A car towing a trailer, two masses over a pulley, a block dragged across a table by a falling weight: when particles are joined by a light inextensible string or a rigid rod, they share one acceleration, and Newton's second law for each particle finds both it and the tension.
A light inextensible string has the same tension all along it (also over a smooth pulley), and the particles it joins have accelerations of equal size. Write \(F = ma\) for EACH particle in its own direction of motion, then solve the equations together. A tow-bar can be in tension or in thrust.
Fig. 1: a 3 kg block on a rough table (μ = 0.2) joined by a light inextensible string over a smooth pulley to a hanging 2 kg block. The tension T is the same on both sides; both blocks have the same acceleration a
Section 01
Car and trailer.
For a car towing a trailer in a straight line, you can write \(F = ma\) for the car, for the trailer, or for the whole system. The tension is internal to the whole system, so it cancels there; use the trailer on its own to find it.
Worked example Tension in a tow-rope
A car of mass 1200 kg tows a trailer of mass 400 kg along a straight horizontal road with a light rope. The driving force is 2500 N, and the resistances are 300 N on the car and 200 N on the trailer. Find the acceleration and the tension.
(Check with the car: \(2500 - 300 - 700 = 1500 = 1200 \times 1.25\).)
Answer: \(a = 1.25\ \text{m s}^{-2}\), \(T = 700\) N.
Worked example Braking with a tow-bar
The same car and trailer are now joined by a light rigid tow-bar. The engine is switched off and the car brakes with a force of 1800 N; the resistances are unchanged. Find the force in the tow-bar.
Trailer: let the tow-bar exert a forward force \(P\) on it: \(P - 200 = 400(-1.4375) = -575\).
\(P = -375\): the tow-bar pushes BACKWARDS on the trailer.
Answer: A thrust of 375 N in the tow-bar.
Examiner trap
A rope can only pull. A tow-bar can push, and when the car brakes hard it does. If your tension comes out negative, the connector is in thrust (for a rope, that would mean it goes slack).
Section 02
Strings over pulleys.
Over a smooth pulley the tension is the same on both sides, but the two particles move in different directions. Write \(F = ma\) for each particle in its OWN direction of motion, then add the equations to eliminate \(T\).
Worked example On a rough table (Fig. 1)
A block A of mass 3 kg rests on a rough horizontal table, \(\mu = 0.2\). A light inextensible string joins A, over a smooth pulley at the edge, to a block B of mass 2 kg hanging freely. Find the acceleration and the tension.
A: \(R = 30\) N, friction \(= 0.2 \times 30 = 6\) N. Along the table: \(T - 6 = 3a\).
B, downwards: \(20 - T = 2a\).
Add: \(14 = 5a\), so \(a = 2.8\ \text{m s}^{-2}\).
\(T = 20 - 2(2.8) = 14.4\) N.
Answer: \(a = 2.8\ \text{m s}^{-2}\), \(T = 14.4\) N.
Worked example Two hanging masses
Particles of mass 5 kg and 3 kg hang on either side of a smooth pulley, connected by a light inextensible string, and are released from rest. Find the acceleration, the tension, and the force the string exerts on the pulley.
5 kg (down): \(50 - T = 5a\). 3 kg (up): \(T - 30 = 3a\).
Add: \(20 = 8a\), so \(a = 2.5\ \text{m s}^{-2}\).
\(T = 30 + 3(2.5) = 37.5\) N.
Both sides of the string pull down on the pulley: \(2T = 75\) N.
Answer: \(2.5\ \text{m s}^{-2}\); 37.5 N; 75 N on the pulley.
Worked example Plane and pulley
A particle A of mass 4 kg lies on a smooth plane inclined at \(30^\circ\). A string from A runs up the plane, over a smooth pulley at the top, to a particle B of mass 6 kg hanging freely. Find the acceleration and the tension.
B (down): \(60 - T = 6a\).
A (up the plane): \(T - 40\sin 30^\circ = 4a\), i.e. \(T - 20 = 4a\).
Add: \(40 = 10a\), so \(a = 4\ \text{m s}^{-2}\).
\(T = 60 - 24 = 36\) N.
Answer: \(a = 4\ \text{m s}^{-2}\), \(T = 36\) N.
Examiner trap
The tension is NOT the weight of the hanging particle. If it were, B would have no resultant force and could not accelerate. In Fig. 1, \(T = 14.4\) N, less than B's 20 N weight.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Two particles are joined by a light inextensible string that stays taut. What is true of their accelerations?
Quick check+10 XP
Over a smooth pulley, how do the tensions on the two sides of a light string compare?
Quick check+10 XP
A car brakes while towing a trailer with a rigid tow-bar. What force can the tow-bar exert on the trailer?
Quick check+10 XP
Particles of 5 kg and 3 kg hang either side of a smooth pulley. Find the acceleration.
Section 03
When the string goes slack.
If a falling particle hits the floor, the string goes slack and the tension becomes zero. The other particle carries on with the velocity it had at that instant, now under its own forces only (for a free particle, just gravity).
Worked example Greatest height after impact
In the 5 kg and 3 kg system above, the 5 kg particle starts 1.2 m above the floor and does not rebound. Find the greatest height the 3 kg particle reaches above its starting point (it does not reach the pulley).
While connected: \(v^2 = 2(2.5)(1.2) = 6\) when the 5 kg particle hits the floor; the 3 kg particle has risen 1.2 m.
Then the string is slack and the 3 kg particle moves freely: \(a = -10\).
\(0 = 6 - 20s\), so it rises a further \(s = 0.3\) m.
Greatest height \(= 1.2 + 0.3\).
Answer: 1.5 m above its starting point.
Your turn
Particles of mass 0.6 kg and 0.4 kg hang on either side of a smooth pulley and are released from rest. Find the acceleration and the tension.
\(6 - T = 0.6a\) and \(T - 4 = 0.4a\). Adding: \(2 = a\), so \(a = 2\ \text{m s}^{-2}\) and \(T = 4 + 0.8 = 4.8\) N.
Examiner trap
After the string goes slack, the rising particle does NOT stop at once; it keeps moving up and decelerates at \(g\). Forgetting this second stage gives 1.2 m instead of 1.5 m.
Stage 2 · Exam
Exam-style questions
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A 1200 kg car tows a 400 kg trailer with a light rope. Driving force 2500 N; resistances 300 N (car) and 200 N (trailer). Find the tension.
Exam style+20 XP
The same vehicles are joined by a rigid tow-bar; the car brakes with force 1800 N (no driving force), resistances unchanged. Find the force in the tow-bar.
Exam style+20 XP
A 3 kg block on a rough table (\(\mu = 0.2\)) is joined over a smooth pulley at the edge to a hanging 2 kg block. Find the tension.
Exam style+20 XP
Particles of 5 kg and 3 kg hang either side of a smooth pulley, released from rest with the 5 kg particle 1.2 m above the floor. It hits the floor and does not rebound. Find the greatest height of the 3 kg particle above its start.
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