Unbalanced forces can change the motion of an object in two ways. When unbalanced forces act on an object at rest, the object will move.
Second, when unbalanced forces act on a moving object, the velocity of the object will change.
A force can act on an object without causing it to accelerate if other forces cancel the push or pull of the force.
Unbalanced forces produced change in motion (acceleration) and the receivers of the forces - the piano and the rope - moved. Unbalanced forces are necessary to cause a nonmoving object to start moving.
That force is a net force, meaning it's the result of the sum of force vectors on an abject. Basically, if they're unbalanced forces, it will not be zero.
So, if your object has mass, and it has an unbalanced force, it has an acceleration.