Potential energy of a particle is `1/2momega^(2)x^(2)`, where m and `omega` are constants. Prove that the motion of the particle is simple harmonic.

The kinetic energy of a particle is 1/2momega^(2)(A^(2)-x^(2)) , where m, omega and A are constants. Prove that the motion of the particle is simple harmonic.

The equation x= A sin omega t gives the relation between the time t and the corresponding displacement x of a moving particle where A and omega are constant. Prove that the acceleration of the particle is proportional to its displacement and is directed opposite to it.

The speed of a body of mass m, when it is at a distance x from the orign, is v. Its total energy is 1/2mv^(2)+1/2kx^(2) , where k is constant. Prove that the body executes a simple harmonic motion.

The displacement x of a particle in rectilinear motion at time t is represented as x^(2) = at^(2) +2bt+c , where a ,b, and c are constants. Show that the acceleration of this particle varies (1)/(x^(3)) .

Derive an expression for the total energy of a particle undergoing simple harmonic motion.

A particle of mass m is located in a one dimensional potential field where potential energy is given by V(x) = A(1- cos Px) , where A and P are constant . The period of small oscillations of the particle is

Find expressions for potential and kinetic energies of a particle executing simple harmonic motion.

Displacement (x) and time (t) of a particle in motion are related as x = at + bt^(2) -ct^(3) where a,b, and c, are constants. Velocity of the particle when its acceleration becomes zero is