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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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.
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.
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.
Free printable worksheet
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