Two forces of 5·0 N each are acting on a point mass. If the angle between the forces is 60°, then the net force acting on the point mass has magnitude close to :
- (a)8·6 N
- (b)4·3 N
- (c)5·0 N
- (d)6·7 N
Correct — A, about 8·6 N. Two forces are combined by the parallelogram (law of cosines) rule: the resultant of forces P and Q with angle θ between them has magnitude √(P² + Q² + 2PQcosθ). With P = Q = 5 N and θ = 60° (so cos60° = 0·5): R = √(25 + 25 + 2 × 5 × 5 × 0·5) = √(25 + 25 + 25) = √75 ≈ 8·66 N, which rounds to 8·6 N.
- (b)4·3 N — This is roughly half the resultant, as if the forces partly cancelled. Two 5 N forces only 60° apart largely reinforce, giving about 8·6 N, not 4·3 N.
- (c)5·0 N — The resultant equals a single 5 N force only if the two forces were 120° apart. At 60° they reinforce more, giving √75 ≈ 8·6 N.
- (d)6·7 N — This does not match √(25 + 25 + 25) = √75. A value near 6·7 N would need a wider angle (smaller cosθ); at 60° the answer is about 8·6 N.
When two forces act at a point, their combined effect is a single resultant found by the parallelogram law. Its magnitude is √(P² + Q² + 2PQcosθ), where θ is the angle between them. The smaller the angle, the more the forces reinforce; at 0° they simply add, and at 180° they subtract.
Substitute straight into √(P² + Q² + 2PQcosθ). Equal 5 N forces at 60° give three equal terms of 25 under the root, so R = √75 ≈ 8·66 N. Checking the limits helps: at 0° it would be 10 N, at 90° about 7·1 N, and at 120° exactly 5 N — 8·6 N sits sensibly between the 0° and 90° values.
- Resultant of two forces: R = √(P² + Q² + 2PQcosθ).
- For P = Q = 5 N and θ = 60°: R = √(25 + 25 + 25) = √75 ≈ 8·66 N.
- Equal forces at 60° reinforce strongly — the resultant (8·6 N) is close to their sum (10 N).
- Two equal forces give a resultant equal to one of them only when the angle between them is 120°.
- Simply adding the magnitudes to 10 N and ignoring the angle.
- Using the wrong angle in the cosine — it is the angle between the two forces.
Two forces of given magnitude act at a stated angle and you find the resultant — apply R = √(P² + Q² + 2PQcosθ) and sanity-check against the 0°/90°/180° limits.
Which one of the following is a vector quantity?
- (a) Momentum
- (b) Pressure
- (c) Energy
- (d) Work
Answer(a) Momentum
Force, like momentum, is a vector with both magnitude and direction — which is exactly why two forces must be combined by the parallelogram law, accounting for the angle, rather than added as plain numbers.
- practice — not a real PYQ
Two equal forces of 10 N act at 120° to each other. Their resultant is:
- (a)20 N
- (b)10 N
- (c)0 N
- (d)17·3 N
Answer(b) 10 N — for equal forces a 120° angle gives a resultant equal to one of them: √(100 + 100 + 2 × 100 × (−0·5)) = √100 = 10 N.
- practice — not a real PYQ
Two forces of 3 N and 4 N act at right angles to each other. Their resultant is:
- (a)1 N
- (b)5 N
- (c)7 N
- (d)12 N
Answer(b) 5 N — at 90°, R = √(3² + 4²) = √25 = 5 N.