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

Question

Accepted Answer

Correct answer: A. (a) To find the angle between F and p, we first need to understand the relationship between force and momentum. The force F acting on a particle is related to the rate of change of its linear momentum p with respect to time, as described by Newton's second law of motion: F= dp dt Given the expression for the momentum p (t )=i cos ⁡ ( kt )-j sin ⁡ ( kt ), we can find F by differentiating p with respect to t : dp dt=-ki sin ⁡ ( kt )-kj cos ⁡ ( kt ) So, F=-ki sin ⁡ ( kt )-kj cos ⁡ ( kt ). Now, to find the angle between F and p, we use the dot product formula: F D= |F | |D | cos ⁡ ( ) where is the angle between F and p. However, in this case, it's more insightful to see if F and p are orthogonal (at a 2 angle to each other), because the dot product of two perpendicular vectors is zero. The dot product of F and p is: (-ki sin ⁡ ( kt )-kj cos ⁡ ( kt ) ) (i cos ⁡ ( kt )-j sin ⁡ ( ki- ksin ⁡ ( kt ) cos ⁡ ( kt )+ kcos ⁡ ( kt ) sin ⁡ ( kt )=0 The result is zero, indicating that the angle between F and p is indeed 2.

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

Solution

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