Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20245 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
5 April
Shift
Evening Shift
Subject
Physics
Chapter
Mathematics in Physics

JEE Main 2024 · 5 April · Evening Shift · Physics · Mathematics in Physics

A particle moves in x-y plane under the influence of a force F such...

A particle moves in x−yx-y plane under the influence of a force F⃗\vec{F} such that its linear momentum is p→(t)=i^cos⁡(kt)−j^sin⁡(kt)\overrightarrow{\mathrm{p}}(\mathrm{t})=\hat{i} \cos (\mathrm{kt})-\hat{j} \sin (\mathrm{kt}). If k\mathrm{k} is constant, the angle between F→\overrightarrow{\mathrm{F}} and p→\overrightarrow{\mathrm{p}} will be :

Options

  1. Aπ4\frac{\pi}{4}
  2. Bπ6\frac{\pi}{6}
  3. Cπ2\frac{\pi}{2}
  4. Dπ3\frac{\pi}{3}

Correct answer

C. π2\frac{\pi}{2}

Step-by-step solution

P→=cos⁡(kt)i^−sin⁡(kt)j^;∣P→∣=1∵P→=mv→∴P^=v^⇒v^=cos⁡(kt)i^−sin⁡(kt)j^a^=−ksin⁡(kt)i^−kcos⁡(kt)j^k⇒a^=−sin⁡kti^−cos⁡ktj^∵F^=a^=−sin⁡kti^−cos⁡ktj^cos⁡θ=F^⋅P^∣F^∣∣P^∣=−sin⁡ktcos⁡t+sin⁡ktcos⁡t1×1=0⇒θ=π2\begin{aligned} & \overrightarrow{\mathrm{P}}=\cos (\mathrm{kt}) \hat{\mathrm{i}}-\sin (\mathrm{kt}) \hat{\mathrm{j}} ;|\overrightarrow{\mathrm{P}}|=1 \\ & \because \overrightarrow{\mathrm{P}}=\mathrm{m} \overrightarrow{\mathrm{v}} \\ & \therefore \hat{\mathrm{P}}=\hat{\mathrm{v}} \\ & \Rightarrow \hat{\mathrm{v}}=\cos (\mathrm{kt}) \hat{\mathrm{i}}-\sin (\mathrm{kt}) \hat{\mathrm{j}} \\ & \hat{\mathrm{a}}=\frac{-\mathrm{k} \sin (\mathrm{kt}) \hat{\mathrm{i}}-\mathrm{k} \cos (\mathrm{kt}) \hat{\mathrm{j}}}{\mathrm{k}} \\ & \Rightarrow \hat{\mathrm{a}}=-\sin k t \hat{\mathrm{i}}-\cos \mathrm{kt} \hat{\mathrm{j}} \\ & \because \hat{\mathrm{F}}=\hat{\mathrm{a}}=-\sin \mathrm{kt} \hat{\mathrm{i}}-\cos \mathrm{kt} \hat{\mathrm{j}} \\ & \cos \theta=\frac{\hat{\mathrm{F}} \cdot \hat{\mathrm{P}}}{|\hat{\mathrm{F}}||\hat{\mathrm{P}}|}=-\frac{\sin \mathrm{kt} \cos \mathrm{t}+\sin \mathrm{kt} \cos \mathrm{t}}{1 \times 1}=0 \\ & \Rightarrow \theta=\frac{\pi}{2}\end{aligned}
Practice Mathematics in Physics on Quantrex Academy →

More from Mathematics in Physics

The density ρ of a uniform cylinder is determined by measuring its mass m , length l and… JEE Main 2026Four persons measure the length of a rod as 20.00 ~cm, 19.75 ~cm, 17.01 ~cm and 18.25 cm.… JEE Main 2026In an experiment the values of two spring constants were measured as k_ 1 =(10 ± 0.2) N /… JEE Main 2026Length, breadth and thickness of a strip having a uniform cross section are measured to… JEE Advanced 2025A quantity Q is formulated as X^ -2 Y^ +(3)/(2) Z^ -(2)/(5) · X, Y and Z are independent… JEE Main 2025A physical quantity C is related to four other quantities p, q, r and s as follows C=… JEE Main 2025A physical quantity Q is related to four observables a, b, c, d as follows : Q= ab ^4 ~cd… JEE Main 2025For an experimental expression y=(32.3 × 1125)/(27.4) , where all the digits are… JEE Main 2025 Full Mathematics in Physics list All JEE Main PYQs