力

Scalar 标量: magnitude only — distance, speed, mass, energy. Vector 矢量: magnitude and direction — displacement, velocity, force, weight, momentum. A vector is drawn as an arrow: length = magnitude, direction = direction.

A force is a push or pull from the interaction with another object:

• contact 接触 forces (touching): friction, air resistance, tension, normal contact force;
• non-contact 非接触 forces (separated): gravitational, electrostatic, magnetic.

Gravity: weight 重力 is the force on an object due to gravity; it acts at the centre of mass 质心 and is measured with a calibrated spring-balance (newtonmeter):

• $W$ weight in N; $m$ mass in kg; $g$ gravitational field strength in N/kg (given, usually 9.8 near Earth).
• Weight and mass are directly proportional ($W \propto m$).

Resultant force 合力: the single force replacing several forces with the same effect. Collinear: add same-direction forces, subtract opposite ones. (HT) Use free-body diagrams 自由体图 (only the forces on the chosen object), resolve a force into perpendicular components, and find resultants by scale drawing.

Worked example. A 65 kg person stands on Mars where $g = 3.7$ N/kg.

Worked scale drawing (HT; teacher-written). Use 1 cm for 10 N. Draw 4.0 cm east, then 3.0 cm north. The resultant joins the first tail to the last head: 5.0 cm represents 50 N, about 37° north of east. The equilibrant has equal magnitude in the opposite direction.

Exam demand. AQA June2025 8463/2H Q05.5 uses 240 N upwards and 200 N left. A scale triangle or parallelogram gives about 310 N, 40° left of vertical (official ranges 300–320 N and 38–42°). The diagram, arrow directions and scale are part of the method.

More than one force is needed to stretch, bend or compress a stationary object (a single force would just move it). Elastic deformation 弹性形变 is recovered when the forces are removed; inelastic 非弹性 is not.

Below the limit of proportionality:

• $k$ spring constant in N/m (stiff spring → large $k$); $e$ extension = stretched length − original length (or compression).
• Work done on the spring = elastic energy stored (if not inelastically deformed).

Required practical 6: hang masses on a spring, measure extension for each (ruler at eye level), plot force against extension. The linear section's gradient is $k$; beyond the limit of proportionality the line curves. Hooke's-law reasoning: doubling the force doubles the extension only below the limit.

Worked example (AQA June2025 Q02.7). A force of 4.0 N produces extension 0.064 m. Choose $F=ke$ in the proportional region, then rearrange:

A plot of total length has a non-zero intercept because the unloaded spring has a non-zero length. A curve away from the proportional line means $F$ and $e$ are no longer proportional; unload the spring to test for permanent deformation. Secure the stand, limit loading, keep the ruler vertical and close, and use a pointer at eye level.

• $M$ moment 力矩 in N·m; $d$ is the perpendicular distance from the pivot to the line of action of the force.
• Balance: total clockwise moment = total anticlockwise moment.

Levers and gears transmit the rotational effect of a force. A longer lever arm produces a larger moment for the same perpendicular force. In an ideal pair of meshed gears, the teeth exert equal forces at the contact: a larger driven gear turns more slowly with a larger moment. The meshed gears turn in opposite directions.

Worked example. Choose the pivot and equate clockwise and anticlockwise moments:

For AQA June2024 Q02.6, convert the perpendicular distance 7.5 cm to 0.075 m: $M=Fd=2.0\ \text{N}\times0.075\ \text{m}=0.15\ \text{N\,m}$. In a pair of meshed gears, the teeth produce a force and moment on the other gear; adjacent gears rotate in opposite directions.

• $p$ pressure in Pa; $F$ force normal to the surface; $A$ area in m².
• (HT) $h$ column height in m, $\rho$ liquid density in kg/m³. Pressure grows with depth and density.
• A submerged object feels greater pressure on its bottom than its top → a resultant upthrust 浮力. Floating at rest: upthrust = weight. If weight initially exceeds upthrust, a released object accelerates downwards; a sinking object can later move at constant speed when upthrust plus drag balances its weight.
• Atmospheric pressure decreases with height: fewer air molecules above a surface as you climb, so less weight of air; the atmosphere gets less dense with altitude.

Worked example (teacher-written; HT). A 2.0 m water column has density 1000 kg/m³; $g=9.8$ N/kg. Its pressure, additional to that at the free surface, is:

Floating at rest requires a complete force balance. A sinking object can reach constant velocity when upthrust + drag = weight; sinking does not always mean downward acceleration.

• Distance 路程 (scalar): how far. Displacement 位移 (vector): straight-line distance and direction.
• Speed 速率 (scalar) — typical values: walking ≈ 1.5 m/s, running ≈ 3 m/s, cycling ≈ 6 m/s, sound in air ≈ 330 m/s. Velocity 速度 (vector): speed in a given direction.
• $s = vt$ (constant speed); average speed = total distance ÷ total time.
• Distance–time graph: gradient = speed; (HT) a tangent gives instantaneous speed of an accelerating object.
• Acceleration 加速度: $a = \Delta v / t$, in m/s²; deceleration means slowing down. With the initial direction chosen positive, its acceleration is negative. Estimate everyday accelerations.
• Velocity–time graph: gradient = acceleration; (HT) signed area gives displacement. Add the magnitudes of areas above and below zero to find total distance. If velocity stays positive, area also gives distance.
• Uniform acceleration: $v^2 - u^2 = 2as$. Free fall near Earth: $a \approx 9.8$ m/s².

Worked example (graph). A v–t graph rises straight from 0 to 20 m/s in 8 s, then stays flat for 12 s.

• Acceleration (gradient): $a=\Delta v/\Delta t=(20-0)/8=2.5$ m/s².
• (HT) Positive-velocity areas: $s=s_1+s_2=\tfrac12\Delta t_1v+v\Delta t_2=\tfrac12\times8\times20+20\times12=320$ m.

Terminal velocity 末速度: a falling object accelerates (weight > drag 空气阻力); as speed grows, drag grows until resultant force = 0 — constant speed = terminal velocity. Skydiver: fast terminal before the chute, slow after; interpret the v–t curve shape.

Worked tangent example (HT; teacher-written). At a chosen instant, a tangent to a distance–time curve passes through (2 s, 3 m) and (6 s, 15 m):

This is instantaneous speed at the point of tangency. A chord over a time interval instead gives an average rate.

Exam demand. AQA June2025 Q05.2 gives mean acceleration 0.64 m/s² from rest for 2.5 minutes. Convert time to 150 s, then:

Q05.3 needs the linked terminal-velocity explanation: speed rises → drag rises → drag equals weight → resultant and acceleration become zero. On opening a parachute, drag initially exceeds weight: upward acceleration slows the still downward-moving skydiver.

• First law: zero resultant force → stationary stays stationary; moving keeps the same velocity. For motion in a straight line at steady speed, driving force = resistive forces. (HT) Inertia 惯性: the tendency to keep the state of motion.
• Second law: $a \propto F$, $a \propto 1/m$, so:

(HT) Inertial mass = force ÷ acceleration — resistance to change of velocity.

Required practical 7: trolley on a runway — vary the driving force by transferring masses from the trolley to its hanging holder, keeping the total moving mass constant. For the combined trolley–hanger system, the driving force is the hanger's weight when resistance is negligible or compensated; the string tension on the trolley is a different force. Then keep hanger mass constant and add mass to the trolley. Measure acceleration with light gates; plot $a$ against driving force at fixed total mass, or $a$ against $1/m$ where $m$ is total moving mass.

• Third law: two interacting objects exert equal and opposite forces on each other — same type, opposite directions, on different objects.

Worked uncertainty example (AQA June2024 Q05.4). Three accelerations are 1.36, 1.39 and 1.33 m/s². The range is 0.06 m/s²; using half the range, report uncertainty ±0.03 m/s². Repeats reveal spread; a mean reduces random variation but does not remove a common calibration error.

From rest under uniform acceleration, acceleration can also be found from distance and time: average speed is $s/t$, final speed is twice the average, and acceleration is final speed divided by time. State the rest/uniform-acceleration assumptions.