How can the x-component of force applied to a sphere be found along with the rotational component?

Question

I'm trying to model a sphere having force applied at position P in the following diagram:

I know that this applied force will have an x and y component; the y component will propel it upwards, but what I am confused about is the x component of the force. I know that the x component will propel it forwards (linear velocity) and make it spin (angular velocity). I can calculate the rotational force as follows: $$F_{rot.}=ma_{tangential}$$ $$F_{rot.}=mr\alpha=\frac{mr\omega}{\Delta t}$$ However, what I don't know is the proportion of $F_{app.\,x}$ that contributes to the rotational force and the proportion that contributes to linear force in the x direction. It doesn't make sense if I try to think of both being proportional. That would only work if the sphere were rolling on the floor without slipping, in which case $V_{Center\,of\,mass}=r\omega$.

Any ideas on the topic would be greatly appreciated. Maybe I'm missing something crucial, as I'm still a high school student. Thank you.

Answer 1

If you reorient your axes to have one axis, say the x-axis, be in the direction of the force, then the problem will be much easier. At that point you can just use Newton's second law for both linear motion (F=ma) and rotational motion (T=Ia/R) where T is the net torque and I is the moment of inertia about the axis of rotation.