Newton's laws as they are stated are only applicable for point mass bodies. To apply them for rigid bodies, we need a point on the body that we can consider all of its mass to be distributed in.

As per axis crossing center of mass, it's useful for a moment of inertia calculations (see parallel axis theorem) but it's not useful for talking about motion. It's not something which we can define consistently.. as in out of all the axis passing through, you'd need to specify a new axis such that your equations hold.

For example, imagine the ball having a torque applied in a direction such that it is rotated around the $y$ axis, suddenly the axis you've shown in the question(assuming it is the z) won't work so well.

See this answer for more details

And, it may be good to know that this extension was actually done by Euler (read here)

Note: point of this answer was to highlight the importance of the center of mass.

Newton's laws as they are stated are only applicable for point mass bodies. To apply them for rigid bodies, we need a point on the body that we can consider all of its mass to be distributed in.

As per axis crossing center of mass, it's useful for a moment of inertia calculations (see parallel axis theorem) but it's not useful for talking about motion. It's not something which we can define consistently. One of the reasons is that out of all the axis passing through, you'd need to specify a new axis such that your equations hold.

For example, imagine the ball having a torque applied in a direction such that it is rotated around the $y$ axis, suddenly the axis you've shown in the question(assuming it is the z) won't work so well.

See this answer for more details

And, it may be good to know that this extension was actually done by Euler (read here)

Note: point of this answer was to highlight the importance of the center of mass.

Newton's laws as they are stated are only applicable for point mass bodies. To apply them for rigid bodies, we need a point on the body that we can consider all of its mass to be distributed in.

See this answer for more details

And, it may be good to know that this extension was actually done by Euler (read here)

Newton's laws as they are stated are only applicable for point mass bodies. To apply them for rigid bodies, we need a point on the body that we can consider all of its mass to be distributed in.

As per axis crossing center of mass, it's useful for a moment of inertia calculations (see parallel axis theorem) but it's not useful for talking about motion. It's not something which we can define consistently.. as in out of all the axis passing through, you'd need to specify a new axis such that your equations hold.

For example, imagine the ball having a torque applied in a direction such that it is rotated around the $y$ axis, suddenly the axis you've shown in the question(assuming it is the z) won't work so well.

See this answer for more details

And, it may be good to know that this extension was actually done by Euler (read here)

Note: point of this answer was to highlight the importance of the center of mass.