When a branch of a tree is shaken vigorously, ripened fruits get detached and fall due to cause of

An object will continue to be in the state of rest or in a state of motion unless an external force acts on it is called the law of inertia 
When a branch of a tree is shaken vigorously, ripened fruits get detached and fall due to the cause of inertia of rest, When the branch comes into motion but the fruits at rest tend to remain at rest due to inertia of rest and get detached, then gravity plays its role in making the fruits fall.