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The potential energy U in joule of a particle of mass 1 kg moving in x-y plane obeys the law U=3x+4y where (x,y) are the coordinates of the particle in metre. If the particle is at rest at (6,4) at time t=0 then:(A) the particle has constant acceleration(B) the particle has zero acceleration(C) the speed of the particle when it crosses x y axis is 10 m/s(D) coordinates of particle at t=1 sec are (4.5,2) Its a multiple correct answer type question

The potential energy U in joule of a particle of mass 1 kg moving in x-y plane obeys the law U=3x+4y where (x,y) are the coordinates of the particle in metre. If the particle is at rest at (6,4) at time t=0 then:(A) the particle has constant acceleration(B) the particle has zero acceleration(C) the speed of the particle when it crosses x y axis is 10 m/s(D) coordinates of particle at t=1 sec are (4.5,2) Its a multiple correct answer type question

Grade:11

1 Answers

Anand
16 Points
5 years ago
 
A particle of mass m = 1 kg moves in x-y plane, where there is a vector field E and vector force F.
      U = P.E. = 3 x + 4 y  Joules  in SI units
x component of force on the particle at (x,y):
      Fx = - dU/dx = - 3 Newtons
      Fy = - dU/dy = - 4 N
Magnitude of the force = 5 N
The direction of force on the particle at (x,y) = π + Tan⁻¹ (4/3)  with x axis or,   - (π - Tan⁻¹ 4/3).   The acceleration of the particle is in the same direction for all (x, y).
The displacement of the particle is along the path (straight line) with slope equal to :    tangent of that angle, ie., 4/3.
So the equation of path of the particle :  y = 4/3 x + c
As (6, 4) lies in this path:      4 = 4/3 * 6 +  c
                                              =>   c = - 4
=> equation of the path:  y = 4x/3 - 4    or   3 y - 4x + 12 = 0
This line meets y axis at x = 0, ie. ,  y = -4.
The point of crossing with y-axis:   (0, -4).
====
Acceleration of the particle in the field:  5 m/sec^2 in the direction in the 3rd quadrant.
We can also find the velocity and energy of the particle as it crosses the y-axis.
The displacement between :  (6 ,4) and (0, -4) = 10 m
so  v² = u² + 2 a s
          = 0 + 2 * 5 * 10 = 100
       v = 10 m/s
 Put y=2 you get x=4.5 there is no mean of time

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