The necessary condition for simple harmonic motion is option 3: displacement and acceleration are proportional.

In simple harmonic motion, the object oscillates back and forth around its equilibrium position. The displacement is the distance from the equilibrium position at any given time, and the acceleration is the rate of change of velocity.

When the displacement and acceleration are proportional, it means that as the displacement increases, the acceleration increases in the same proportion, and as the displacement decreases, the acceleration decreases in the same proportion. This relationship is described by Hooke`s law, which states that the force acting on an object is directly proportional to the displacement of the object from its equilibrium position.

Option 1, constant period, is not a necessary condition for simple harmonic motion. While simple harmonic motion does have a constant period, it is not a requirement for the motion to be considered simple harmonic.

Option 2, constant acceleration, is not correct. The acceleration in simple harmonic motion is not constant, but rather varies sinusoidally.

Option 4, displacement and torque being proportional, is not a necessary condition for simple harmonic motion. Torque is not typically involved in simple harmonic motion.

Overall, option 3, displacement and acceleration being proportional, is the necessary condition for simple harmonic motion.