Mechanics: Forces and Newton's Laws
Forces and Newton's laws is the A Level Mechanics topic covering how the forces acting on a particle produce acceleration, using Newton's first, second and third laws together with F = ma. It covers resolving forces on horizontal and inclined planes, connected particles joined by strings, pulleys and friction. It is examined in the compulsory Mechanics content of every A Level Maths specification.
Before you start
Make sure you're comfortable with these topics first:
Method
- Draw a clear force diagram showing every force acting on the particle (weight, normal reaction, tension, friction, applied forces), with directions and labels.
- Choose a positive direction, usually the direction of motion, and resolve every force along and perpendicular to it.
- Apply Newton's second law, F = ma, in the direction of motion, using the sum of the resolved components as the resultant force.
- For connected particles, write a separate equation of motion for each particle, or one whole-system equation if you only need the acceleration, then solve simultaneously for any unknown tension.
- If friction is involved, resolve perpendicular to the surface first to find the normal reaction R, then use F = mu R for limiting friction or F <= mu R for equilibrium.
- Check that the units are consistent and that the sign of your answer makes sense with the positive direction you chose.
Worked example
A crate of mass 20 kg is pulled in a straight line across rough horizontal ground by a horizontal rope with tension 60 N. The resistance to motion is a constant 12 N. Take g = 9.8 m/s^2. Find (a) the acceleration of the crate and (b) the normal reaction between the crate and the ground.
- Draw the force diagram: weight mg acts down, normal reaction R acts up, tension T = 60 N acts in the direction of motion, and resistance 12 N opposes motion.
- Resolve horizontally: resultant force = 60 - 12 = 48 N.
- Apply Newton's second law: 48 = 20a, so a = 2.4 m/s^2.
- Resolve vertically: since the rope is horizontal it has no vertical component, so R = mg = 20 x 9.8 = 196 N.
- Final answer: acceleration = 2.4 m/s^2, normal reaction R = 196 N.
Practice questions
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Q1A box of mass 4 kg is pushed along smooth horizontal ground by a horizontal force of 20 N. Find its acceleration.Show answer
Answer: a = 5 m/s^2 (F = ma: 20 = 4a)
Q2A resultant force of 18 N acts on a particle of mass 3 kg. Find the acceleration produced.Show answer
Answer: a = 6 m/s^2 (18 = 3a)
Q3Tom, of mass 60 kg, stands in a lift that accelerates upward at 2 m/s^2. Take g = 9.8 m/s^2. Find the normal reaction force the lift floor exerts on Tom.Show answer
Answer: R = 708 N (R = 60(9.8 + 2))
Q4Two blocks, A (mass 6 kg) and B (mass 4 kg), connected by a light inextensible string, are pulled across a smooth horizontal surface by a horizontal force of 40 N applied to A, with B behind A. Find the acceleration of the system and the tension in the string.Show answer
Answer: a = 4 m/s^2 (40 = 10a); T = 16 N (equation of motion for B: T = 4 x 4)
Q5A block of mass 5 kg rests in equilibrium on a smooth plane inclined at 25 degrees to the horizontal, held by a light string parallel to the slope. Take g = 9.8 m/s^2. Find the tension in the string.Show answer
Answer: T = 20.7 N (T = 5 x 9.8 x sin25)
Q6Particles P (mass 7 kg) and Q (mass 5 kg) are connected by a light inextensible string passing over a smooth fixed pulley and hang freely on either side, released from rest. Take g = 9.8 m/s^2. Find the acceleration of the system and the tension in the string.Show answer
Answer: a = 1.63 m/s^2 (g(M-m)/(M+m) = 9.8 x 2/12); T = 57.2 N (T = 5(9.8 + 1.63))
Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
A crate of mass 15 kg rests on a rough plane inclined at 18 degrees to the horizontal. The coefficient of friction between the crate and the plane is 0.4. Take g = 9.8 m/s^2. Show that the crate remains in equilibrium on the plane, and state the friction force acting on it.
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Particle A, of mass 5 kg, lies on a rough plane inclined at 30 degrees to the horizontal. A is connected by a light inextensible string, passing over a smooth pulley fixed at the top of the plane, to particle B, of mass 4 kg, which hangs freely below the pulley. The coefficient of friction between A and the plane is 0.2. The system is released from rest, with B moving downward and A moving up the plane. Take g = 9.8 m/s^2. Find the acceleration of the system.
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A particle P of mass 4 kg moves on a smooth horizontal plane under the action of two constant forces, F1 = (5i + 3j) N and F2 = (-2i + j) N, where i and j are perpendicular unit vectors in the plane. Given that P starts at rest, find (a) the acceleration of P and (b) the speed of P after 3 seconds, giving your answer to 3 significant figures.
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Free printable worksheet
Want more practice on paper? Download the mechanics: forces and newton's laws worksheet pack - 16 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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