If the court awarded custody of the child to the father on what grounds would the court give custody to the mother if she took him to court?

Most courts use the best interest of the child to determine custody. If the court awarded custody to the father, it determined that living with the father is better for the child than living with the mother. In most states, you can file a new suit to modify custody after the fact. You generally have to wait for a certain period of time after the first award, and you'll need to prove that there has been a sufficient change in circumstances such to warrant changing the prior order. At that point, you would then have to prove that the child's needs would be better served by being moved to the mother's custody.

Custody modifications are difficult, tedious and expensive. If you are serious about this, you need to start talking to a good lawyer.

Harm alone does not reward you the custodial parent status. In most instances where harm is apparent, Child Services will intervene and correct the issue before it becomes an abusive situation. In many different ways, you must also prove the parent unfit. The process is and will be expensive, complicated and lengthly. If you have the funds and evidence to pour into a complex case, it may be in the child's best interests for you to pursue. Depending on the state the child resides, this case starts at $3500-$5000 to retain representation. You will have to petition in the state in which the child resides. I caution you, do not attempt to pursue for your own well being. The attorney representing the father will have you extensively evaluated to determine your true motive. In most cases, the original court order will not be overturned.

if you could prove harm is being done to the child.