If two balls, being identical in volume, but different in density (one ball is made of iron, the other of aluminum) roll down from an inclined plane, which will reach the bottom first and which will cover a larger distance after having reached the bottom?
IMPORTANT NOTE: please take into account that I need to be able to solve similar question conceptually. I really wonder about the theory, but the practical view on the matter is utmost important as I am not allowed to use a calculator and paper when answering this and similar questions.
Moment of inertia (or rotational inertia) (being I = 2/5MR^2 for a solid sphere)
Torque = Ffriction*Radiusofball = I*angularvelocity,
Vector calculations for the forces
The Attempt at a Solution
Regarding weight and volume: [/B]I think both weight and volume is important when it comes to the velocity the balls will reach and the distance they will cover.
Regarding air resistance: since the volumes are equal, I assume the air resistance (drag) force acting on it will be equal in size as drag is dependent on area (Fd = ½ ρ * v^2 * Cd * A, with FD = Drag force, ρ = fluid density, v = Relative velocity between the fluid and the object, Cd = Drag coefficient and A = Transversal area or cross sectional area). I even think, that air resistance won’t have any significant effect, because of the low velocity. To reach a high velocity, very long distances would be needed, which seems to me illogical for just an inclined plane.
Regarding the gravity force: The x (or friction) and y component of the gravity is different in size for each ball, but the ratio between them is equal for both. Therefore, the “relative” friction is the same for both (I wonder if the impact of friction in that case is the same for both balls being of the same volume. Anyone who would like to explain?).
Regarding the moment of inertia (or rotational inertia): rotating objects have a moment of inertia (I). The “I” differs per body type. For a solid sphere, I = 2/5MR^2. I think the iron ball with the greater density, having a greater mass, will have a greater “I” compared to the aluminum ball. Torque, as we know, defines acceleration and thus velocity. Since torque can be expressed as Ffriction*Radiusofball = I*angularvelocity, “I” must play a role in defining the acceleration and the velocity at which an object will roll down an inclined plane. Thus, the bigger “I”, the bigger the Torque, the greater the acceleration, the greater the velocity. Therefore, the ball with the greater mass will have a bigger “I”, wherefore it will roll down faster.
For the same reason, a ball with a bigger Volume, having a larger radius, would roll down faster than a ball with an equal mass. Moreover, experiments have proven that a larger ball (identical masses) will roll down faster and further. Is the reason for the greater distance coverage, the greater inertia? Could a greater contact surface between ball and ground also have a significant effect on this?
My question to you: Is it right to conclude it like this or am I underestimating/overseeing determinative factors. I especially care about your conceptual/practical view on the matter. Hoping to receive your valuable guidance. Thanks in advance!