4.0$	$16.0$	$36.0$	$64.0$
$F_{D}\ /\ \mathrm{N}$	$0.030$	$0.120$	$0.270$	$0.480$

(a) Explain why a graph of $F_{D}$ against $v^{2}$ should be a straight line through the origin, and state what its gradient represents.解释为何 $F_{D}$ 对 $v^{2}$ 的图应为过原点的直线，并说明其斜率代表什么。 [3]

(b) Using widely spaced data points, calculate the gradient and hence the constant $b$.用相距较远的数据点，计算斜率，由此求常数 $b$。 [3]

(c) Calculate the terminal velocity of the sphere when it falls freely through this liquid (ignore buoyancy).计算小球在此液体中自由下落的收尾速度（忽略浮力）。 [3]

(d) The drag reading at $v = 8.0\ \mathrm{m\,s^{-1}}$ has an absolute uncertainty of $\pm 0.02\ \mathrm{N}$. Calculate its percentage uncertainty, and state one experimental reason the points might not lie exactly on a straight line.$v = 8.0\ \mathrm{m\,s^{-1}}$ 处的阻力读数绝对不确定度为 $\pm 0.02\ \mathrm{N}$。计算其百分比不确定度，并写出数据点可能不恰好共线的一个实验原因。 [3]

Extended Structured Problems长结构问题

Set up each problem with a clear free-body diagram and labelled axes. For collisions, write total momentum before equals total momentum after. Carry intermediate values to extra figures and round only the final answer.每题先画清晰的受力图并标注坐标轴。碰撞题写"碰前总动量等于碰后总动量"。中间值多保留几位，仅在最终答案处取舍有效数字。

Q8HARD Paper 2 collisions, energy, explosion碰撞、能量、爆炸 [12 marks]

A $1200\ \mathrm{kg}$ car moving at $18\ \mathrm{m\,s^{-1}}$ collides head-on into the rear of a stationary $800\ \mathrm{kg}$ car. The two vehicles lock together. Treat the road as horizontal and external friction during the brief impact as negligible.一辆 $1200\ \mathrm{kg}$ 的汽车以 $18\ \mathrm{m\,s^{-1}}$ 追尾撞上静止的 $800\ \mathrm{kg}$ 汽车，两车锁在一起。视路面水平，且碰撞短暂瞬间的外部摩擦可忽略。

(a) State the principle of conservation of momentum and the condition under which it applies.陈述动量守恒原理及其适用条件。 [2]

(b) Calculate the common velocity of the wreckage immediately after the collision.计算碰后残骸的共同速度。 [3]

(c) Calculate the kinetic energy before and after the collision, and hence the fraction of kinetic energy lost. Classify the collision.计算碰前与碰后的动能，由此求动能损失的比例，并判定碰撞类型。 [4]

(d) In a separate event a stationary $5.0\ \mathrm{kg}$ object explodes into two pieces: a $2.0\ \mathrm{kg}$ piece moving at $6.0\ \mathrm{m\,s^{-1}}$ and a $3.0\ \mathrm{kg}$ piece. Calculate the speed of the $3.0\ \mathrm{kg}$ piece and state its direction.在另一事件中，一个静止的 $5.0\ \mathrm{kg}$ 物体爆炸为两块：$2.0\ \mathrm{kg}$ 的一块以 $6.0\ \mathrm{m\,s^{-1}}$ 运动，另一块 $3.0\ \mathrm{kg}$。计算 $3.0\ \mathrm{kg}$ 块的速率并写出其方向。 [3]

Q9HARD Paper 2 circular motion: flat vs banked圆周运动：平直与倾斜弯道 [10 marks]

A $1100\ \mathrm{kg}$ car rounds a curve of radius $60\ \mathrm{m}$ at a constant speed of $20\ \mathrm{m\,s^{-1}}$.一辆 $1100\ \mathrm{kg}$ 的汽车以恒定速率 $20\ \mathrm{m\,s^{-1}}$ 通过半径 $60\ \mathrm{m}$ 的弯道。

(a) Calculate the centripetal acceleration and the centripetal force required.计算所需的向心加速度与向心力。 [3]

(b) The curve is flat (unbanked). Calculate the minimum coefficient of static friction needed, and explain why the result is independent of the car's mass.该弯道为平直（无倾斜）。计算所需的最小静摩擦系数，并解释结果为何与汽车质量无关。 [4]

(c) The curve is now banked so that no friction is required at this speed. Determine the banking angle.现将弯道倾斜，使在此速率下不需要摩擦力。求倾斜角。 [3]

Q10HARD Paper 2 HL ONLY $\Sigma F = \mathrm{d}p/\mathrm{d}t$, variable mass$\Sigma F = \mathrm{d}p/\mathrm{d}t$ 与变质量 [8 marks]

Sand falls vertically from a hopper onto a horizontal conveyor belt at a steady rate of $3.0\ \mathrm{kg\,s^{-1}}$. The belt moves horizontally at a constant $1.5\ \mathrm{m\,s^{-1}}$. The sand has no horizontal velocity before it lands.沙子从料斗竖直落到水平传送带上，速率恒为 $3.0\ \mathrm{kg\,s^{-1}}$。传送带以恒定 $1.5\ \mathrm{m\,s^{-1}}$ 水平运动。沙子落下前没有水平速度。

(a) Starting from $\Sigma F = \mathrm{d}p/\mathrm{d}t$, show that the horizontal force needed to drive the belt at constant speed is $F = v\,\dfrac{\mathrm{d}m}{\mathrm{d}t}$.从 $\Sigma F = \mathrm{d}p/\mathrm{d}t$ 出发，证明使传送带匀速运行所需的水平力为 $F = v\,\dfrac{\mathrm{d}m}{\mathrm{d}t}$。 [3]

(b) Calculate this horizontal force.计算该水平力。 [2]

(c) Calculate the rate at which the motor does work to accelerate the sand, and compare it with the rate of gain of kinetic energy of the sand. Comment on the difference.计算电机用于使沙子加速做功的功率，并与沙子动能的增长率比较。对两者的差异作出说明。 [3]

Forces and Momentum力与动量

Short Structured Items短结构题

Graph and Data Questions图像与数据题

Extended Structured Problems长结构问题