Forces of 5 N and 3 N act on a particle with an angle of 60° between them. What is the size of their resultant?
- Use R2 = A2 + B2 + 2AB cos θ.
- R2 = 52 + 32 + 2 × 5 × 3 × cos 60° = 25 + 9 + 15 = 49.
- R = √49 = 7.
Answer: 7 N
Vectors and equilibrium
NUST NET (Engineering) · Physics · Measurement, vectors and equilibrium. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
The idea
A vector has size and direction. The fast way to handle vectors is by components: split each one into parts along i, j and k, then add or subtract the matching parts.
Two vectors can be multiplied in two ways. The scalar product A · B gives a number and is zero when the vectors are perpendicular. The vector product A × B gives a vector at right angles to both and is zero when they are parallel. Work is a scalar product; torque is a vector product.
A body is in equilibrium when the forces on it add to zero and the torques about any point add to zero.
Rules to remember
- Components: Ax = A cos θ, Ay = A sin θ (θ measured from the x-axis); A = √(Ax2 + Ay2), tan θ = Ay/Ax
- Resultant of A and B with angle θ between them: R = √(A2 + B2 + 2AB cos θ); largest A + B, smallest |A − B| (the difference of the magnitudes)
- Scalar product: A · B = AB cos θ = AxBx + AyBy + AzBz; i · i = 1, i · j = 0
- Vector product: |A × B| = AB sin θ = area of the parallelogram on A and B (the triangle is half of it)
- i × j = k, j × k = i, k × i = j; reversing the order changes the sign: A × B = −(B × A)
- Torque τ = r × F, size rF sin θ = force × perpendicular distance from the pivot (N m)
- Equilibrium: sum of forces = 0 and sum of torques = 0 (clockwise moments = anticlockwise moments)
Common mistake
Swapping the order in a vector product. j × i = −k, not +k, so in τ = r × F always write r first and keep the signs of each term.
Worked example
Practice questions
Try each one, then open the answer.
1. Two sides of a triangle are represented by the vectors A = 3i m and B = 4j m. The area of the triangle is
- A6 m2
- B12 m2
- C7 m2
- D3.5 m2
Show answer
Answer: A. Area of the triangle = ½|A × B| = ½ × 3 × 4 × sin 90° = 6 m2. 12 m2 is |A × B| itself, the area of the parallelogram.
2. For two vectors A and B, |A × B| = √3 (A · B). The angle between them is
- A30°
- B45°
- C90°
- D60°
Show answer
Answer: D. AB sin θ = √3 AB cos θ gives tan θ = √3, so θ = 60°. Swapping sin and cos gives tan θ = 1/√3, that is 30°.
3. A force F = (i + 3j) N acts at the point r = (2i + j) m. The torque τ = r × F about the origin is
- A5k N m
- B7k N m
- C−5k N m
- D6k N m
Show answer
Answer: A. τ = (2i + j) × (i + 3j) = 6(i × j) + 1(j × i) = 6k − k = 5k N m. Adding the two products gives 7k; working out F × r instead of r × F gives −5k.
Read the full notes
Keep going
← MeasurementMotion and force →All rules on one pageStuck? Ask a question