A particle of mass m in the xy -plane experiences a two-dimensional restoring force given by F = -k(x i + y j ) , where k is a positive constant. The particle is displaced to a point P(x₀, y₀) , where x₀ and y₀ are non-zero and unequal, and is released from rest at t=0 . The shape of the trajectory of the particle will be a/an:

Step-by-step solution

The force acting on the particle is F = -kx i - ky j . This can be resolved into two independent components: F_x = -kx and F_y = -ky . These represent independent simple harmonic motions along the x and y axes with the same angular frequency = k m . The general equations for the displacements are x(t) = A_x ( t + _x) and y(t) = A_y ( t + _y) . Since the particle is released from rest at t=0 from the point (x₀, y₀) , its initial velocity components are zero. This means it starts from the extreme positions for both x