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Displacement and Force in Two Dimensions

Displacement and Force in Two Dimensions — American System Physics Revision Notes & Topical Questions

5 min read 6 parts Lesson 1 of 5 1 quiz

📚 American System Physics Hub / Topical Revision
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Lesson summary

Vectors, Friction and Inclined Planes — AMERICAN SYSTEM PHYSICS notes: Add vectors graphically and by components and use friction and component methods to solve problems on flat surfaces and slopes.A scalar has size only; a vector has size and direction.

  • equilibrant
  • Tip-to-tail method
  • Perpendicular vectors
  • coefficient of static friction
  • parallel component of weight
  • angle of repose

Section 1 of 6Scalars, vectors and adding vectors

A scalar has size only; a vector has size and direction. Distance, speed, mass, time and energy are scalars; displacement, velocity, acceleration, force and momentum are vectors. The resultant is the vector sum of two or more vectors. The equilibrant is the force equal and opposite to the resultant (equilibrant = −resultant), producing equilibrium.

Scalars, vectors and adding vectors
Tip-to-tail method
Tip-to-tail method draw each vector to scale so the tail of each starts at the tip of the previous one; the resultant runs from the tail of the first to the tip of the last.
Perpendicular vectors
Perpendicular vectors use the Pythagorean theorem, R = √(A2 + B2).
Components
Rx = Ax + Bx + ..., Ry = Ay + By + ..., then R = √(Rx2 + Ry2).

Useful valuessin 30° = 0.5 and cos 30° ≈ 0.866.

Section 2 of 6Friction

Walking relies on static friction, and lubricants (for example oil between surfaces) reduce friction by separating them.

Friction
Static friction is a responsive force: if you push a heavy box gently with 50 N and it does not move, static friction is exactly 50 N.
Push harder and it rises to its maximum, 0 ≤ Fs ≤ μsFN; any greater push starts the box sliding.
Kinetic friction is then roughly constant, Fk = μkFN, and usually smaller than maximum static friction, which is why a box "jerks" into motion.

The coefficient of static friction is μs = Fs,max/FN; it has no unit and depends on the two surfaces.

Friction
Friction depends on the normal force and the nature of the surfaces, but to a good approximation not on contact area or sliding speed.
On a level surface with only weight and normal force vertical, FN = mg, so Fk = μkmg.

Section 3 of 6Inclined planes and forces in two dimensions

On a slope at angle θ the parallel component of weight is Fg∥ = mg sin θ (down the slope) and the normal force is FN = mg cos θ.

Angle of repose
An object is just about to slide when μs = tan θ, the angle of repose, and slides at constant velocity when μk = tan θ.
Two ropes at angle θ each side of the vertical holding weight W satisfy 2T cos θ = W, so T = W/(2 cos θ).

The method for any slope problemdraw the free-body diagram, choose axes along and perpendicular to the slope, resolve the weight into mg sin θ (along) and mg cos θ (perpendicular), set the perpendicular components equal to find FN, then use Fnet = mg sin θ − μkFN along the slope and a = Fnet/m.

Inclined planes and forces in two dimensions
A steeper slope increases the parallel component and reduces the normal force, so friction falls while the pull down the slope grows.
The same component method handles any set of forces at angles: resolve each into x and y, sum each direction, then combine the resultant with Pythagoras and tan−1.

Section 4 of 6Worked examples

  1. A 10 kg box on a floor with μk = 0.30 slides. Find the friction force (g = 9.8 m/s2). [2]2 marks

  2. A 10 kg box is pulled horizontally by 60 N and μk = 0.40. Find its acceleration. [3]3 marks

  3. A 5.0 kg block slides down a frictionless 30° slope. Find the acceleration and normal force. [3]3 marks

  4. The same block is on the 30° slope with μk = 0.20. Find its acceleration down the slope. [3]3 marks

  5. A 20 N sign hangs from two cables each at 30° to the vertical. Find the tension in each. [2]2 marks

Section 5 of 6Key terms

TermMeaning
ResultantThe vector sum of two or more vectors.
EquilibrantForce equal and opposite to the resultant, producing equilibrium.
Tip-to-tail methodGraphical vector addition where each vector starts at the tip of the previous one.
Coefficient of static frictionThe ratio of maximum static friction to the normal force; no unit.
LubricantA substance placed between surfaces to reduce friction.
Parallel component of weightThe component of weight down the slope, mg sin θ.
Angle of reposeThe steepest angle at which an object stays at rest on a slope; tan θ = μs.

Section 6 of 6Exam technique

1
Adding vector magnitudes directly — add by tip-to-tail or components.
2
Using weight instead of the normal force in F = μFN on a slope — FN = mg cos θ.
3
Thinking static friction is always at its maximum — it equals the applied force until the limit is reached.
4
Saying friction depends on contact area — to a good approximation it depends on normal force and surfaces.
5
Confusing the equilibrant with the resultant — it is equal in size but opposite in direction.

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