The physics behind engineering and the physical sciences: motion, forces, energy, momentum, torque, circuits, induction, waves and heat, with Python exercises and a home pendulum lab.

Physics for Engineering & Science

## Why this matters

Forces and accelerations tell you what happens from moment to moment. Energy lets you skip the middle and jump straight from the start to the end. How fast is a roller coaster at the bottom of a drop? How much electricity does a pump need to lift water to a tank? How much heat do brakes have to absorb? Energy answers these without tracking every instant, and conservation of energy is one of the most reliable rules in all of physics.

## Work

When a force pushes an object through a displacement, it does **work**:

```text
W = F · d · cos θ
```

F is the force, d is the distance moved, and θ is the angle between the force and the direction of motion. Work is measured in **joules** (J), and 1 J = 1 N·m.

- Pushing a cart 5 m with 20 N along the direction of motion: W = 20 × 5 = 100 J.
- Pulling a sled with 20 N on a rope at 60° above the ground for 5 m: W = 20 × 5 × cos 60° = 50 J. Only the part of the force along the motion does work.
- A force perpendicular to the motion does no work (cos 90° = 0). The normal force on a car driving on a flat road does no work.
- Friction usually points against the motion (θ = 180°), so it does **negative** work: it takes energy out of the motion and turns it into heat.

## Kinetic and potential energy

**Kinetic energy** is the energy of motion:

```text
KE = ½ · m · v²
```

A 1,000 kg car at 20 m/s has KE = ½ × 1000 × 20² = 200,000 J = 200 kJ. Because v is squared, the same car at 40 m/s has 800 kJ: four times as much energy for the brakes to absorb.

**Gravitational potential energy** is energy stored by height:

```text
PE = m · g · h
```

h is height above a reference level you choose. Only changes in height matter, so pick whatever reference is convenient (often the lowest point).

A stretched or compressed spring also stores energy, ½·k·x², where k is the spring constant and x the stretch.

## The work-energy theorem

The net work done on an object equals its change in kinetic energy:

```text
W_net = ΔKE = ½·m·v² − ½·m·v₀²
```

This is Newton's second law rewritten in energy terms. It is often the fastest route to a speed.

## Conservation of energy

If only gravity (and springs) do work, the total mechanical energy stays constant:

```text
KE₀ + PE₀ = KE + PE
```

When friction or air drag act, some energy leaves the motion as heat. Energy is still conserved overall; we just have to account for it:

```text
KE₀ + PE₀ = KE + PE + W_friction
```

where W_friction is the energy lost to friction (a positive number).

## Worked example 1: a skateboarder

A 60 kg skateboarder starts from rest at the top of a ramp 4.0 m high. Ignoring friction, how fast is she going at the bottom?

```text
m·g·h = ½·m·v²     →     v = √(2·g·h) = √(2 × 9.81 × 4.0) = 8.86 m/s
```

The mass cancels: a heavy and a light skater reach the same speed (without friction). The shape of the ramp does not matter either, only the height.

## Worked example 2: with friction

Now suppose friction and air drag take 500 J out of her motion on the way down.

```text
PE at top      = 60 × 9.81 × 4.0  = 2354 J
energy left    = 2354 − 500       = 1854 J  (all kinetic at the bottom)
v              = √(2 × 1854 / 60) = 7.86 m/s
```

## Power

**Power** is the rate of doing work, measured in **watts** (W): 1 W = 1 J/s.

```text
P = W / t
```

Lifting a 50 kg load 3.0 m takes W = 50 × 9.81 × 3.0 = 1,472 J. Doing it in 10 s needs 147 W; doing it in 2 s needs 736 W. The work is the same; the power is not. Motors, engines and people are rated by power. Real machines are less than 100% efficient, so the input power is always larger than the useful output.

## In code

```python
import math

def speed_at_bottom(height, g=9.81):
    return math.sqrt(2 * g * height)

print(round(speed_at_bottom(4.0), 2))   # 8.86
```

## Common mistakes

- **Forgetting to square v.** KE is ½mv², not ½mv.
- **Mixing kJ and J,** or grams and kilograms. Convert everything to SI first.
- **Treating work as a vector.** Work and energy are scalars: they have a size and sign, but no direction.
- **Ignoring losses.** Real systems lose energy to friction, drag and heat. Frictionless results are an upper bound on speed.

## Quick recap

- W = F·d·cos θ, in joules.
- KE = ½mv², PE = mgh.
- Net work = change in KE.
- Total energy is conserved; friction converts some of it to heat.
- Power = work per unit time, in watts.

## Try it

In the **Do** tab, use an energy balance to find the speed at the bottom of a ramp, with and without friction losses.


Calculate work, kinetic and potential energy and power, and apply conservation of energy with and without friction.

Work and energy

Work, Energy, and Power: Crash Course Physics #9

## Why this matters

Collisions happen too quickly, and with forces too complicated, to track with F = m·a moment by moment. Momentum lets you ignore what happens during the crash and still predict what comes out of it. Crash reconstruction, the design of airbags and crumple zones, rocket propulsion, particle physics and even pool shots all rest on one fact: in an isolated system, total momentum does not change.

## Momentum

The **momentum** of an object is its mass times its velocity:

```text
p = m · v        (units: kg·m/s)
```

Momentum is a vector: it points the same way as the velocity. In one dimension, a sign is enough. A 1,000 kg car moving east at 20 m/s has p = +20,000 kg·m/s; the same car moving west has −20,000 kg·m/s.

## Impulse

A force acting for a time changes momentum. The product is called **impulse**:

```text
J = F · Δt = Δp
```

(For a force that changes during the impact, F is the average force.) The same change in momentum can come from a big force for a short time or a smaller force for a longer time.

A 0.145 kg baseball arrives at 40 m/s and is stopped by a glove in 0.020 s:

```text
Δp = 0.145 × 40 = 5.8 kg·m/s
F  = 5.8 / 0.020 = 290 N
```

If the catcher lets the glove move back so the stop takes 0.040 s, the average force halves to 145 N. Airbags, crumple zones, helmet foam and bending your knees when you land all work this way: they stretch out Δt to cut the force.

## Conservation of momentum

When two objects collide, Newton's third law says they push on each other with equal and opposite forces for the same time. So the impulse on one is exactly the opposite of the impulse on the other, and the **total** momentum is unchanged:

```text
m₁·v₁ + m₂·v₂ = m₁·v₁' + m₂·v₂'
```

(Primes mean "after the collision".) This holds as long as outside forces such as friction are small during the brief collision, which is usually a good approximation.

## Types of collisions

Momentum is always conserved in a collision. Kinetic energy is not always conserved:

- **Elastic:** kinetic energy is conserved too. Close to true for billiard balls and for collisions between atoms.
- **Inelastic:** some kinetic energy turns into heat, sound and deformation.
- **Perfectly inelastic:** the objects stick together and move off as one. This loses the most kinetic energy that momentum conservation allows.

## Worked example 1: two cars lock together

A 1,000 kg car moving at 20 m/s rear-ends a 1,500 kg car stopped at a light. The bumpers lock. How fast do they move just after impact?

```text
before:  1000 × 20 + 1500 × 0 = 20,000 kg·m/s
after:   (1000 + 1500) × v'   = 20,000
         v' = 20,000 / 2,500  = 8.0 m/s
```

Kinetic energy check:

```text
before:  ½ × 1000 × 20²  = 200,000 J
after:   ½ × 2500 × 8.0² =  80,000 J
```

60% of the kinetic energy, 120 kJ, went into crumpling metal, heat and sound. Crash reconstruction works backwards from skid marks and final positions using exactly these two ideas, plus a lot of careful measurement.

## Worked example 2: an elastic collision

For a one-dimensional elastic collision, combining momentum and kinetic-energy conservation gives:

```text
v₁' = ((m₁ − m₂)·v₁ + 2·m₂·v₂) / (m₁ + m₂)
v₂' = ((m₂ − m₁)·v₂ + 2·m₁·v₁) / (m₁ + m₂)
```

Two equal billiard balls (m₁ = m₂), the first moving at 2.0 m/s and the second at rest: v₁' = 0 and v₂' = 2.0 m/s. The moving ball stops dead and the other takes all its velocity, which is exactly what you see in a straight pool shot.

## In code

```python
def total_momentum(masses, velocities):
    return sum(m * v for m, v in zip(masses, velocities))

print(total_momentum([1000, 1500], [20, 0]))   # 20000
```

Writing the before and after totals as numbers you can compare is a good habit: if they differ, you have made a sign or arithmetic error.

## Common mistakes

- **Dropping the signs.** Objects moving in opposite directions have momenta of opposite sign. A head-on collision of equal cars at equal speeds has zero total momentum.
- **Assuming kinetic energy is conserved.** Only in elastic collisions. Momentum is the reliable one.
- **Using conservation when a big outside force acts.** A car hitting a wall is not isolated: the wall (and the Earth) take the momentum.
- **Confusing momentum and kinetic energy.** p = mv is a vector and grows with v; KE = ½mv² is a scalar and grows with v².

## Quick recap

- p = m·v; impulse F·Δt = Δp.
- Longer stopping time means smaller average force.
- In a collision, total momentum before equals total momentum after.
- Perfectly inelastic: v' = (m₁v₁ + m₂v₂) / (m₁ + m₂).
- Kinetic energy is conserved only in elastic collisions.

## Try it

In the **Do** tab, write the function for a perfectly inelastic collision and use it to measure how much kinetic energy is lost.


Use impulse and conservation of momentum to solve elastic and perfectly inelastic collisions.

Momentum and collisions

Collisions: Crash Course Physics #10

Elastic and Inelastic Collisions

## Why this matters

Many of the things engineers build turn: shafts, gears, wheels, turbines, motors, doors and wrenches. Even structures that do not move, such as beams, cantilevers and cranes, must not *start* turning, so their designers balance **moments** (another name for torques) as carefully as forces. This lesson covers the rotational versions of the ideas you already know: position, velocity, acceleration and force.

## Describing rotation

- **Angle** θ, in **radians**. One full turn is 2π rad (360°).
- **Angular velocity** ω, in rad/s: how fast the angle changes.
- **Angular acceleration** α, in rad/s².

Machines are usually rated in revolutions per minute (rpm). To convert:

```text
ω (rad/s) = rpm × 2π / 60
3000 rpm  = 3000 × 2π / 60 = 314 rad/s
```

A point at distance r from the axis moves along its circle at speed v = r·ω. The rim of a 0.30 m radius wheel turning at 10 rad/s moves at 3.0 m/s. The constant-acceleration equations from the kinematics lesson carry over directly, with θ, ω and α in place of x, v and a.

## Torque

A force makes something rotate only if it acts at a distance from the pivot and has a part perpendicular to the lever arm. **Torque** measures that turning effect:

```text
τ = r · F · sin θ
```

- r is the distance from the pivot to where the force acts.
- θ is the angle between the lever arm and the force.
- The units are **newton-metres** (N·m). (Joules are also N·m, but torque is not energy; always write torque as N·m.)

Equivalently, τ = F × (perpendicular distance from the pivot to the force's line of action). That perpendicular distance is the **moment arm**.

By convention, counterclockwise torques are positive and clockwise torques negative.

### Worked example 1: a wrench

You pull with 40 N on a wrench, 0.25 m from the bolt.

```text
pulling at 90° to the handle:  τ = 0.25 × 40 × sin 90° = 10 N·m
pulling at 30° to the handle:  τ = 0.25 × 40 × sin 30° =  5 N·m
pulling along the handle:      τ = 0.25 × 40 × sin 0°  =  0 N·m
```

A longer wrench gives more torque for the same pull, which is the whole point of a breaker bar. Bolts on real equipment have specified tightening torques from the manufacturer and are tightened with a calibrated torque wrench; do not estimate them by feel.

## Rotational equilibrium

An object is in **static equilibrium** when it is not accelerating in any direction and not starting to rotate:

```text
ΣF = 0     and     Στ = 0   (about any pivot you choose)
```

### Worked example 2: a seesaw

A 30 kg child sits 2.0 m from the pivot of a seesaw. Where must a 40 kg child sit on the other side to balance it?

```text
counterclockwise:  30 × 9.81 × 2.0 = 588.6 N·m
clockwise:         40 × 9.81 × d
balance:           40 × 9.81 × d = 588.6   →   d = 1.5 m
```

The g cancels, so in practice you can balance mass × distance: 30 × 2.0 = 40 × 1.5. The heavier child sits closer.

Choosing the pivot at an unknown force removes that force from the torque equation, because its moment arm is zero. This trick makes beam and bracket problems much easier, and you will use it constantly in statics.

## Newton's second law for rotation

The rotational version of ΣF = m·a is:

```text
Στ = I · α
```

I is the **moment of inertia** (kg·m²): rotational mass. It depends not only on the mass but on how far that mass sits from the axis. A solid disc of mass M and radius R has I = ½·M·R²; a thin ring of the same mass and radius has I = M·R², twice as much, because all of its mass is at the rim. That is why flywheels put their mass far from the axis.

## Rotational power

For a rotating shaft, power is torque times angular velocity:

```text
P = τ · ω
```

A motor delivering 10 N·m at 3000 rpm (314 rad/s) supplies 10 × 314 ≈ 3,140 W, about 3.1 kW. This relation is how motor and engine data sheets link torque, speed and power.

## In code

```python
import math

def rpm_to_rad_s(rpm):
    return rpm * 2 * math.pi / 60

print(round(10 * rpm_to_rad_s(3000)))   # 3142 (watts)
```

## Common mistakes

- **Using the full length when the force is at an angle.** Only the perpendicular part counts: multiply by sin θ, or use the moment arm.
- **Forgetting sign conventions.** Decide which way is positive before you add torques.
- **Mixing rpm and rad/s.** Formulas such as v = r·ω and P = τ·ω need rad/s.
- **Writing torque in joules.** Use N·m.

## Engineering note

Balancing moments on a seesaw is physics. Sizing a real beam, crane, shaft or bolted joint involves materials, fatigue, dynamic loads, codes and safety factors, and anything that affects public safety must be reviewed and signed off by a licensed engineer.

## Quick recap

- τ = r·F·sin θ, in N·m.
- Equilibrium: ΣF = 0 and Στ = 0.
- Στ = I·α; mass far from the axis means a large I.
- P = τ·ω, with ω in rad/s.

## Try it

In the **Do** tab, write the torque function and use it to balance a seesaw.


Compute torques and moments, balance them for equilibrium, and link torque, angular speed and power.

Rotation and torque

Torque: Crash Course Physics #12

Rotational Motion: Crash Course Physics #11

Use conservation of energy and momentum to skip the details of a motion, and extend the ideas to rotation, torque and shaft power.