Please do have the patience to read on since this is pretty long. Okay, so I'm pretty bad at physics due my basic understanding of the subject not being strong and am trying to improve my understanding of the subject since it frustrates me to not be able to comprehend something in class. So here goes a very basic question which I think has been asked many many times: * Why do objects with different masses accelerate at the same rate towards the center of the earth, despite having different masses? * Also, but less importantly, why are both acceleration due to gravity and gravitational field strength given the symbols 'g' despite having different units and different meanings? *So here is what I have gathered and am struggling with: Gravitational field strength is the gravitational force per unit mass, as in the force per kg. Thus, the gravitational force on a heavier object is larger. So, according to Newton's second law of motion, F=ma, wouldn't a heavier object have a greater acceleration towards the earth when air resistance is negligible of course? But I know very well that isn't the case and would just like to know why in hell it is what it is? If you have read till here, thank you. Here's a thumbs up just for you- :thumbs:
There is a faq on your first question. Check it out; https://www.physicsforums.com/showthread.php?t=511172 Try it yourself. You know F=ma. You also know (or should know) the force law for gravity. Equate the two and solve for "a". For your second question... The acceleration due to gravity on earth and the gravitational constant do not have the same symbol. One is "g" the other is "G". Symbols in math and science are (or should be) case sensitive.
More mass means a larger force of gravity, but it also means more inertia. You're right when you say "the gravitational force on a heavier object is larger" You're wrong when you say "wouldn't a heavier object have a greater acceleration towards the earth" Mass is essentially a "measure of resistance to acceleration" (m=F/a) so even though it has a larger force, it has more inertia and so the larger force does not result in a larger acceleration. (If gravity weren't stronger for more massive objects then more massive objects would accelerate slower.)
Ask yourself: What is an "object"? Why should two apples, dropped side by side, fall differently based on whether they are separated (two lighter objects) or connected (one heavier object)? They have the same units and are the same thing. No. If you increase m and F by the same factor, a stays the same, according to the formula you cited.
Potential energy ( mgh) is transferred to kinetic energy ( .5mv^2) so the mass on each side of the equation cancels out and the velocity at any particular moment is independent of mass.
Here's an intuitive answer to your first question, which expands a little on what AT said. At the Earth's surface, gravity has a fixed strength. By this, we mean: gravity is able to accelerate every tiny "bit" of matter by the same amount--9.81 m/s^2. If you take two tiny bits of matter, gravity attempts to accelerate each little bit by the same amount. This is true regardless of whether the two different tiny bits of matter are (a) side by side or (b) attached to each other so as to form a larger object. Imagine dropping a golf ball at the same time as and side-by-side with a much more massive bowling ball. The golf ball has fewer "bits" of matter than the bowling ball does. But, the golf ball is made up of tiny bits of matter, just like the bowling ball. Each bit of matter in the golf ball gets pulled by the same amount as each bit of matter in the bowling ball. In essence, the (hopefully) intuitive reason why bigger objects don't fall faster is that bigger objects have more matter that gravity must pull on. I think it was Galileo who famously came up with a thought experiment along these lines. (If you tie together a smaller brick with a bigger brick and drop the combo, what happens? Does the smaller brick fall more slowly and retard the descent of the bigger brick because they're tied together? Or does the combo fall more quickly because together they make an even bigger object which falls every more quickly? Logic would lead us to conclude both, and hence all objects must fall at the same rate regardless of mass.) To your second question--gravitational field strength and acceleration due to gravity are, in fact, different. You are correct that they have the same variable, little g. You are also correct that, mathematically, they are always equal. But they have different conceptual bases and different mathematical definitions. Acceleration due to gravity describes how fast an object changes its velocity when falling toward the ground under the influence of gravity only. Once the object in free fall reaches the ground, the object no longer is accelerating due to gravity. Gravitational field strength, on the other hand, is a measure of how hard gravity pulls on each kg of mass at a point in space. At all points near the Earth's surface, gravity pulls on each kilogram of mass (each "bit" of matter) with the same strength. This is true regardless of whether the object is falling or is stationary. So, there is an important conceptual difference between gravitational field strength and acceleration due to gravity: gravitational field strength exists for points in space regardless of whether objects are located at the points (much less moving), whereas acceleration due to gravity specifically describes a feature of a falling object's motion/movement. Mathematically, the two quantities are always equal, but they have different defining equations: acceleration due to gravity = (change in velocity of object falling under influence of gravity alone) ÷ (time for the change in velocity to occur) gravitational field strength at some point P in space = (the gravitational force generated on a small hypothetical test mass placed at the point P) ÷ (how many kilograms the test mass has)
Sometimes we use g to refer to the gravitational field in newtons/kilogram, analogously to E for the electric field in newtons/coulomb. This is numerically equal to the "gravitational acceleration" in m/s^{2}. I think this is what the OP refers to.
These aren't two independent definitions. "Acceleration due to gravity" follows from "gravitational field strength" and Newtons 2nd Law.
My understanding is that g field strength is defined as F/m whereas every acceleration is define as velocity change over time. Acceleration due to gravity is a special case of acceleration: the case where the velocity change is produced only by gravity. The two different defining equations are F/m and (delta v)/time.
Exactly. Your "acceleration due to gravity (only)" is just a special case, and follows from "gravitational field strength" and Newtons 2nd.
In other words, acceleration due to gravity does follow from Newton's second law, as you say. However, Newton's second law does not set forth the definition of acceleration due to gravity. Rather, the definition (from which the concept derives) of any acceleration is change in velocity over time.
Perhaps the distinction I'm making is that just because acceleration "follows from" Newton's second law does not mean the concept is "defined" by Newton's second law.
There is actually something deep going on here, Einstein's principle of equivalence, i.e., the equivalence between 1) inertial mass, a quantitative measure of inertia, the ##m## in Newton's second law ##F=ma##, and 2) active gravitational mass, gravitational "charge", the ##m## in Newton's law of gravity, a measure of how strongly an object couples to a gravitational field, $$G \frac{m M}{r^2}.$$ Compare to Newton's second law combined with Coulomb's law of electrostatics, $$ ma = k \frac{q Q}{r^2}. $$ Here, ##m## is inertial mass and ##q## is electric charge, a quantitative measure of how strongly an object couples to an electric field. Dividing by one of the charges results in force per unit charge on the left, and electric field on the right. Do the same for gravity: $$ \frac{m_i a}{m_g} = G \frac{M}{r^2}. $$ where ##m_i## is inertial mass and ##m_g## is active gravitational mass (i.e., gravitational "charge"). Again, we have (gravitational) force per unit (gravitational) charge on the left, and gravitational field on the right. It is only because of a deep principle of nature, the (weak) principle of equivalence ##m_i = m_g## that we have acceleration equal to gravitational field.
Thanks George! An insightful way to approach this question. A follow-up question to help me better understand this principle: if we didn't have this principle of equivalence, ##m_i = m_g##, would the force from Newton's second law still equal the force from Newton's law of gravitation? In other words, in the quest to prove that a = g, do we use the fact of the (weak) principle of equivalence twice: first when setting the "inertial force" equal to gravitational force equal and second when canceling out little m? (Is there any term that specifies the force from Newton's second law?)
The equality of gravitational and inertial mass was observed and experimentally confirmed centuries before the equivalence principle was advanced (and thoughtful people did wonder about this remarkable coincidence and the apparent uniqueness of the gravitational force in its relationship to inertial mass). Early in the 20th century general relativity showed us how the equivalence principle would explain the "remarkable coincidence"; but this was definitely a case of long-standing observational results validating the equivalence principle and not the other way around.