my physics teacher said that in simple harmonic motion of simple pendulum the direction of torque acting is opposite to angular displacement now i am wondering how ?as angular displacement is either anticlockwise or clockwise and torque direction is outward or inward to the plane of paper so how their directions are opposite?please answer i am really confused thanks.

It is a question much easier to explain with a diagram.

On the other hand, you probably already understand that in a simple harmonic motion (SHM), the restoring force is proportional to the angular displacement, and the direction is opposite to it (for small oscilations of a pendulum).

This restoring force multiplied by the length of the pendulum creates what your teacher calls a torque. Think of using a wrench to tighten a pipe, you are applying a force perpendicular to the rotation of the pipe, so the restoring force is also perpendicular to the axis of rotation, namely the fixed end of the pendulum string.

Hope that clears up a little.

In the case of a simple pendulum undergoing simple harmonic motion, the direction of torque acting on the pendulum is indeed opposite to its angular displacement. Let's break it down step by step:

1. First, let's clarify the direction of angular displacement. When a simple pendulum swings, it moves back and forth along a circular arc. The angular displacement of the pendulum is defined as the angle between the pendulum's position and its equilibrium position. This angular displacement can be either clockwise or counterclockwise, depending on the direction in which the pendulum swung.

2. Now, let's consider the torque acting on the pendulum. Torque is the rotational equivalent of force, and it causes an object to rotate. In the case of the simple pendulum, the force of gravity acts on the mass of the pendulum bob (the weight) and creates a torque. This torque tends to restore the pendulum to its equilibrium position.

3. The key to understanding why the torque direction is opposite to the angular displacement lies in the fact that the torque is restoring the pendulum to its equilibrium position. When the pendulum is displaced from its equilibrium position, a restoring torque is created that tries to bring the pendulum back.

4. Let's consider an example. Suppose the pendulum is initially displaced to the right (clockwise direction). As it swings back to the left, the torque acts in the opposite direction, trying to rotate the pendulum in the clockwise direction. This restoring torque helps bring the pendulum back to its equilibrium position.

5. Conversely, if the pendulum is initially displaced to the left (counterclockwise direction), the torque will act in the opposite direction, trying to rotate the pendulum in the counterclockwise direction. Again, the torque acts opposite to the angular displacement, working to restore the pendulum's equilibrium position.

So, while the angular displacement is either clockwise or counterclockwise, the torque acting on the pendulum is opposite to that direction, aiming to restore the pendulum to its equilibrium position. This relationship holds true throughout the entire motion of the pendulum during simple harmonic motion.