Motion, forces and energy

Length and volume

Use a ruler 直尺 to measure length. Read the scale with your eye straight in front of the mark to avoid a reading error.

The water an irregular solid pushes up gives its volume

Read a measuring cylinder at the bottom of the meniscus, with your eye level

Use a measuring cylinder 量筒 to measure the volume of a liquid. Read the scale at the bottom of the curved surface (the meniscus 弯月面), with your eye level with it.

Measuring small amounts

A single small length or a short time is hard to measure well. The trick is to measure many and divide 测多个再相除:

• To find the thickness of one page, measure 100 pages and divide by 100.
• To find the time for one swing of a pendulum 摆, measure the time for 20 swings and divide by 20. One full swing is called the period 周期.

This makes the uncertainty 不确定度 (the size of the error) much smaller.

Scalars and vectors

A scalar 标量 has size (magnitude 大小) only. A vector 矢量 has size and direction.

• Scalars: distance, speed, time, mass 质量, energy 能量, temperature 温度.
• Vectors: force 力, weight 重力, velocity 速度, acceleration 加速度, momentum 动量.

To add two vectors at right angles (90°), draw them as two sides of a rectangle. The resultant 合矢量 is the diagonal. Find its size with Pythagoras and its direction with trigonometry.

Two vectors at right angles add to a resultant $R$, the diagonal of the rectangle

Speed and velocity

Speed 速率 is the distance travelled per unit time.

Velocity is speed in a stated direction. So velocity is a vector but speed is a scalar.

Acceleration

Acceleration is the change in velocity per unit time.

Here $\Delta v$ means "the change in velocity". A deceleration 减速 (slowing down) is a negative acceleration.

Worked example. A car speeds up from $8\ \text{m/s}$ to $20\ \text{m/s}$ in $4.0\ \text{s}$. Find its acceleration.

Motion graphs

A distance–time graph 距离-时间图 shows how far an object has gone:

• A flat (horizontal) line means the object is at rest 静止.
• A straight slope means constant speed. The gradient 斜率 (steepness) is the speed.
• A curve that gets steeper means the object is speeding up.

The shape of a distance–time line: flat is at rest, a straight slope is constant speed, a rising curve is speeding up

A speed–time graph 速度-时间图 shows how fast an object is going:

• A flat line means constant speed.
• A straight slope means constant acceleration. The gradient is the acceleration.
• The area under the line 线下面积 is the distance travelled.

On a velocity–time graph the gradient is the acceleration and the area under the line is the distance travelled

Falling objects

Near the Earth, all objects speed up as they fall at the same rate. This is the acceleration of free fall 自由落体加速度, $g \approx 9.8\ \text{m/s}^2$.

When an object falls through air, air resistance 空气阻力 (a drag force) acts upward. As it speeds up, air resistance grows. When air resistance equals the weight, the resultant force is zero and the object stops speeding up. It then falls at a steady terminal velocity 终极速度.

At first the weight is bigger than the air resistance, so the object speeds up; at terminal velocity the two are equal and the speed is steady

Mass is the amount of matter 物质 in an object. It is measured in kilograms (kg) and does not change when you move the object.

Weight is the force of gravity on a mass. It is measured in newtons (N). Weight can change: it is smaller on the Moon because the Moon's gravity is weaker.

Gravitational field strength 重力场强度 is the force per unit mass:

This $g$ has the same value as the acceleration of free fall ($\approx 9.8\ \text{N/kg}$). You can compare masses with a balance 天平.

Density is the mass per unit volume.

The symbol $\rho$ is the Greek letter "rho". The unit is $\text{kg/m}^3$ or $\text{g/cm}^3$.

To find density: measure the mass with a balance, find the volume, then divide.

• Regular solid 规则固体 (like a box): measure the sides and calculate the volume.
• Irregular solid 不规则固体 (a strange shape): lower it into water in a measuring cylinder. The rise in water level is its volume. This is the displacement method 排水法.

An object floats 漂浮 if its density is less than the density of the liquid. It sinks if its density is greater.

Worked example. A stone of mass $54\ \text{g}$ is lowered into a measuring cylinder. The water level rises from $20\ \text{cm}^3$ to $40\ \text{cm}^3$. Find the density of the stone.

The volume of the stone is the rise in water level, $40 - 20 = 20\ \text{cm}^3$, so

A force is a push or a pull. A force can change the shape 形状, the speed, or the direction of an object.

Stretching (Hooke's law)

When you hang a load on a spring, it stretches. The stretch is called the extension 伸长量.

On a load–extension graph 载荷-伸长图 the line is straight at first: extension is proportional to load. The point where the line stops being straight is the limit of proportionality 比例极限.

Load is proportional to extension until the limit of proportionality, then the line curves

The spring constant 弹簧常数 is the force per unit extension:

A large $k$ means a stiff spring.

Resultant force and Newton's laws

Add forces on a straight line to get the resultant force 合力 (forces one way are positive, the other way negative).

• If the resultant force is zero, the object stays at rest or keeps moving in a straight line at constant speed. (Newton's first law.)
• If the resultant force is not zero, the object accelerates in the direction of the force:

A free-body diagram shows every force on the box as a labelled arrow

Worked example. A $1200\ \text{kg}$ car has a $3000\ \text{N}$ driving force and $600\ \text{N}$ of friction. Find its acceleration.

First find the resultant force: $3000 - 600 = 2400\ \text{N}$. Then

Friction

Friction 摩擦力 is the force between two surfaces that touch. It tries to stop motion and makes things heat up. Drag (friction in a liquid or gas, like air resistance) also slows objects down.

Moments — the turning effect

The moment 力矩 of a force is its turning effect about a pivot 支点.

The unit is the newton metre (N m).

Principle of moments 力矩原理: when an object is balanced (in equilibrium 平衡),

An object is in equilibrium when there is no resultant force and no resultant moment.

The beam balances when the clockwise moment equals the anticlockwise moment

Worked example. A $30\ \text{N}$ weight sits $0.20\ \text{m}$ to the left of a pivot. How far to the right must a $20\ \text{N}$ weight sit to balance the beam?

Balanced, so anticlockwise moment $=$ clockwise moment:

Centre of gravity

The centre of gravity 重心 is the single point where all the weight of an object seems to act.

For a flat shape (lamina 薄片), hang it from a pin and let it settle; draw a vertical line down from the pin using a plumb line. Repeat from another point. The centre of gravity is where the lines cross.

An object is more stable 稳定 when its centre of gravity is low and its base is wide. It tips over when the centre of gravity passes outside the base.

Momentum is mass times velocity. It is a vector.

Impulse 冲量 is the force times the time it acts, and it equals the change in momentum:

So the resultant force is the change in momentum per unit time:

Conservation of momentum 动量守恒: when objects collide and no outside force acts, the total momentum before equals the total momentum after.

(Here $u$ is a velocity before and $v$ is a velocity after.)

Worked example. A $2.0\ \text{kg}$ trolley moving at $3.0\ \text{m/s}$ hits a stationary $1.0\ \text{kg}$ trolley and they stick together. Find their common velocity afterwards.

Total momentum is conserved, so

Energy stores

Energy can be stored 储存 in different ways: kinetic 动能, gravitational potential 重力势能, chemical 化学能, elastic (strain) 弹性势能, nuclear 核能, electrostatic 静电能, and internal (thermal) 内能.

Energy is transferred 转移 between stores by forces (mechanical work), by electric currents, by heating, and by waves (such as light and sound).

Wind turbines transfer energy from the kinetic store of the moving air to electricity

Kinetic and potential energy

Kinetic energy is the energy of a moving object:

The change in gravitational potential energy when an object goes up or down by a height $\Delta h$:

Conservation of energy

The principle of conservation of energy 能量守恒定律 says energy is never made or destroyed; it only moves between stores. A falling object turns gravitational potential energy into kinetic energy. A Sankey diagram 桑基图 shows how the input energy splits into useful energy and wasted 浪费 energy.

Energy moves between gravitational potential energy (at the highest points) and kinetic energy (at the lowest point)

On a roller-coaster, gravitational potential energy stored at the top of the hill becomes kinetic energy as the cars speed down

Worked example. A $0.50\ \text{kg}$ ball is dropped from a height of $1.8\ \text{m}$. Ignoring air resistance, find its speed just before it lands. (Take $g = 10\ \text{m/s}^2$.)

All the gravitational potential energy becomes kinetic energy, so $\tfrac{1}{2}mv^2 = mg\,\Delta h$. The mass cancels, leaving $v^2 = 2g\,\Delta h$:

Because the mass cancels, every object dropped from this height would land at the same speed.

Work

Work done 做功 equals the energy transferred. When a force moves an object:

The unit of work and energy is the joule 焦耳 (J).

Power

Power 功率 is the work done (or energy transferred) per unit time.

The unit is the watt 瓦特 (W). $1\ \text{W} = 1\ \text{J/s}$.

Efficiency

Efficiency 效率 tells you how much of the input energy becomes useful energy.

Efficiency is always less than 100% because some energy is always wasted (usually as heat).

Worked example. A motor is supplied with $200\ \text{J}$ of electrical energy and lifts a load, giving it $150\ \text{J}$ of gravitational potential energy. Find its efficiency.

A Sankey diagram for this motor: the band widths are proportional to the energy, so the 200 J input splits into 150 J useful and 50 J wasted as heat

Energy resources

We generate electricity from many energy resources. Most spin a turbine that drives a generator – often by boiling water to make steam (fossil fuels, nuclear fuel, geothermal, biofuels), or directly by moving water or air (hydroelectric, waves, tides, wind). Solar cells make electricity from sunlight directly; solar panels heat water.

Most of these trace back to the Sun (fossil fuels are ancient stored sunlight; wind and waves come from solar heating) – the exceptions are geothermal, nuclear, and tidal energy. The Sun itself is powered by nuclear fusion 核聚变, joining hydrogen nuclei into helium.

Pressure is the force per unit area.

The unit is the pascal 帕斯卡 (Pa). $1\ \text{Pa} = 1\ \text{N/m}^2$.

A small area gives a large pressure (a sharp knife cuts well). A large area gives a small pressure (snowshoes stop you sinking).

Pressure in a liquid

In a liquid, pressure increases with depth 深度 and with the liquid's density:

This is why a dam is built thicker at the bottom, where the water pressure is greatest. Pressure in a liquid acts in all directions.

The deeper the liquid, the greater the pressure, so the outward push on the wall grows with depth

A real dam is built much thicker at the bottom, because the water pushes hardest where it is deepest

Worked example. Find the extra pressure at a depth of $2.0\ \text{m}$ in water (density $1000\ \text{kg/m}^3$, $g = 10\ \text{N/kg}$).

• On a distance–time graph the gradient is the speed; on a speed–time graph the gradient is the acceleration and the area under the line is the distance travelled. Never read one graph as if it were the other.
• Mass (kg) is the amount of matter and is the same everywhere; weight (N) is the force of gravity, $W = mg$. On the Moon your mass is unchanged but your weight is smaller.
• Speed is a scalar; velocity is a vector. Something moving at steady speed around a curve is still accelerating, because its direction keeps changing.
• In Hooke's law, load is proportional to extension only up to the limit of proportionality. Use the extension (stretched length − original length), never the whole length.
• Momentum $p = mv$ is conserved in a collision. Give each velocity a $+$ or $-$ sign for its direction before you add them.
• Convert units first (cm → m, g → kg). Use $p = F/A$ for a solid pressing on a surface, but $p = \rho g h$ for the pressure inside a liquid — they are different formulas.