Which object will reach the ground first if both roll down an incline, one with I1 = I and the other with I2 = 2I?

When two objects roll down an incline, their motion is influenced by both their mass and their moment of inertia. The moment of inertia determines how easily an object can rotate, and as such, it affects its acceleration down the incline.

The object with a lower moment of inertia relative to its mass will generally accelerate faster down the incline. In this case, the object with I1 = I has a moment of inertia that is half the moment of inertia of the other object with I2 = 2I. This means that the object with I1 requires less torque to accelerate and therefore rolls down the incline faster.

The acceleration of a rolling object is given by the formula that incorporates both translational and rotational motion. For an object rolling without slipping, the linear acceleration can be shown to depend on the fraction of the gravitational force that goes into translational acceleration versus rotational acceleration. A smaller moment of inertia (like I1) means more of the gravitational force contributes to linear speed instead of rotational speed.

Thus, the object with I1 = I will reach the bottom of the incline before the object with I2 = 2I, leading to the conclusion that the first option is indeed correct.