Ch 4 · 4.4 Newton's Laws of Motion · A Level Mathematics 9709
Topic 4 of 5 in Mechanics — A Level Mathematics 9709 — 62 lessons.
Introduction
Section 4.1 dealt with particles that do not accelerate. Here the forces do not balance, and Newton's second law links the resultant force to the acceleration. Combine it with the suvat formulae and you can model lifts, sledges, cars towing trailers and masses hanging over pulleys. (Syllabus section 4.4, Paper 4.)
Key ideas — Newton's laws
Weight W = mg, with g = 10 m s⁻². Air resistance is only included if the question says so.
🧑🏫 Teacher's explanation
Imagine you are pushing a heavy shopping trolley. If you push it with a constant force and nothing is resisting you, the trolley will keep moving at the exact same speed forever. This is the First Law. If you suddenly push harder, the trolley speeds up; if you push less, it slows down. This relationship between your push, the trolley's mass, and its change in speed is the Second Law.
📖 Definitions
🔍 Worked example — Easy
A particle of mass 5 kg is acted upon by a constant resultant force of 20 N in the direction of its motion. Calculate its acceleration.
🔍 Worked example — Standard
A block of mass 8 kg is pulled up a rough inclined plane at a constant velocity. The component of its weight acting down the plane is 40 N. Calculate the magnitude of the friction force acting down the plane.
🔍 Worked example — Exam-style
Two particles, A (mass 3 kg) and B (mass 5 kg), are connected by a light inextensible string passing over a smooth, fixed pulley. Particle A is released from rest. Calculate the acceleration of the system and the tension in the string. Use g = 10 m s^-2.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — A method that always works
🧑🏫 Teacher's explanation
Imagine you are pushing a heavy supermarket trolley. If you push it with a force of 50 N but friction resists you with 10 N, the trolley doesn't just feel 50 N of force; it feels the 'net' result, which is 40 N. This net force is what actually causes the trolley to speed up according to F=ma.
📖 Definitions
🔍 Worked example — Easy
A particle of mass 3 kg is pulled up a smooth inclined plane at an angle of 30^° to the horizontal by a force of 20 N acting up the plane. Calculate the acceleration of the particle. Use g=10 m s^-2.
🔍 Worked example — Standard
Two particles, A of mass 2 kg and B of mass 3 kg, are connected by a light inextensible string passing over a smooth, fixed pulley. Calculate the acceleration of the system when released from rest. Use g=10 m s^-2.
🔍 Worked example — Exam-style
A car of mass 1200 kg is towing a trailer of mass 500 kg by a light rope. The car exerts a constant forward thrust of 4000 N. The total resistance (friction/air) acting on the car and trailer combined is 1000 N. Calculate the acceleration of the car and trailer. Use g=10 m s^-2.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Inclined planes
Resolve parallel and perpendicular to the plane. On a rough plane, friction F = μ R opposes the motion, so its direction changes when the particle changes direction. Moving up: weight component and friction both act down the slope, so the deceleration is large. Moving down: friction acts up the slope, so the acceleration is smaller. A particle that stops on a slope only slides back if mgsinα > μ R, i.e. tanα > μ.
🧑🏫 Teacher's explanation
Imagine you are sitting on a slide. If the slide is very steep, you zoom down quickly. If the slide is shallow, you might move slowly or even stay still. This is because gravity pulls you down, but the surface of the slide pushes back against you (the normal reaction) and rubs against you (friction).
📖 Definitions
🔍 Worked example — Easy
A particle of mass 5 kg rests on a smooth plane inclined at an angle of 30^° to the horizontal. Calculate the acceleration of the particle down the plane.
🔍 Worked example — Exam-style
A particle of mass m is on a rough plane inclined at angle α. If the particle is on the point of sliding down the plane, show that tanα = μ.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Connected particles
Particles joined by a light inextensible string have the same speed and size of acceleration, and the tension is the same throughout the string. Over a smooth pulley the tension is the same on both sides. Write F = ma for each particle separately, then solve simultaneously. You may also treat the whole system as one body (the tension is then an internal force and cancels), which quickly gives a.
🧑🏫 Teacher's explanation
Imagine two people playing tug-of-war with a rope. If the rope is 'light' and 'inextensible', it means the rope itself has no weight to move and it doesn't stretch like a rubber band. Because it doesn't stretch, both people must move at the exact same speed and acceleration. If one person moves 1 meter, the other moves 1 meter at the same time.
📖 Definitions
🔍 Worked example — Easy
Two particles, A and B, with masses 3 kg and 5 kg respectively, are connected by a light inextensible string passing over a smooth, fixed pulley. Calculate the acceleration of the system when released from rest. Use g = 10 m s^-2.
🔍 Worked example — Standard
A car of mass 1200 kg is towing a trailer of mass 400 kg by a light, inextensible rope. The car accelerates at 1.5 m s^-2 on a horizontal road. Calculate the tension in the rope. Use g = 10 m s^-2.
🔍 Worked example — Exam-style
Two particles P and Q are connected by a light inextensible string over a smooth pulley. Particle P has mass 4 kg and particle Q has mass 6 kg. A horizontal friction force of 10 N acts on particle Q opposing its motion. Calculate the acceleration of the system. Use g = 10 m s^-2.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: up and down a rough plane
A 2 kg particle is projected at 8 m s⁻¹ up a plane inclined at 30^°, with μ = 0.2. Find how far it travels up the plane and its speed when it returns to the start.
🔍 Worked example — Easy
A 5 kg particle is on a rough plane inclined at 20^°. The coefficient of friction is μ = 0.3. Calculate the maximum frictional force.
🔍 Worked example — Standard
A 2 kg particle is projected up a plane inclined at 30^° with speed 8 m s^-1. The coefficient of friction is μ = 0.2. Find the distance traveled up the plane. (Use g=10)
🔍 Worked example — Exam-style
A 2 kg particle is projected up a plane inclined at 30^° at 8 m s^-1 with μ = 0.2. Find its speed when it returns to the start of the plane. (Use g=10)
⚠️ Common errors
Worked examples — Example 2: particles over a pulley
Particles of mass 3 kg and 5 kg hang from the ends of a light inextensible string over a smooth pulley. Released from rest:
🔍 Worked example — Easy
Two masses, 2 kg and 4 kg, are connected by a light inextensible string over a smooth pulley. Find the acceleration of the system when released from rest. Use g = 10 m s⁻².
🔍 Worked example — Standard
A 3 kg mass and a 5 kg mass hang over a smooth pulley. Find the tension T in the string when the system is released from rest. Use g = 10 m s⁻².
🔍 Worked example — Exam-style
A 5 kg mass is connected to a 10 kg mass by a light inextensible string over a smooth pulley. A frictional force of 10 N resists the motion of the 10 kg mass as it moves upwards. Find the acceleration of the system. Use g = 10 m s⁻².
⚠️ Common errors
Worked examples — Example 3: table and hanging mass
Block A (4 kg) is on a rough horizontal table (μ = 0.25), joined by a string over a smooth pulley at the edge to B (2 kg), which hangs freely.
🔍 Worked example — Easy
A 3 kg mass A is on a smooth horizontal table, connected by a light inextensible string over a smooth pulley to a 2 kg mass B hanging vertically. Calculate the acceleration of the system and the tension in the string. (Use g = 10 m s^-2)
🔍 Worked example — Standard
Block A (4 kg) is on a rough horizontal table (μ = 0.25) joined by a string over a smooth pulley to mass B (2 kg) hanging freely. Find the acceleration a and tension T. (Use g = 10 m s^-2)
🔍 Worked example — Exam-style
A particle A of mass 5 kg is connected by a light inextensible string over a smooth pulley to particle B of mass 3 kg. Particle A is on a rough inclined plane making an angle of 30^° with the horizontal. The coefficient of friction between A and the plane is μ = 0.4. Calculate the acceleration of the system if B is released from rest. (Use g = 10 m s^-2)
⚠️ Common errors
Worked examples — Example 4: car and trailer
A 1200 kg car tows a 400 kg trailer with a tow-bar. The driving force is 3000 N; resistances are 600 N on the car and 200 N on the trailer.
🔍 Worked example — Easy
A 1000 kg car pulls a 500 kg trailer. The driving force is 4000 N and the total resistance is 500 N. Calculate the acceleration of the system.
🔍 Worked example — Standard
A 1200 kg car tows a 400 kg trailer with a tow-bar. The driving force is 3000 N; resistances are 600 N on the car and 200 N on the trailer. Find the tension in the tow-bar.
🔍 Worked example — Exam-style
A car of mass 1500 kg pulls a trailer of mass 500 kg. The driving force is 5000 N. The resistance on the car is 400 N and the resistance on the trailer is 200 N. Calculate the tension in the tow-bar.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) T - 6000 = 600(1.5) ⇒ T = 6900 N. (2) 25 - 15 = 5a ⇒ a = 2 m s⁻². (3) a = 10sin 20^° = 3.42; v = 10.3 m s⁻¹. (4) a = 60 - 40/10 = 2; v^2 = 2(2)(1) ⇒ v = 2 m s⁻¹. (5) Trailer: forces backwards = 400 × 2 = 800 N; resistance gives 200 N, so the tow-bar pushes back with 600 N: a thrust of 600 N.