Week 4 — Forces and Newton's Laws

Overview

Week 4 transitions from kinematics to dynamics: the study of forces that cause motion. Students learn to identify the common forces acting on an object, represent them on a free body diagram (FBD), and apply Newton’s three laws — particularly Newton’s 2nd law, ${\vec{F}}_{net}=m\vec{a}$ — to determine unknown forces or accelerations. The lecture is the foundation for Portfolio 3 and for next week’s deeper look at friction and inclined surfaces.

Key concepts

• Force — A push or pull. Acts on an object, requires an identifiable agent, and is a vector (magnitude + direction). Unit: newton, $N=\text{kg}\cdot {\text{m/s}}^2$.
• Contact force — The agent physically touches the object (e.g., bat hits ball).
• Long-range force — Acts without contact (e.g., gravity).
• Gravity (${\vec{F}}_G$ or ${\vec{F}}_W$, weight) — The pull of a planet on an object near its surface. Long-range. Always points vertically down. On Earth, $F_G=mg$ with $g=9.81{\text{m/s}}^2$.
• Spring force (${\vec{F}}_S$) — The push (compressed) or pull (stretched) a spring exerts on a contacting object. Magnitude $F_S=kx$.
• Spring constant ($k$) — Stiffness of the spring. Units: $\text{N/m}$.
• Spring displacement ($x$) — Distance the spring is stretched or compressed from its natural length. Units: $m$.
• Normal force (${\vec{F}}_N$) — Push of a surface on an object resting on it. Contact force, perpendicular ($90^{\circ }$) to the surface.
• Tension force (${\vec{F}}_T$) — Force exerted by a string, rope, or wire pulling on an object. Always directed along the rope.
• Friction force (${\vec{F}}_{fric}$) — Force from a surface that opposes or prevents motion. Acts parallel to the surface. $F_{fric}=F_N\mu$.
• Kinetic friction (${\mu }_k$) — Coefficient when surfaces are sliding. Force vector points opposite to velocity vector.
• Static friction (${\mu }_s$) — Coefficient when surfaces are not sliding. Force points opposite to the direction the object would move without friction. $F_{fric}\le F_N{\mu }_s$.
• Thrust (${\vec{F}}_{thrust}$) — Force on a rocket/jet from expelling exhaust at high speed. Acts opposite to the direction of exhaust expulsion.
• Drag (${\vec{F}}_{drag}$) — Resistive force on an object moving through a fluid (liquid or gas). Always opposite to direction of motion. Ignore unless explicitly stated.
• Buoyancy (${\vec{F}}_B$) — Normal force for fluids; from the lecture notes for Example 2, treated as the upward fluid force balancing weight.
• Free body diagram (FBD) — Pictorial representation showing the object as a dot at the origin of a coordinate system, with every force on it drawn as a labelled vector.
• Net force (${\vec{F}}_{net}$) — Vector sum of every force on the object: ${\vec{F}}_{net}={\vec{F}}_A+{\vec{F}}_B+{\vec{F}}_C+\dots$
• Equilibrium — State where ${\vec{F}}_{net}=0$; object is not accelerating (either at rest or moving at constant velocity).
• Universal gravitational constant ($G$) — $G=6.67\times 10^{-11}\text{N}{\text{m}}^2/{\text{kg}}^2$.

Core formulas

Newton’s law of universal gravitation — the force between any two masses: $F_{A\text{on}B}=F_{B\text{on}A}=\frac{G{m}_A{m}_B}{r^2}$

• $G=6.67\times 10^{-11}\text{N}{\text{m}}^2/{\text{kg}}^2$
• $m_A,m_B$ — masses of the two objects ($\text{kg}$)
• $r$ — distance between centres ($\text{m}$)

Weight (gravity near a planetary surface): $F_G=mg$

• $m$ — mass ($\text{kg}$); $g=9.81{\text{m/s}}^2$ on Earth.

Spring (Hooke’s law form used in the lecture): $F_S=kx$

• $k$ — spring constant ($\text{N/m}$); $x$ — displacement from natural length ($\text{m}$).

Friction (magnitude): ${\vec{F}}_{fric}=F_N\mu$

• Kinetic: $F_{fric}=F_N{\mu }_k$
• Static: $F_{fric}\le F_N{\mu }_s$

Net force (component form): $F_{\text{Net},x}={\sum }_{i=1}^i{F}_{1,x}+F_{2,x}+F_{3,x}+\cdots +F_{i,x}$ $F_{\text{Net},y}={\sum }_{i=1}^i{F}_{1,y}+F_{2,y}+F_{3,y}+\cdots +F_{i,y}$

Newton’s 1st Law (inertia): ${\vec{F}}_{net}=0⟺\text{object at rest or moving at constant velocity}$

Newton’s 2nd Law: ${\vec{F}}_{net}=m\vec{a}\text{or}\sum \vec{F}=m\vec{a}\text{or}\vec{a}=\frac{{\vec{F}}_{net}}{m}$ The acceleration vector points in the same direction as the net force.

Newton’s 3rd Law (action–reaction): ${\vec{F}}_{A\text{on}B}=-{\vec{F}}_{B\text{on}A}$

Worked examples

Example 1 — Newton’s 2nd law on a train

A train has mass $m_T=1.2\times 10^6\text{kg}$.

(a) What force is required to accelerate it at $a_x=2.4{\text{m/s}}^2$?

FBD of train: ${\vec{F}}_N$ up, ${\vec{F}}_W=mg$ down, ${\vec{F}}_{app}$ horizontal in the $+x$ direction. Vertical forces cancel. Apply Newton’s 2nd law in $x$: $\sum F_x=m{a}_x$ $F_{app}=(1.2\times 10^6)(2.4)=2.88\times 10^6\text{N}$

(b) Train is unloaded, $m_T=9.8\times 10^5\text{kg}$, with the same applied force. Find new acceleration. $a_x=\frac{F_{app}}{m_T}=\frac{2.88\times 10^6}{9.8\times 10^5}=2.94{\text{m/s}}^2$

Example 2 — Motorboat (buoyancy + drag)

A $960\text{kg}$ motorboat accelerates from a dock at $a_x=2.1{\text{m/s}}^2$ with thrust $F_{thrust}=4.1\text{kN}=4100\text{N}$. Find buoyancy $F_B$ and drag $F_{drag}$.

FBD of boat: ${\vec{F}}_{thrust}$ forward ($+x$), ${\vec{F}}_{drag}$ backward ($-x$), ${\vec{F}}_W=mg$ down, ${\vec{F}}_B$ up.

$y$-plane (no vertical acceleration, $a_y=0$): $\sum F_y=0:F_B-F_W=0$ $F_B=mg=960\times 9.8=9408\text{N}$

$x$-plane ($a_x=2.1{\text{m/s}}^2$): $\sum F_x=m{a}_x:F_{thrust}-F_{drag}=m{a}_x$ $F_{drag}=F_{thrust}-m{a}_x=4100-960\times 2.1=2084\text{N}$

Example 3 — Spring (stretch and compression)

Spring with $k=270\text{N/m}$.

(a) Force to stretch the spring by $x=48\text{cm}=0.48\text{m}$: $|F_S|=k|x|=270\times 0.48=129.6\text{N}$

(b) Held vertically with $m=5\text{kg}$ placed on top — how far does it compress?

FBD of mass: ${\vec{F}}_S$ up (spring pushes back to natural length when compressed), ${\vec{F}}_W=mg$ down. Mass is in equilibrium, $a_y=0$: $\sum F_y=0:F_S-F_W=0$ $kx-mg=0$ $x=\frac{mg}{k}=\frac{5\times 9.8}{270}=0.181\text{m}$

Gravity detour (from slide 14 / handwritten notes)

Case 1 — Two $1\text{kg}$ masses, $1\text{m}$ apart: $F=\frac{(6.67\times 10^{-11})(1)(1)}{1^2}=6.67\times 10^{-11}\text{N}$ (Why we ignore mutual gravitation between small objects.)

Case 2 — $1\text{kg}$ mass on Earth’s surface ($r_{earth}=6.37\times 10^6\text{m}$, $m_{earth}=5.98\times 10^{24}\text{kg}$): $F=\frac{(6.67\times 10^{-11})(5.98\times 10^{24})(1)}{(6.37\times 10^6{)}^2}=9.8\text{N}$ This recovers $g\approx 9.8{\text{m/s}}^2$.

Things to practise

From Tutorial 4 (Wolfson Chapters 4 & 5):

1. Exercise 1 — A $910\text{kg}$ racing car starts from rest and covers $350\text{m}$ in $4.85\text{s}$ at constant acceleration. (a) Find the force on the car. (b) Find its final velocity. (Combine kinematics + Newton’s 2nd law.)
2. Exercise 2 — A $52\text{kg}$ passenger in an elevator accelerating downward at $2.4{\text{m/s}}^2$. (a) What force does the floor exert on the passenger? (b) If the elevator now accelerates upward at $1.8{\text{m/s}}^2$, what force does the passenger exert on the floor? (Apparent weight / Newton’s 3rd law.)
3. Exercise 3 — An elevator moving upward at constant speed decelerates to a stop in $0.53\text{s}$. Find the maximum speed such that the passengers remain on the floor (normal force does not vanish).
4. Exercise 4 — A spring with $k=340\text{N/m}$ is used to weigh a $6.7\text{kg}$ fish. How far does the spring stretch? (Equilibrium with $kx=mg$.)
5. Exercise 5 — A $10\text{kg}$ block on a table with ${\mu }_s=0.4$, ${\mu }_k=0.2$. Find the block’s acceleration if (a) a horizontal force of $20\text{N}$ is applied; (b) $100\text{N}$ is applied. (Check whether static friction is overcome first.)
6. Exercise 6 — A skier is pulled up a $25^{\circ }$ slope at constant velocity by a tow bar (parallel to slope). Skier mass $55\text{kg}$, ${\mu }_k=0.120$. Find the tow-bar force magnitude. (Inclined-plane FBD; resolve gravity into components along and normal to slope.)

Also: Mastering Physics “Forces” set, Wolfson Chapter 3 (additional reading), and Portfolio 3 in the Workshop class.

Common pitfalls

From the lecture’s “Common Errors” slide and the productive-failure framing:

• Forces not in vector form — Every force on an FBD must have both magnitude and direction. Write $\vec{F}$ not just $F$.
• Wrong forces on the wrong body — Isolate the body of interest. Only draw forces acting on that body (not forces it exerts on others).
• Missing weight or normal force — Even if the question doesn’t mention them, ask whether they act. They almost always do for anything resting on a surface.
• Misidentifying action–reaction pairs — The normal force and gravity on the same object are not a Newton’s 3rd-law pair. (Gravity’s reaction is the pull of the object on the Earth; the normal’s reaction is the push of the object on the surface.)
• Ignoring drag/air resistance unless told to — Ignore air resistance unless the problem explicitly says to include it.
• Friction direction — Kinetic friction opposes the velocity vector; static friction opposes the impending motion, not necessarily any current motion.
• Spring sign convention — When the spring is compressed by $x$, ${\vec{F}}_S$ pushes back toward the natural length; when stretched, ${\vec{F}}_S$ pulls back. The handwritten Example 3(b) notes: “Spring is compressed → $F_S$ acts to push back to uncompressed state.”
• Units — Convert $\text{cm}\to \text{m}$ before plugging into $F_S=kx$; check $\text{N}=\text{kg}\cdot {\text{m/s}}^2$ at the end. Always Assess the result: is it believable? Are the units right? (Step 4 of the Model–Visualise–Solve–Assess approach.)

Source citations

• Lecture4_CTP1.pdf — slide 1 (title), slides 3–8 (productive failure framing), slide 9 (problem-solving approach Model/Visualise/Solve/Assess), slide 11 (definition of force), slide 12 (gravity, spring), slide 13 (universal gravitation formula and constants), slide 14 (gravity case studies), slide 15 (normal, tension), slide 16 (friction kinetic/static), slide 17 (thrust, drag), slides 19–20 (FBD construction + common errors), slides 21–22 (worked FBD example: safe/furniture/pulley), slide 24 (Newton’s three laws), slide 25 (net force in component form), slide 26 (Newton’s 2nd law forms), slide 27 (analysis approach), slide 28 (Example 1 train), slide 29 (Example 2 motorboat), slide 30 (Example 3 spring), slide 32 (Week 4 activities incl. Portfolio 3), slide 33 (Wolfson reference).
• EGD102 - Lecture4 - Notes.pdf — pp. 1–4 handwritten worked solutions for the gravity detour (Cases 1 & 2), Example 1 (train, both parts), Example 2 (motorboat — full $x$/$y$ decomposition giving $F_B=9408\text{N}$, $F_{drag}=2084\text{N}$), and Example 3 (spring stretch and compression, giving $x=0.181\text{m}$).
• Tutorial 4.pdf — slides 2–7 recap of common forces, FBD, Newton’s laws and analysis approach; Exercises 1–6 (slides 8, 9, 10, 11, 13, 14); slide 15 lists Mastering Physics, Wolfson Ch. 4, Pre-Lab 1 questions, and Portfolio 4 preparation.