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Measurement, Units and Vectors

Physics describes nature with numbers. Before any law can be used, every quantity needs a unit, a sensible size and, when it has a direction, a vector. This chapter builds that toolkit, and every later chapter uses it.

What you will learn: 1. What makes physics an experimental science, and what a physical quantity is. 2. The International System of Units (SI), its seven base units, and the prefixes from nano to giga. 3. How to estimate orders of magnitude, and how to check any equation with dimensional analysis. 4. Vectors: how they differ from scalars, how to write them with unit vectors and components, and how to add and multiply them. A thread to follow: every number you write in physics has three parts. - a value - a unit - for vectors, a direction Drop any one of them and the answer is wrong, even if the arithmetic is right.

The Scientific Method and Physical Quantities

Physics is an experimental science. A theory is accepted only while its predictions agree with measurement, and one reproducible experiment that disagrees is enough to overturn it.

The scientific method, as a cycle 1. Observe a phenomenon. 2. Formulate a hypothesis, ideally as a mathematical model. 3. Use the model to predict the result of a new experiment. 4. Do the experiment and measure. 5. Compare. If prediction and measurement agree within experimental uncertainty, the model survives, for now. If not, revise it. Key ideas - A model is a simplified description. A falling apple treated as a point particle with no air resistance is a model, useful within its range of validity. - A physical law summarises many experiments (for example Newton's second law). A theory is a broader framework that explains many laws. - No theory is proved true for ever. Newton's mechanics works for everyday speeds, but fails near the speed of light and at atomic scales. Physical quantities - A physical quantity is anything that can be measured: length, time, mass, temperature, force. - Measuring means comparing the quantity with a standard of the same kind, the unit. "The table is 1.5 m long" means it is 1.5 times the metre standard. - Every measurement has an uncertainty. Writing 1.50 m claims precision to the centimetre; 1.5 m claims only a tenth of a metre. Significant figures express this. Significant figures, the rules - Multiplication and division: the result has as many significant figures as the least precise factor. - Addition and subtraction: the result keeps as many decimal places as the least precise term. Exam trap: a calculator answer like 3.14159265 m for a length measured to the millimetre is wrong. Round to the precision your data support.

SI Units, Prefixes and Scales

The SI is built on seven base units. Every other unit (newton, joule, pascal) is derived from them, which is why a correct equation always balances in base units.

The seven SI base units - length: metre (m) - mass: kilogram (kg) - time: second (s) - electric current: ampere (A) - temperature: kelvin (K) - amount of substance: mole (mol) - luminous intensity: candela (cd) How the standards are defined today - Since 2019 all base units are fixed by defined constants of nature. - The second: from the frequency of a caesium-133 atomic transition. - The metre: the distance light travels in vacuum in 1/299 792 458 of a second, so the speed of light is exact by definition. - The kilogram: fixed through Planck's constant (it used to be a platinum-iridium cylinder in Paris). Derived units, examples - velocity: m/s - acceleration: m/s² - force: newton, 1 N = 1 kg·m/s² - pressure: pascal, 1 Pa = 1 N/m² - energy: joule, 1 J = 1 N·m = 1 kg·m²/s² Unit prefixes (powers of ten) - giga G = 10⁹ - mega M = 10⁶ - kilo k = 10³ - centi c = 10⁻² - milli m = 10⁻³ - micro μ = 10⁻⁶ - nano n = 10⁻⁹ - pico p = 10⁻¹² Unit conversion, the safe method 1. Write the quantity with its unit. 2. Multiply by a conversion factor equal to 1, for example (1 km / 1000 m). 3. Cancel units like algebra. Example: 72 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 20 m/s. Scales in nature and orders of magnitude - The order of magnitude is the power of ten nearest to a value. It lets you check whether an answer is sensible. - Sizes: proton about 10⁻¹⁵ m; atom about 10⁻¹⁰ m; red blood cell about 10⁻⁵ m (7-8 μm); human about 10⁰ m; Earth radius about 10⁷ m. - Times: a heartbeat about 1 s; a human life about 2 × 10⁹ s. Clinical tip: medicine lives on prefixes. Doses in mg and μg, cell sizes in μm, wavelengths in nm, and the order of magnitude catches the dangerous error of a factor of 1000 between mg and μg. Exam trap: km/h to m/s means divide by 3.6, not multiply.

Dimensional Analysis

Every physical equation must be dimensionally consistent: the two sides, and every term that is added, must have the same dimensions. This single check catches a large share of algebra mistakes.

Dimensions versus units - A dimension is the kind of quantity: length [L], mass [M], time [T]. - A unit is a particular size of that dimension: metre, foot, millimetre are all [L]. Dimensions of common quantities - velocity: [L][T]⁻¹ - acceleration: [L][T]⁻² - force: [M][L][T]⁻² - energy: [M][L]²[T]⁻² The rules 1. You can add or subtract only quantities with the same dimensions (you cannot add metres to seconds). 2. Both sides of an equation must have the same dimensions. 3. Arguments of functions like sin, cos, exp and log must be dimensionless. Angles in radians are dimensionless (arc length divided by radius). Worked check: x = x₀ + v₀t + ½at² - x and x₀: [L] - v₀t: [L][T]⁻¹ × [T] = [L] - ½at²: [L][T]⁻² × [T]² = [L] All terms are [L], so the equation is consistent. (The ½ is a pure number and has no dimensions.) Using dimensions to find a formula - Suppose the period T of a pendulum depends on its length l and on g. Try T = k·lᵃ·gᵇ. - Dimensions: [T] = [L]ᵃ × ([L][T]⁻²)ᵇ, so a + b = 0 and −2b = 1, giving b = −½, a = ½. - So T = k·√(l/g). Experiment gives k = 2π. Limits of the method - It cannot give pure numbers such as 2π or ½. - A dimensionally correct equation can still be wrong. A dimensionally wrong equation is always wrong. Exam focus: when you are unsure of a formula in an exam, check its dimensions before using it.

Vectors

Many quantities (displacement, velocity, force) need a direction as well as a size. Vectors are the language for them, and the rules for adding and multiplying vectors are not the rules of ordinary numbers.

This section covers: - scalars versus vectors, notation, magnitude and the unit vector (versor) - components in a Cartesian frame, and adding vectors graphically and by components - the two products: the scalar (dot) product, which gives a number, and the vector (cross) product, which gives a vector Every kinematics and dynamics problem later reduces to these operations. It is worth practising them until they are automatic.

Scalars, Vectors and Unit Vectors

A scalar is fully described by one number with a unit; a vector needs a magnitude and a direction. A unit vector (versor) has magnitude 1 and carries only direction.

Scalars - Examples: mass, time, temperature, energy, speed, distance. - They add like ordinary numbers. Vectors - Examples: displacement, velocity, acceleration, force. - Written in bold (A) or with an arrow over the letter. The magnitude is written |A| or simply A, and is always positive or zero. - Drawn as an arrow: the length is proportional to the magnitude, the arrowhead gives the direction. - Two vectors are equal if they have the same magnitude and direction, wherever they are drawn. A vector can be moved parallel to itself without changing it. - The negative vector −A has the same magnitude and the opposite direction. Displacement versus distance - Displacement is a vector from start point to end point. It ignores the path taken. - Distance travelled is a scalar along the path. - Walk once around a running track: distance 400 m, displacement zero. Multiplying a vector by a scalar - cA has magnitude |c|·A. - If c is positive, the direction is the same; if c is negative, the direction is reversed. The unit vector (versor) - Any vector can be written as its magnitude times a unit vector in its direction: A = A·û, where û = A / A. - The Cartesian unit vectors are î, ĵ and k̂, along x, y and z. Exam trap: speed is a scalar and velocity a vector. A car going round a bend at a constant 50 km/h has constant speed but changing velocity, so it is accelerating.

Components and Vector Addition

Resolving a vector into components along perpendicular axes turns vector algebra into ordinary algebra: add the x parts, add the y parts, add the z parts.

Components in two dimensions - A vector A at angle θ from the positive x axis has components: - Ax = A·cos θ - Ay = A·sin θ - So A = Ax·î + Ay·ĵ. - Components can be negative. The sign gives the direction along the axis. Back from components to magnitude and direction - Magnitude: A = √(Ax² + Ay²). In 3D: A = √(Ax² + Ay² + Az²). - Direction: tan θ = Ay / Ax. Exam trap: a calculator returns arctan in the range −90° to +90° only. Always check the quadrant from the signs of Ax and Ay. A vector with Ax = −3 and Ay = −3 points at 225°, not 45°. Adding vectors graphically - Head-to-tail: draw B starting at the head of A. The resultant R = A + B goes from the tail of A to the head of B. - Parallelogram: draw both from the same point. The diagonal is R. - Addition is commutative (A + B = B + A) and associative. - Subtraction: A − B = A + (−B). Adding vectors by components (the method to use) 1. Resolve every vector into components. 2. Add all x components: Rx = Ax + Bx + ... 3. Add all y components: Ry = Ay + By + ... 4. Find R = √(Rx² + Ry²) and its direction from tan θ = Ry / Rx, checking the quadrant. Worked example - A = 3 m east, B = 4 m north. - R = √(3² + 4²) = 5 m, at θ = arctan(4/3) = 53° north of east. Exam focus: the magnitude of a sum is not the sum of the magnitudes. |A + B| lies between |A − B| and A + B, depending on the angle between them.

Scalar and Vector Products

There are two ways to multiply vectors. The scalar product A·B gives a number and measures how much the vectors point the same way. The vector product A×B gives a vector perpendicular to both, with magnitude equal to the area of the parallelogram they span.

Scalar (dot) product - Definition: A·B = A·B·cos φ, where φ is the angle between them. - In components: A·B = AxBx + AyBy + AzBz. - Unit vector rules: î·î = ĵ·ĵ = k̂·k̂ = 1; î·ĵ = ĵ·k̂ = k̂·î = 0. - Properties: - commutative: A·B = B·A - positive if φ < 90°, zero if the vectors are perpendicular, negative if φ > 90° - A·A = A² - Uses: - finding the angle between two vectors: cos φ = A·B / (AB) - the component of A along B: A·B̂ - in later physics, work W = F·d Vector (cross) product - Magnitude: |A×B| = A·B·sin φ. - Direction: perpendicular to the plane of A and B, given by the right-hand rule (fingers from A towards B, thumb gives A×B). - Unit vector rules: - î×ĵ = k̂, ĵ×k̂ = î, k̂×î = ĵ (cyclic order gives +) - ĵ×î = −k̂ (reversed order gives −) - î×î = 0 (any vector crossed with itself is zero) - Properties: - anticommutative: A×B = −(B×A) - zero if the vectors are parallel, maximum if perpendicular - In components: - (A×B)x = AyBz − AzBy - (A×B)y = AzBx − AxBz - (A×B)z = AxBy − AyBx The easiest way to remember these is as the determinant of a 3×3 matrix with rows (î ĵ k̂), (Ax Ay Az), (Bx By Bz). - Uses in later physics: torque (r×F), angular momentum, magnetic force. Worked example - A = 2î + 3ĵ, B = 4î − ĵ. - A·B = 8 − 3 = 5. - A×B = (2·(−1) − 3·4) k̂ = −14 k̂. Exam trap: the dot product of two vectors is a scalar, so "A·B = 5î" is always wrong. The cross product order matters. The dot product order does not.

Check Yourself: Measurement, Units and Vectors

Try every question before opening the answers. Retrieval, not rereading, is what fixes knowledge in memory.

1. Why is physics called an experimental science? What happens to a theory when a reproducible experiment disagrees with it? 2. List the seven SI base units and the quantity each measures. 3. Convert 90 km/h to m/s. 4. Write 0.000 005 m with a prefix, and give the order of magnitude of a red blood cell's diameter. 5. What are the dimensions of force and of energy? 6. Show that v² = v₀² + 2a(x − x₀) is dimensionally consistent. 7. Can a dimensionally correct equation be wrong? Can a dimensionally incorrect one be right? 8. A runner completes one full lap of a 400 m track. Give the distance and the displacement. 9. A vector of magnitude 10 m points at 150° from the positive x axis. Find its components. 10. A has components (−4, −3). Find its magnitude and direction. 11. A = 3î + 4ĵ and B = −î + 2ĵ. Find A + B, A − B, and |A + B|. 12. Find the angle between A = î + ĵ and B = î. 13. Compute î×ĵ, ĵ×î and k̂×k̂. 14. For A = 2î + 3ĵ and B = 4î − ĵ, find A·B and A×B. 15. Two forces of 3 N and 4 N act on a point. What are the largest and smallest possible magnitudes of their sum?

Check only after trying. Reread any section where you were wrong and retry tomorrow.

1. Theories are accepted only while their predictions agree with measurement. A reproducible disagreement forces the theory to be revised or limited to a narrower range of validity. 2. Metre (length), kilogram (mass), second (time), ampere (current), kelvin (temperature), mole (amount of substance), candela (luminous intensity). 3. 90 / 3.6 = 25 m/s. 4. 5 μm. A red blood cell is about 7-8 μm across, order of magnitude 10⁻⁵ m. 5. Force: [M][L][T]⁻². Energy: [M][L]²[T]⁻². 6. v² and v₀² are [L]²[T]⁻². a(x − x₀) is [L][T]⁻² × [L] = [L]²[T]⁻². All terms match. 7. Yes: dimensions cannot detect wrong pure numbers or missing dimensionless terms. No: a dimensionally inconsistent equation is always wrong. 8. Distance 400 m. Displacement 0. 9. Ax = 10 cos 150° = −8.66 m. Ay = 10 sin 150° = 5.00 m. 10. Magnitude 5. Both components are negative, so the vector is in the third quadrant: arctan(3/4) = 36.9°, and the direction is 180° + 36.9° = 216.9°. 11. A + B = 2î + 6ĵ. A − B = 4î + 2ĵ. |A + B| = √40 = 6.32. 12. A·B = 1. |A| = √2, |B| = 1. cos φ = 1/√2, so φ = 45°. 13. î×ĵ = k̂. ĵ×î = −k̂. k̂×k̂ = 0. 14. A·B = 8 − 3 = 5. A×B = (2·(−1) − 3·4) k̂ = −14 k̂. 15. Largest 7 N (same direction). Smallest 1 N (opposite directions). At 90° the sum is 5 N.

Motion in One Dimension

Kinematics describes motion without asking what causes it. In one dimension every vector reduces to a signed number, which makes this the cleanest place to master position, velocity and acceleration and the calculus that links them.

What you will learn: 1. Displacement and average velocity; instantaneous velocity as a derivative; speed versus velocity. 2. Average and instantaneous acceleration, and how to read x-t and v-t graphs. 3. The constant-acceleration equations: where they come from and when they apply. 4. Free fall as the most important constant-acceleration case. 5. Going backwards by integration: from acceleration to velocity to position. A thread to follow: position, velocity and acceleration are linked by derivatives going down (x, then v = dx/dt, then a = dv/dt) and by integrals going up. Every problem in this chapter uses this chain in one direction or the other.

Displacement, Time and Velocity

Average velocity is displacement divided by elapsed time. Instantaneous velocity is its limit as the time interval shrinks to zero, which is the derivative of position with respect to time and the slope of the x-t graph.

Setting up - Treat the body as a particle moving along the x axis. Choose an origin and a positive direction. - Position x(t) is a signed number. The sign says which side of the origin. - Displacement: Δx = x₂ − x₁. It is positive if the motion is in the +x direction. Average velocity - v_av = Δx / Δt = (x₂ − x₁) / (t₂ − t₁). - On an x-t graph it is the slope of the straight line (secant) joining the two points. - It depends only on start and end positions, not on what happened in between. Instantaneous velocity - v = lim (Δt → 0) Δx/Δt = dx/dt. - On an x-t graph it is the slope of the tangent at that instant. - Reading x-t graphs: - steeper slope means faster - positive slope means moving in +x - zero slope means momentarily at rest - negative slope means moving in −x Speed versus velocity - Instantaneous speed is the magnitude of instantaneous velocity, always positive or zero. - Average speed is total distance divided by time. It is not the magnitude of average velocity. A round trip has average velocity zero, but a non-zero average speed. Worked example (the cheetah type of problem) - x(t) = 20 m + (5.0 m/s²) t². - Displacement between t = 1 s and t = 2 s: x(2) − x(1) = 40 − 25 = 15 m. - Average velocity over that interval: 15 m/s. - Instantaneous velocity: v = dx/dt = (10 m/s²) t. At t = 1 s it is 10 m/s; at t = 2 s it is 20 m/s. Exam trap: "average velocity equals the mean of the initial and final velocities" is true only when acceleration is constant.

Acceleration

Acceleration is the rate of change of velocity. Instantaneous acceleration is dv/dt = d²x/dt²: the slope of the v-t graph, and the curvature of the x-t graph.

Definitions - Average acceleration: a_av = Δv / Δt. Units m/s². - Instantaneous acceleration: a = dv/dt = d²x/dt². Reading graphs - v-t graph: - the slope is the acceleration - the area under the curve is the displacement - x-t graph: - curving upward (concave up) means a > 0 - curving downward means a < 0 - a straight line means a = 0 Speeding up or slowing down depends on signs - Speeding up: v and a have the same sign. - Slowing down: v and a have opposite signs. - "Negative acceleration" does not automatically mean slowing down. A car moving in −x that speeds up has a < 0 and |v| increasing. Worked example - v(t) = 60 m/s + (0.50 m/s³) t². - Average acceleration from t = 1 s to t = 3 s: v(1) = 60.5 m/s, v(3) = 64.5 m/s, so a_av = 4.0 / 2 = 2.0 m/s². - Instantaneous: a = dv/dt = (1.0 m/s³) t. At t = 1 s it is 1.0 m/s²; at t = 3 s it is 3.0 m/s². Clinical tip: injury in trauma depends mainly on deceleration, not speed. A stop from 50 km/h over 1 m (crumple zone, airbag) is far safer than over 10 cm. The velocity change is the same, but the acceleration and so the force on the tissues is ten times smaller.

Motion with Constant Acceleration

When acceleration is constant, four equations describe the motion completely. Each links four of the five variables (x − x₀, v₀, v, a, t), so pick the one that omits the quantity you neither know nor need.

The four equations (a constant) 1. v = v₀ + a·t (no x) 2. x = x₀ + v₀·t + ½·a·t² (no v) 3. v² = v₀² + 2a·(x − x₀) (no t) 4. x − x₀ = ½·(v₀ + v)·t (no a) Where they come from - a constant means the v-t graph is a straight line of slope a, which gives equation 1. - The displacement is the area under that line: a rectangle (v₀·t) plus a triangle (½·a·t²). That is equation 2. - Eliminating t between 1 and 2 gives equation 3. - With linear v, the average velocity is (v₀ + v)/2, which gives equation 4. Special case, constant velocity (a = 0) - x = x₀ + v·t. The x-t graph is a straight line; the v-t graph is horizontal. Problem-solving method 1. Draw the axis and choose the positive direction and the origin. 2. List the knowns with signs, and the unknown. 3. Choose the equation that contains the unknown and only knowns. 4. Solve symbolically first, then substitute numbers with units. 5. Check the sign and size of the answer. Worked example - A car at 20 m/s brakes at 5 m/s² to a stop. What is the stopping distance? - Known: v₀ = 20, v = 0, a = −5. Time is not needed, so use equation 3. - 0 = 400 − 10·(x − x₀), so x − x₀ = 40 m. Exam focus: stopping distance scales with v². Doubling the speed quadruples the braking distance. Exam trap: these equations are valid only for constant a. If a depends on time, you must integrate (see "Velocity and position by integration").

Free Fall

Neglecting air resistance, every body near the Earth's surface falls with the same constant acceleration g ≈ 9.8 m/s², directed downward, whatever its mass. Free fall is simply constant-acceleration motion with a = −g if +y points up.

Key facts - g = 9.80 m/s² at the Earth's surface. It varies slightly with latitude and altitude. - Galileo showed that, without air, a heavy and a light body fall together. The Leaning Tower of Pisa is the classic example. - "Free fall" includes a body thrown upward. It is in free fall from the moment it leaves the hand, on the way up too. Equations (y axis up, a = −g) - v = v₀ − g·t - y = y₀ + v₀·t − ½·g·t² - v² = v₀² − 2g·(y − y₀) Results worth knowing - At the highest point v = 0 but a is still −g. The body is momentarily at rest, yet still accelerating. - Time to the top: t = v₀ / g. Maximum height above the launch point: h = v₀² / (2g). - Returning to the launch height, it has the same speed it started with, directed downward, and the time down equals the time up. - Dropped from rest, the distance fallen is d = ½·g·t², so it grows with t². Worked example (the coin from the tower) - A coin is dropped from rest. After 3.0 s: - v = −9.8 × 3 = −29.4 m/s (downward) - it has fallen ½ × 9.8 × 9 = 44.1 m Worked example (ball thrown up) - A ball is thrown up at 15 m/s. - Maximum height: 15² / (2 × 9.8) = 11.5 m, reached after 1.53 s. Exam trap: when solving y(t) = value, the quadratic often gives two roots. Decide which time is physical. The negative root, or the one on the way up when you need the way down, is discarded.

Velocity and Position by Integration

When acceleration is not constant, the four equations fail. Go back to the definitions and integrate: velocity is the integral of acceleration, position is the integral of velocity, each plus its initial value.

The integral relations - v(t) = v₀ + ∫ a dt, from 0 to t. - x(t) = x₀ + ∫ v dt, from 0 to t. - Geometrically: - the change in velocity is the area under the a-t graph - the displacement is the area under the v-t graph Check: with a constant, v = v₀ + a·t, and x = x₀ + v₀·t + ½·a·t². The constant-acceleration equations are just a special case. Worked example - A car accelerates with a(t) = 2.0 m/s² − (0.10 m/s³) t, starting from rest at x₀ = 0. - v(t) = 2.0t − 0.05t² - x(t) = 1.0t² − (0.05/3) t³ - Maximum velocity when a = 0: t = 20 s, v = 40 − 20 = 20 m/s. Method 1. Write a(t) with units. 2. Integrate once and add v₀. 3. Integrate again and add x₀. 4. Use conditions such as v = 0 (turning point) or a = 0 (maximum speed) to answer the question. Exam focus: the initial conditions x₀ and v₀ are the "constants of integration". A problem is only fully solved when both are fixed.

Check Yourself: Motion in One Dimension

Try every question before opening the answers. Retrieval, not rereading, is what fixes knowledge in memory.

1. Define average velocity and instantaneous velocity. What does each mean on an x-t graph? 2. A cyclist rides 3 km east in 10 min and returns 3 km west in 20 min. Find the average velocity and the average speed. 3. x(t) = 4t − t² (SI units). When is the particle at rest? Where is it then? 4. What does the area under a v-t graph represent? And the slope? 5. Can a body have zero velocity and non-zero acceleration? Give an example. 6. A car moving in the −x direction slows down. What is the sign of its acceleration? 7. Write the four constant-acceleration equations. Which variable does each omit? 8. A train at 30 m/s brakes at 1.5 m/s². What is its stopping distance and time? 9. A stone is dropped from 20 m. How long does it take to land, and at what speed? 10. A ball is thrown vertically up at 20 m/s. Find the maximum height and the total time in the air. 11. Why do the constant-acceleration equations fail when a = kt? 12. a(t) = 6t (SI units), v₀ = 0, x₀ = 0. Find v(t) and x(t). 13. Stopping from 50 km/h, why does a longer stopping distance reduce injury?

Check only after trying. Reread any section where you were wrong and retry tomorrow.

1. v_av = Δx/Δt, the slope of the secant. v = dx/dt, the slope of the tangent. 2. Displacement 0, so the average velocity is 0. Average speed: 6 km / 0.5 h = 12 km/h. 3. v = 4 − 2t = 0 at t = 2 s. x(2) = 8 − 4 = 4 m. 4. Area: displacement. Slope: acceleration. 5. Yes. A ball at the top of its flight has v = 0 and a = −g. 6. Positive. Velocity is negative and slowing down means a has the opposite sign. 7. v = v₀ + at (no x); x = x₀ + v₀t + ½at² (no v); v² = v₀² + 2a(x − x₀) (no t); x − x₀ = ½(v₀ + v)t (no a). 8. t = 30/1.5 = 20 s. d = 30²/(2 × 1.5) = 300 m. 9. t = √(2 × 20/9.8) = 2.02 s. v = 9.8 × 2.02 = 19.8 m/s. 10. h = 400/19.6 = 20.4 m. Up time 2.04 s, so 4.08 s in total. 11. They are derived assuming a constant. With a = kt, you must integrate. 12. v = 3t². x = t³. 13. The same Δv spread over a longer time and distance means a smaller deceleration, and so a smaller force on the body.

Motion in Two and Three Dimensions

In two or three dimensions position, velocity and acceleration become true vectors. The key idea is that motion along perpendicular axes is independent, so a 2D problem splits into two 1D problems linked only by time.

What you will learn: 1. Position, displacement, velocity and acceleration vectors, and moving between them by differentiation and integration. 2. Projectile motion: uniform along x, free fall along y. 3. Circular motion: - polar, cylindrical and spherical coordinates - angular position, velocity and acceleration - centripetal and tangential acceleration 4. Relative velocity: how motion looks from different moving frames. A thread to follow: in 2D, acceleration can change the direction of velocity as well as its size. Uniform circular motion has constant speed but non-zero acceleration, and this idea sets up the dynamics chapter.

Position, Velocity and Acceleration Vectors

In 3D the position is a vector r(t) = x î + y ĵ + z k̂. Velocity v = dr/dt is always tangent to the path, and acceleration a = dv/dt describes any change in the size or the direction of v.

Definitions - Position vector: r = x î + y ĵ + z k̂, from the origin to the particle. - Displacement: Δr = r₂ − r₁. - Average velocity: v_av = Δr / Δt. - Instantaneous velocity: v = dr/dt = (dx/dt) î + (dy/dt) ĵ + (dz/dt) k̂. - Each component is a 1D velocity. - The speed is |v| = √(vx² + vy² + vz²). - v is tangent to the trajectory at every point. - Instantaneous acceleration: a = dv/dt = d²r/dt². Components of acceleration relative to the path - Parallel to v: changes the speed (tangential acceleration). - Perpendicular to v: changes the direction (normal or centripetal acceleration). - So a curved path always means acceleration, even at constant speed. From motion law to acceleration: differentiate each component twice. From acceleration to motion law: integrate each component twice. - v(t) = v₀ + ∫ a dt - r(t) = r₀ + ∫ v dt - If a is constant: r = r₀ + v₀t + ½at², applied separately to each component. Special cases - Straight line, constant velocity: a = 0, r = r₀ + v·t. - Straight line, constant acceleration: a parallel to v₀. Worked example - r(t) = (2t) î + (t²) ĵ (SI units). - v = 2 î + 2t ĵ; a = 2 ĵ, constant. - At t = 1 s: speed √8 = 2.83 m/s at 45°. - The path is y = x²/4, a parabola. Exam focus: the direction of v is along the path. The direction of a is not along the path in general; it points towards the concave side of a curve.

Projectile Motion

A projectile moves under gravity alone. Horizontally there is no acceleration, so vx is constant. Vertically it is in free fall with a = −g. The two motions share only time, and together they trace a parabola.

Setup: launch from the origin with speed v₀ at angle θ above the horizontal. - v₀x = v₀ cos θ - v₀y = v₀ sin θ Equations - Horizontal: - x = v₀ cos θ · t - vx = v₀ cos θ (constant) - Vertical: - y = v₀ sin θ · t − ½ g t² - vy = v₀ sin θ − g t Derived results (launch and landing at the same height) - Time to maximum height: t_top = v₀ sin θ / g. - Maximum height: H = v₀² sin²θ / (2g). - Total flight time: T = 2 v₀ sin θ / g. - Range: R = v₀² sin 2θ / g. - The range is largest at θ = 45°. - Complementary angles (30° and 60°) give the same range. - Trajectory: y = x tan θ − g x² / (2 v₀² cos²θ), a parabola. Key insights - At the top, vy = 0 but vx is not, so the speed is at its minimum and not zero. Acceleration is still g downward everywhere. - A ball dropped and a ball fired horizontally from the same height land at the same time. The vertical motions are identical. - Air resistance is ignored. Real ranges are shorter and the best angle is below 45°. Method 1. Resolve v₀. 2. Write x(t) and y(t). 3. Use the vertical equation to find the time (to the top, or to landing). 4. Put that time into the horizontal equation. Exam trap: if the launch and landing heights differ (a cliff, a table), the range formula does not apply. Solve y(t) = y_landing for t, then find x.

Circular Motion

Circular motion is best described with angles rather than x and y. Once you describe position by an angle, the circle becomes a 1D problem in θ, with angular versions of velocity and acceleration.

This section covers: - the non-Cartesian coordinate systems (polar, cylindrical, spherical) that make circles and spheres simple - angular kinematics (θ, ω, α) and how they link to linear quantities (s = rθ, v = rω, a_t = rα) - uniform circular motion, with centripetal acceleration v²/r, and non-uniform circular motion, where a tangential component is added The dynamics chapter uses the centripetal acceleration from here to explain why cars need friction on a bend and why a centrifuge separates blood.

Polar, Cylindrical and Spherical Coordinates

Choose coordinates that match the symmetry of the problem. Polar coordinates (r, θ) suit circles in a plane, cylindrical (r, θ, z) suit axes and tubes, and spherical (r, θ, φ) suit anything centred on a point.

Plane polar coordinates (r, θ) - r: distance from the origin; θ: angle from the +x axis. - To Cartesian: x = r cos θ, y = r sin θ. - From Cartesian: r = √(x² + y²), tan θ = y/x (check the quadrant). - On a circle of radius R, only θ changes. That is why circular motion is 1D in θ. Cylindrical coordinates (r, θ, z) - Polar coordinates in the xy plane, plus the height z. - x = r cos θ, y = r sin θ, z = z. - Natural for helices, pipes and blood vessels (flow along z, velocity profile depending on r). Spherical coordinates (r, θ, φ) - r: distance from the origin. The two angles fix the direction. - Convention (physics): θ is the polar angle from the +z axis, φ is the azimuth in the xy plane. - x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. - Natural for radiation from a point source and for central forces such as gravity. Exam trap: books differ on which angle is called θ and which φ in spherical coordinates. State your convention and stick to it.

Angular Kinematics

Angular position θ, angular velocity ω = dθ/dt and angular acceleration α = dω/dt mirror x, v and a exactly. On a circle of radius r they link to linear quantities by s = rθ, v = rω and a_t = rα, with angles in radians.

Angular quantities - Angular position θ, in radians. 1 rad is the angle whose arc equals the radius; 2π rad = 360°. - Angular velocity ω = dθ/dt, in rad/s. - Also expressed as frequency f (revolutions per second, Hz) and period T. - ω = 2πf = 2π/T. - Angular acceleration α = dω/dt, in rad/s². Linear and angular links (point at radius r) - arc length: s = r θ - speed: v = r ω - tangential acceleration: a_t = r α - These hold only with θ in radians. Constant angular acceleration: the same four equations as 1D - ω = ω₀ + α t - θ = θ₀ + ω₀ t + ½ α t² - ω² = ω₀² + 2α (θ − θ₀) - θ − θ₀ = ½ (ω₀ + ω) t From motion law to angular acceleration: differentiate θ(t) twice. From angular acceleration to motion law: integrate α(t) twice, adding ω₀ and θ₀. Worked example - A centrifuge speeds up from rest to 3000 rpm in 10 s. - ω = 3000 × 2π / 60 = 314 rad/s. - α = 31.4 rad/s². - Revolutions turned: θ = ½ × 31.4 × 100 = 1570 rad, which is 250 revolutions. Exam trap: rpm must be converted to rad/s (multiply by 2π/60) before using v = rω.

Uniform and Non-Uniform Circular Motion

In uniform circular motion the speed is constant but the direction changes, so there is an acceleration of magnitude v²/r = ω²r pointing to the centre (centripetal). If the speed also changes, a tangential component a_t = dv/dt is added.

Uniform circular motion (constant speed) - Velocity is tangent to the circle, with constant magnitude. - Acceleration: a_c = v²/r = ω² r, directed towards the centre. - Period T = 2πr / v. - Where the result comes from: in time Δt the velocity vector turns by Δθ, so |Δv| = v Δθ. Dividing by Δt gives v·ω = v²/r. Non-uniform circular motion - Total acceleration a = a_c + a_t. - a_c = v²/r, radial, towards the centre. - a_t = dv/dt, tangent to the path. - The magnitude is √(a_c² + a_t²). - The total acceleration points inside the circle but not at the centre. Worked examples - A car takes a bend of radius 50 m at 20 m/s: a_c = 400/50 = 8 m/s², close to g. - A centrifuge at 314 rad/s with r = 0.10 m: a_c = 314² × 0.1 ≈ 9900 m/s², about 1000 g. Clinical tip: a centrifuge works because the enormous centripetal acceleration (the relative centrifugal force, RCF, quoted in multiples of g) makes denser components sediment fast. Red cells go to the bottom, then the buffy coat, with plasma on top. Lab protocols give RCF (× g), not rpm, because RCF depends on the rotor radius. Exam trap: "centrifugal force" is not a real force in an inertial frame. The only real requirement is a net inward force supplying m·v²/r (see Dynamics of circular motion).

Relative Velocity

Velocity depends on the observer. The velocity of P relative to A equals its velocity relative to B plus B's velocity relative to A: v_P/A = v_P/B + v_B/A. In 1D this is simple addition; in 2D it is vector addition.

Notation - v_P/A means "the velocity of P as measured by observer A". - Reversing the subscripts changes the sign: v_A/B = −v_B/A. One dimension - v_P/A = v_P/B + v_B/A, with signs. - Example: you walk forward at 1.5 m/s in a train moving at 20 m/s. Relative to the ground you move at 21.5 m/s. Walking backward, at 18.5 m/s. - Two cars approach head-on at 25 m/s each. Each sees the other closing at 50 m/s. Two and three dimensions - The same equation, as vectors. Add by components. - Classic cases: - a boat crossing a river with a current - an aircraft in a crosswind - Example: a plane has airspeed 240 km/h heading north, and the wind blows east at 100 km/h. The ground velocity is √(240² + 100²) = 260 km/h, at arctan(100/240) = 22.6° east of north. - To fly due north instead, the pilot heads into the wind: sin φ = 100/240, so φ = 24.6° west of north. The ground speed is then √(240² − 100²) = 218 km/h. Why this matters - Newton's laws hold in every frame moving at constant velocity relative to an inertial frame (Galilean relativity). Relative velocity is the basis of that idea. Exam trap: write the subscripts carefully and check that the inner ones "cancel" (P/B + B/A gives P/A). Most errors are a subscript written the wrong way round.

Check Yourself: Motion in Two and Three Dimensions

Try every question before opening the answers. Retrieval, not rereading, is what fixes knowledge in memory.

1. r(t) = 3t î + (4t − 5t²) ĵ (SI units). Find v(t), a(t) and the speed at t = 0. 2. Why is velocity always tangent to the path, while acceleration generally is not? 3. A ball is kicked at 20 m/s at 30° above the horizontal on level ground. Find the flight time, the maximum height and the range. 4. At the top of its trajectory, what are the projectile's velocity and acceleration? 5. Which launch angle gives the maximum range on level ground? Which pairs of angles give equal ranges? 6. A ball rolls off a 1.25 m table at 3 m/s. How far from the table does it land? 7. Convert 3000 rpm to rad/s. What is the speed of a point 10 cm from the axis? 8. Write the relations between s, v, a_t and θ, ω, α. What condition on the angle is needed? 9. A car takes a bend of radius 80 m at 20 m/s. What is its centripetal acceleration? 10. In non-uniform circular motion, what are the two components of acceleration? Where does the total acceleration point? 11. Give Cartesian x, y and z in terms of spherical r, θ and φ. 12. A swimmer crosses a river at 1.0 m/s relative to the water, heading straight across, and the current is 0.5 m/s. What is the swimmer's velocity relative to the bank? 13. Why do lab protocols specify centrifugation in × g rather than rpm?

Check only after trying. Reread any section where you were wrong and retry tomorrow.

1. v = 3 î + (4 − 10t) ĵ. a = −10 ĵ. At t = 0 the speed is √(9 + 16) = 5 m/s. 2. v = dr/dt follows the change of position along the path. a also includes the change of direction, which points towards the concave side. 3. v₀y = 10 m/s and v₀x = 17.3 m/s. T = 2 × 10/9.8 = 2.04 s. H = 100/19.6 = 5.1 m. R = 17.3 × 2.04 = 35.3 m. 4. Velocity is horizontal (v₀ cos θ); acceleration is g downward. 5. 45°. Complementary angles give equal ranges (for example 30° and 60°). 6. Time to fall: t = √(2 × 1.25/9.8) = 0.505 s. x = 3 × 0.505 = 1.52 m. 7. 314 rad/s. v = 314 × 0.1 = 31.4 m/s. 8. s = rθ, v = rω, a_t = rα. θ must be in radians. 9. 400/80 = 5 m/s². 10. Centripetal (v²/r, towards the centre) and tangential (dv/dt). The total points inside the circle, off the centre. 11. x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ (θ measured from +z). 12. √(1 + 0.25) = 1.12 m/s, at arctan(0.5/1) = 26.6° downstream of straight across. 13. RCF = ω²r depends on the rotor radius, so the same rpm gives different separation on different centrifuges.

Dynamics: Forces and Newton's Laws

Kinematics describes motion; dynamics explains it. Newton's three laws link the forces acting on a body to its acceleration, and with free-body diagrams they solve almost every problem in classical mechanics.

What you will learn: 1. What a force is: contact and long-range forces, how forces are measured, and how they add (superposition). 2. Newton's first law, inertia, and inertial reference frames. 3. Newton's second law, ΣF = m·a; the difference between mass and weight. 4. Newton's third law: action and reaction act on different bodies. 5. The free-body diagram method for applying the laws. 6. Friction: static and kinetic, and why it is easier to keep something sliding than to start it. 7. Dynamics of circular motion: the net force that supplies centripetal acceleration. 8. The four fundamental forces of nature. A thread to follow: in every problem, (1) isolate one body, (2) draw every force acting on it, (3) write ΣF = m·a along each axis. Almost all mistakes come from skipping step 2.

Force and Interactions

A force is an interaction between two bodies that can change a body's motion. Forces are vectors, and when several act on the same body their effect equals that of their vector sum (the net or resultant force).

What a force is - A push or a pull resulting from an interaction between two bodies. - A vector, measured in newtons: 1 N = 1 kg·m/s². Contact forces (the bodies touch) - Normal force n: perpendicular to the surface, it prevents interpenetration. - Friction f: parallel to the surface, it opposes relative sliding. - Tension T: the pull transmitted along a rope, string or tendon. - Also: air resistance and fluid drag, and the elastic (spring) force. Long-range forces (act at a distance) - Gravity (weight, w = m·g). - Electric and magnetic forces. Measuring forces - A spring balance (dynamometer): the extension is proportional to the force (Hooke's law, F = kx). It is calibrated against known weights. Superposition - The net force is R = ΣF = F₁ + F₂ + ..., added as vectors by components. - Rx = ΣFx, Ry = ΣFy. - Only the net force determines the acceleration. Clinical tip: tendons and ligaments are tension elements; bone mostly resists compression. Musculoskeletal biomechanics in Applied Physics builds on exactly this force vocabulary. Exam trap: forces are exerted by something. If you cannot name the body that exerts a force, it is probably not a real force ("the force of motion" does not exist).

Newton's First Law and Inertial Frames

A body on which no net force acts stays at rest or moves in a straight line at constant velocity. This property is inertia. The law holds only in inertial reference frames, which are frames that are not accelerating.

Statement - If ΣF = 0, then v is constant (a = 0). This covers both rest and uniform straight-line motion. - A body does not need a force to keep moving; it needs a force to change its motion. This is the key break from Aristotle's physics. Equilibrium - A body with ΣF = 0 is in equilibrium: static if at rest, dynamic if moving at constant velocity. - In components: ΣFx = 0, ΣFy = 0, ΣFz = 0. Inertial frames - A frame in which the first law holds. - Any frame moving at constant velocity relative to an inertial frame is also inertial. - The Earth's surface is approximately inertial for most everyday problems. - In an accelerating frame (a braking bus, a rotating carousel) bodies seem to accelerate with no force. Passengers lurch forward in a braking bus because their bodies continue at the old velocity, not because a force pushes them. Validity of the first law - It fails in non-inertial frames. To use Newton's laws, always analyse from an inertial frame. Clinical tip: whiplash comes from inertia. In a rear-end collision the car and torso are pushed forward, and the head lags behind by inertia until the neck pulls it. Head restraints limit this relative motion.

Newton's Second Law, Mass and Weight

The net force on a body equals its mass times its acceleration, ΣF = m·a, and the acceleration has the direction of the net force. Mass measures inertia; weight is the gravitational force on the body, w = m·g.

Statement - ΣF = m·a, a vector equation. In components: - ΣFx = m·ax - ΣFy = m·ay - ΣFz = m·az - It applies only to the external forces on the chosen body, in an inertial frame. - 1 N is the net force that gives a 1 kg mass an acceleration of 1 m/s². Mass - A scalar measure of inertia: how strongly a body resists changes in velocity. - An intrinsic property. The same on the Earth, the Moon or in orbit. - Unit: kg. Weight - The gravitational force on a body: w = m·g, directed downward. - It depends on where you are. On the Moon g ≈ 1.6 m/s², so a 70 kg person weighs 686 N on Earth and about 112 N on the Moon, with the same mass. - A bathroom scale measures the normal force, which equals the weight only when there is no vertical acceleration. Apparent weight in an elevator (y axis up) - n − m·g = m·a, so n = m·(g + a). - Accelerating upward: you feel heavier. Accelerating downward: you feel lighter. In free fall (a = −g), n = 0: "weightlessness". Worked example - A 1000 kg car goes from 0 to 20 m/s in 10 s. - a = 2 m/s², so the net force is 2000 N. Exam trap: m·a is not a force to be drawn on the free-body diagram. It is the result of the forces. Draw only real forces, then set their sum equal to m·a.

Newton's Third Law

If body A exerts a force on body B, then B exerts on A a force equal in magnitude and opposite in direction. The two forces of such a pair act on different bodies, so they never cancel each other.

Statement - F_A on B = −F_B on A. - The two forces: - are of the same type (both gravitational, both contact, and so on) - act at the same instant - act on different bodies Examples of action-reaction pairs - You push a wall; the wall pushes you back equally. That is how you can push off it. - The Earth pulls an apple down with force m·g. The apple pulls the Earth up with the same force. The Earth's acceleration is tiny because its mass is enormous. - Walking: your foot pushes the ground backward, and the ground pushes you forward (static friction). - A rocket pushes exhaust gas backward, and the gas pushes the rocket forward. It needs no air to push against. The classic confusion: a book on a table - Forces on the book: its weight (Earth on book) and the normal force (table on book). They are equal and opposite because the book is in equilibrium (first and second laws). They are not a third-law pair: both act on the book and are of different types. - The third-law partner of the weight is the book pulling the Earth up. - The third-law partner of the normal force is the book pushing down on the table. Exam trap: "why does the horse pull the cart forward if the cart pulls the horse back equally?" The two forces act on different bodies. The horse moves because the ground pushes it forward (friction) more than the cart pulls it back.

Free-Body Diagrams and Applying Newton's Laws

A free-body diagram shows one body alone with every external force acting on it, each drawn from the body with its correct direction. It is the step that turns a word problem into equations, and skipping it is the main source of error.

Method for any dynamics problem 1. Choose the body (or bodies) to analyse. Treat each separately. 2. Draw the free-body diagram: the body as a dot or box, with all forces acting on it. - weight (always) - normal forces from each contact surface - friction at each contact where sliding occurs or tends to occur - tension from each rope - any applied push or pull 3. Choose axes. Align one axis with the acceleration if you know its direction (for example along an incline). 4. Resolve every force into components. 5. Write ΣFx = m·ax and ΣFy = m·ay. 6. Add any constraints (a rope of fixed length gives connected bodies the same acceleration magnitude). 7. Solve symbolically, then substitute numbers and check units and limiting cases. Standard cases - Body on a frictionless incline of angle θ (x down the slope, y perpendicular to it): - x: m·g·sin θ = m·a, so a = g·sin θ - y: n − m·g·cos θ = 0, so n = m·g·cos θ - Checks: θ = 0 gives a = 0; θ = 90° gives a = g. - Two blocks joined by a rope over a frictionless, massless pulley (Atwood machine): - a = (m₁ − m₂)·g / (m₁ + m₂) - T = 2·m₁·m₂·g / (m₁ + m₂) - A hanging mass at rest on a rope: T = m·g. Accelerating up: T = m·(g + a). Exam focus: examiners give marks for a correct free-body diagram even when the arithmetic fails. Always draw it. Exam trap: the normal force is not always m·g. It is smaller on an incline (m·g·cos θ), and changes when you push down or pull up at an angle, or when the body accelerates vertically.

Friction

Friction acts along the contact surface and opposes relative sliding. Static friction adjusts itself up to a maximum μs·n to prevent sliding; once sliding starts, kinetic friction is roughly constant at μk·n, and μk is smaller than μs.

Static friction - Acts when the surfaces do not slide relative to each other. - It takes whatever value is needed to prevent sliding, from 0 up to a maximum: fs ≤ μs·n. - Its direction opposes the motion that would occur without friction. Kinetic friction - Acts when the surfaces slide. - fk = μk·n, roughly independent of speed and of contact area. Directed opposite to the relative velocity. Coefficients - μs and μk are dimensionless and depend on the pair of materials. - Usually μk < μs. This is why it takes more force to start pushing a heavy box than to keep it moving. - Rough values: rubber on dry concrete about 1.0 (static) and 0.8 (kinetic); steel on ice about 0.1; synovial joint cartilage about 0.003-0.01. The friction graph - Increase a horizontal pull F on a box from zero: - fs = F (the box does not move) until F reaches μs·n - then the box starts to slide - friction drops to μk·n, and the box accelerates with a = (F − μk·n)/m Worked examples - Box on an incline, the angle at which it starts to slide: m·g·sin θ = μs·m·g·cos θ, so tan θ = μs. - A box sliding on a floor with μk = 0.3 decelerates at a = μk·g = 2.9 m/s². Why friction matters - Walking, driving and gripping all need static friction. The foot or tyre does not slip at the contact point. - ABS braking keeps tyres rolling (static friction, μs) rather than skidding (kinetic, μk), which stops a car in a shorter distance. Clinical tip: healthy articular cartilage lubricated by synovial fluid has one of the lowest friction coefficients known. Osteoarthritis roughens the surface and raises friction and wear. Exam trap: μs·n is the maximum static friction, not its value. If the applied force is smaller, static friction just equals the applied force.

Dynamics of Circular Motion

A body moving on a circle needs a net force pointing to the centre, of size m·v²/r. This "centripetal force" is not a new kind of force; it is supplied by tension, friction, gravity or the normal force, whichever acts inward.

The equation - Radial direction: ΣF_radial = m·v²/r = m·ω²·r, towards the centre. - Tangential direction (if the speed changes): ΣF_tangential = m·a_t. What supplies the inward force, examples - A ball on a string whirled horizontally: tension. - A car on a flat bend: static friction between the tyres and the road. - The maximum safe speed is v_max = √(μs·g·r). - Above it the car skids outward, meaning it continues straight as friction can no longer turn it. - A banked curve with no friction: a component of the normal force. tan β = v²/(r·g) gives the design speed. - The Moon and satellites: gravity. - A vertical loop: at the top, m·g + n = m·v²/r. The minimum speed to stay on the track is v = √(g·r), where n = 0. The "centrifugal force" confusion - In an inertial frame there is no outward force on the body. The passenger in a turning car feels "thrown outward" because by inertia their body tends to move straight, while the door pushes them inward. - Centrifugal force appears only as a fictitious force in the rotating (non-inertial) frame. Worked example - A 1200 kg car takes a bend of radius 50 m at 15 m/s. - Required friction: 1200 × 225/50 = 5400 N. - With μs = 0.8, the maximum friction is 0.8 × 1200 × 9.8 = 9400 N, so the car holds the road. v_max = √(0.8 × 9.8 × 50) = 19.8 m/s. Clinical tip: in a centrifuge the tube wall pushes the fluid inward. Denser particles need more inward force to follow the circle than the surrounding fluid can supply, so they drift outward and sediment. Exam trap: never draw "m·v²/r" as an extra force on the free-body diagram. It is the required net inward force, made up of the real forces you have drawn.

The Fundamental Forces of Nature

Every force in nature reduces to four fundamental interactions: gravitational, electromagnetic, strong nuclear and weak nuclear. All everyday contact forces (normal force, friction, tension) are electromagnetic at the atomic level.

The four interactions, from strongest to weakest 1. Strong nuclear force. - Holds quarks together in protons and neutrons, and holds nucleons together in the nucleus. - Very short range (about 10⁻¹⁵ m, the size of a nucleus). 2. Electromagnetic force. - Between electric charges. Long range (falls as 1/r²). Can attract or repel. - It is responsible for chemistry, for the structure of matter and for all contact forces. 3. Weak nuclear force. - Responsible for beta decay and some other nuclear processes. - Extremely short range. 4. Gravitational force. - Between all masses. Long range (1/r²), always attractive. - By far the weakest. It dominates at astronomical scales only because matter is electrically neutral overall and gravity always adds up. Everyday forces traced back - Normal force, friction, tension, spring and drag forces all come from electromagnetic repulsion and attraction between atoms at the contact. - Weight is gravitational. Unification (awareness only) - Electricity and magnetism are one force (Maxwell). The electromagnetic and weak forces unify at high energies (electroweak theory). Clinical tip: the weak force (beta decay) underlies many radionuclides used in nuclear medicine, and the strong force sets nuclear binding energies. These return in the radiobiology and imaging courses.

Check Yourself: Dynamics

Try every question before opening the answers. Retrieval, not rereading, is what fixes knowledge in memory.

1. State Newton's three laws. 2. What is an inertial frame? Give one example of a non-inertial frame and describe what goes wrong there. 3. A 60 kg person stands in an elevator accelerating upward at 2 m/s². What does the scale read (in N)? 4. What is the difference between mass and weight? How do both change on the Moon? 5. A book rests on a table. Name the forces on it, and the third-law partner of each. 6. If action and reaction are equal and opposite, how can anything accelerate? 7. Draw the free-body diagram and find the acceleration for a block sliding down a frictionless 30° incline. 8. Why is the normal force on a block on an incline less than m·g? 9. A 10 kg box needs 49 N to start moving and 39 N to keep moving at constant speed on a level floor. Find μs and μk. 10. What angle of incline makes a block just start to slide, in terms of μs? 11. Why does ABS braking shorten the stopping distance? 12. What provides the centripetal force for a car on a flat bend? Derive the maximum safe speed. 13. A 0.5 kg ball on a 1 m string is whirled in a horizontal circle at 4 m/s. What is the tension (ignore gravity)? 14. Explain why a passenger feels pushed outward on a bend, using only forces in an inertial frame. 15. List the four fundamental forces in order of strength. Which one underlies friction?

Check only after trying. Reread any section where you were wrong and retry tomorrow.

1. First: with no net force, the velocity stays constant. Second: ΣF = m·a. Third: forces come in equal and opposite pairs acting on different bodies. 2. A frame in which the first law holds (not accelerating). A braking bus: objects seem to accelerate with no force acting on them. 3. n = m(g + a) = 60 × 11.8 = 708 N. 4. Mass is inertia, in kg, and is the same everywhere. Weight is m·g, in N, and depends on g. On the Moon the mass is unchanged and the weight is about 1/6. 5. Weight (partner: the book pulling the Earth up). Normal force (partner: the book pushing down on the table). 6. The pair acts on different bodies. Each body accelerates according to the net force on it alone. 7. The forces are the weight and the normal force. a = g sin 30° = 4.9 m/s² down the slope. 8. Only the perpendicular component m·g·cos θ presses into the surface, and n balances only that component. 9. n = 98 N. μs = 49/98 = 0.5. μk = 39/98 = 0.4. 10. tan θ = μs. 11. The wheels keep rolling without skidding, so static friction (μs > μk) acts, giving a larger decelerating force. 12. Static friction. μs·m·g = m·v²/r, so v_max = √(μs·g·r). 13. T = m·v²/r = 0.5 × 16/1 = 8 N. 14. The body tends to continue straight by inertia; the door pushes it inward to make it turn. No outward force acts on it. 15. Strong, electromagnetic, weak, gravitational. Friction is electromagnetic.