Newton's laws and connected particles
Newton's laws connect forces with motion. In mechanics questions you apply F = ma to a whole system, or to one part of it, to find an acceleration and then the force in a rope or tow bar.
Part 1 of 3: Learn it
In short
- Resultant force = mass × acceleration, with the force in the direction of the acceleration.
- Objects joined by a light, inextensible string have the same acceleration (and the same tension throughout).
- Treat the system as one object to find a; then one part on its own to find the tension.
Where this is in your specification
Spec points: DfE R1 to R6 (AQA 7357 R1 to R6, Edexcel 9MA0 Mechanics topic 8, OCR A H240 3.03)
| Board | Topic: Forces and Newton's laws |
|---|---|
| DfE content | R1-R6 |
| AQA 7357 | R1-R6 |
| Edexcel 9MA0 | Mechanics topic 8 |
| OCR H240 | 3.03 |
Newton's three laws
- First law: an object stays at rest or keeps a constant velocity unless a resultant force acts on it.
- Second law: F = ma, where F is the resultant force in newtons, m the mass in kg and a the acceleration in m/s².
- Third law: if A exerts a force on B, B exerts an equal and opposite force on A.
Forces to include
| Force | Acts |
|---|---|
| Weight, mg | vertically downwards |
| Normal reaction, R | perpendicular to the surface |
| Tension or thrust | along a string (tension pulls) or rod (thrust can push) |
| Resistance or friction | against the direction of motion |
Connected particles
For a car towing a trailer, both have the same acceleration, and the tow bar pulls the trailer forwards with the same force as it pulls the car backwards (third law). Over a smooth pulley, a string has the same tension on both sides, and the heavier particle accelerates down while the lighter one accelerates up at the same rate.
A resultant force of 20 N acts on a 4 kg mass. Find its acceleration.
Show the answer
a = 20 ÷ 4 = 5 m/s².
Part 2 of 3: See it worked
Worked examples
Example 1
A 1200 kg car tows a 400 kg trailer. The driving force is 2400 N, with resistances of 300 N on the car and 100 N on the trailer. Find the acceleration and the tension in the tow bar.
- Whole system: 2400 − 300 − 100 = 1600a, so a = 1.25 m/s²
- Trailer alone: T − 100 = 400 × 1.25 = 500
- T = 600 N
Answer: 1.25 m/s² and 600 N.
Example 2
Particles of 7 kg and 3 kg hang from a light string over a smooth pulley and are released from rest. Find the acceleration and the tension. (g = 9.8 m/s²)
- 7 kg: 7g − T = 7a. 3 kg: T − 3g = 3a
- Adding the two equations: 4g = 10a, giving a = 3.92 m/s²
- T = 3g + 3a = 3 × 13.72 = 41.16 N
Answer: 3.92 m/s² and 41.2 N (3 s.f.).
Example 3
A 600 kg lift accelerates upwards at 0.5 m/s². Find the tension in its cable.
- Up is positive: T − 600g = 600 × 0.5
- T = 600 × (9.8 + 0.5)
Answer: 6180 N.
Common mistakes
- Putting the tension into the whole-system equation. Internal forces cancel out.
- Using weight in kg instead of newtons.
- Writing F = ma with F as one force instead of the resultant force.
- Giving the two particles over a pulley different accelerations.
What does "inextensible" tell you about a string?
Show the answer
It does not stretch, so the particles it joins have the same acceleration.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
A resultant force of 20 N acts on a 4 kg mass. Find its acceleration.
a = 20 ÷ 4 = 5 m/s².
What does "inextensible" tell you about a string?
It does not stretch, so the particles it joins have the same acceleration.
State Newton's third law.
When one object exerts a force on another, the second exerts a force of the same size in the opposite direction on the first.
Jobs that use this
- Mechanical engineer (opens a new tab)
- Aerospace engineer (opens a new tab)
- Structural engineer (opens a new tab)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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