Question

can you please justify the statement . "The net acceleration is directed towards the center in all types of circular motion."

Answer 1

All changes in motion (speed and direction) are due to acting forces and a force can only change the trajectory of a particle in the same direction as the force itself. That is to say, that when for example a ball is thrown and it is moving along its (approximately parabolic) trajectory, gravity that pulls the ball downwards does nothing to change the ball's forward velocity component.

In a similar case, if an object is in circular motion, we can investigate what forces there would have to be in order for the object to maintain its change in velocity. If we look at the object in circular motion and measure its velocity in the direction of travel, then the only forces that can change its speed are such that work tangentially to the circular trajectory, i.e. either pushing the object forward or dragging it in the opposite direction. These forces would have no change in the travelling direction but only the object's relative speed.

No matter if the speed of the object is changing or not, in circular motion the direction of travel is always changing. This means there is always, at any given time, a force acting non-parallel to the motion of the object since any force simply pushing forwards or backwards wouldn't change the particle's trajectory.

Now, non-parallel doesn't necessarily mean perpendicular, i.e. pointing towards the center of circular motion. But any force acting on both direction and velocity of the object (a force that acts "diagonally" on the object) has at least a component towards the center of circular motion. And if the object is moving at a uniform velocity along the circular trajectory, then there can be no velocity-parallel forces, leaving us only with the perpendicular force that makes the object turn and accelerate towards the center.

Did this clear anything up?

Answer 2

A simpler explanation:

Draw a circle and an arrow on the circle indicating direction of motion. Draw a ray tangent to the circle, pointing in the same direction as the direction of motion. In order for any object with momentum on that ray to maintain a path along the circle, the object must accelerate towards the center of the circle.

Note that if you move the angle of the ray, you turn the circle into an ellipse.

This is exactly as Joonas describes; I'm just using a simple visual construction that you can try sketching yourself.

Answer 3

If you twirl a stone on the end of a string around in a circle, the string always pulls the stone toward the center of the circle. Since the acceleration of the stone points in the same direction as that force, the acceleration is directed toward the center of the circle too.

Actually your statement is only true for UNIFORM circular motion. For nonuniform circular motion, the acceleration has a tangential component too.