A particle moves in x-y plane under the influence of a force F⃗ such that its linear momentum is p⃗(t)=î cos (kt)-ĵ sin (kt). If k is constant, the angle between F⃗ and p⃗ will be :

Solution

Physics sir · 4 steps

Force is the rate of change of momentum, so differentiate p⃗ and take a dot product.

1. Find the force. By Newton's second law, F⃗ = dp⃗/dt. Differentiating each component gives F⃗ = -k sin(kt) î - k cos(kt) ĵ.
2. Take the dot product. Multiply matching components of F⃗ and p⃗ and add them.

   F⃗·p⃗ = (-k sin kt)(cos kt) + (-k cos kt)(-sin kt) = 0

3. Why it is zero. The magnitude |p⃗| = √(cos² kt + sin² kt) = 1 never changes. Only the direction of p⃗ turns, and a vector of fixed length can only change at right angles to itself.
4. So. Both vectors are nonzero and their dot product is zero, so cos θ = 0 and the angle is π/2, option (a).

The idea to keep. If a vector keeps a constant magnitude, its rate of change is always perpendicular to it. Check |p⃗| first and you often get the angle without any calculation.

Answer from the NTA final answer key. Solution by PrepWiser.