### Video Transcript

A stone of mass 686 grams is thrown vertically upward from Earth with a force of 45 newtons. What magnitude force does the gravitational field of Earth exert on the stone? Give your answer to one decimal place.

Okay, so in this question, we’re being asked about a stone that’s been thrown vertically upward. Let’s imagine that this pink circle here represents the stone. And we know that the instant after it’s thrown, the stone is traveling vertically upward. Let’s label the mass of the stone as 𝑚. And we’re told in the question that this has a value of 686 grams. Now, we’re also told that this stone is thrown with a force of 45 newtons. However, as soon as this person who’s throwing the stone releases it, this 45-newton force is no longer acting.

In this first part of the question, we’re asked about a different force acting on the stone. Specifically, we’re asked what magnitude of force is exerted on the stone by the gravitational field of Earth. This force on the stone due to gravity will act vertically downward toward the Earth. Let’s label this force as 𝐹 subscript g. We can recall that the force that acts on an object due to gravity is equal to the object mass, 𝑚, multiplied by the gravitational field strength, 𝑔, which is also known as the acceleration due to gravity.

We can further recall that on Earth, 𝑔 has a value of negative 9.8 meters per second squared. Now, the reason for this negative sign here comes from the fact that forces and accelerations are both vector quantities, which means that they have a direction as well as a magnitude. As we’ve seen in the case of the acceleration and the resulting force due to gravity, this direction is vertically downward toward the Earth. So this negative sign in the quantity 𝑔 means that we’ve implicitly taken upward as the positive direction. In other words, forces and accelerations acting upward will be positive while those acting downward will be negative.

We can use this value for 𝑔 along with the mass of the stone that we were given in this equation in order to calculate the force exerted on the stone by Earth’s gravitational field. In order to calculate a force in units of newtons, we need a mass in units of kilograms and a gravitational acceleration in meters per second squared. At the moment, our value for the mass 𝑚 is in units of grams. So we need to convert this to kilograms before we use it in this equation.

Let’s recall that one kilogram is equal to 1000 grams. If we divide both sides of this expression by 1000, then on the right-hand side, the 1000 in the numerator cancels with the 1000 in the denominator. We then have that one gram is equal to one thousandth of a kilogram. This means that to convert a value of mass from units of grams into units of kilograms, we take that value and divide it by 1000. So then in units of kilograms, the mass 𝑚 of this stone is equal to 686 divided by 1000 kilograms. This works out as 0.686 kilograms.

Let’s now take this mass in kilograms along with our value for the gravitational acceleration 𝑔 and substitute them into this equation. When we do this, we find that the force 𝐹 subscript g exerted on the stone by Earth’s gravitational field is equal to 0.686 kilograms multiplied by negative 9.8 meters per second squared. Evaluating this expression gives a force of negative 6.7228 newtons. This negative sign just means that the force is acting vertically downward in the direction we defined as negative. However, the question doesn’t ask us for this force, but rather for the magnitude of the force. In this expression for the force 𝐹 subscript g, the negative sign indicates the direction of the force, while this number, along with the unit of newtons, indicates the force’s magnitude.

We can denote the magnitude of a quantity by writing that quantity with vertical lines placed on either side of it. And in this case then, we have that the magnitude of 𝐹 subscript g is equal to 6.7228 newtons. This value that we’ve calculated is the magnitude of the force exerted on the stone by Earth’s gravitational field, which is what this first part of the question was asking us to find. We can notice though that we were asked to give our answer to one decimal place. Rounding to this precision, we get our answer to this first part of the question that the magnitude force that the gravitational field of Earth exerts on the stone is 6.7 newtons to one decimal place.

Let’s now clear some space on the board and look at the second part of the question.

What magnitude force does the gravitational field of the stone exert on Earth? Give your answer to one decimal place.

So in the first part of the question, we were asked about the force exerted on the stone by the gravitational field of the earth. Now in this second part of the question, the role of Earth and the stone had been flipped over. So we’re being asked about the force exerted on Earth by the stone’s gravitational field. The key to answering this lies in Newton’s third law of motion. This law is often summarized as saying that every action has an equal and opposite reaction.

To understand what Newton’s third law of motion means in practice, let’s consider two objects labeled as A and B. Well, suppose that object A somehow exerts a force on object B. For example, object A could be traveling towards object B and collides with it. Let’s say that this force is equal to 𝐹 subscript A and that it acts toward the right. Now, Newton’s third law tells us that every action has an equal and opposite reaction. This means that if object A exerts a force on object B, then object B also exerts a force on object A. The force exerted by B on A acts in the opposite direction to the force exerted by A on B. So in this case, that’s toward the left.

Let’s label this force exerted by B on A as 𝐹 subscript B. These two forces 𝐹 subscript A and 𝐹 subscript B have the same magnitude. So when Newton’s third law of motion talks about every action having a reaction that is equal and opposite, this means that if an object A exerts a force on an object B, then object B also exerts a force on object A that is equal in magnitude but opposite in direction. Now in this question, the objects we’re being asked about are Earth and a stone. We’ve already calculated the magnitude of the force that the gravitational field of Earth exerts on the stone.

Now from Newton’s third law of motion, we know that the stone must also exert a gravitational force on Earth. This force will act in the opposite direction to the force exerted on the stone by Earth, so that’s vertically upward. The other thing we know from Newton’s third law is that these two gravitational forces must be equal in magnitude. Since we’ve already found that the magnitude of the force exerted by the gravitational field of Earth on the stone is equal to 6.7 newtons, then our answer to the second part of the question is that the magnitude of the force exerted by the gravitational field of the stone on Earth is also 6.7 newtons.

Now let’s clear some space once more and look at the third part of the question.

Which of the following statements most correctly describes the motion of Earth due to the gravitational field of the stone? (A) Earth is not accelerated at all by the gravitational field of the stone. (B) The magnitude of the acceleration of Earth due to the stone’s gravitational field is equal to the mass of the stone divided by Earth’s mass. (C) The magnitude of the acceleration of Earth due to the stone’s gravitational field is equal to the force applied by the gravitational field of the stone divided by Earth’s mass. (D) The magnitude of the acceleration of Earth due to the stone’s gravitational field is equal to the acceleration of the stone but in the opposite direction.

Okay, so in this third part of the question, we’re being asked about the motion or acceleration of Earth due to the gravitational field of the stone. Let’s recall that in the previous part of the question, we use Newton’s third law of motion to deduce that the gravitational field of the stone exerts a force on Earth. In fact, we found that the gravitational force on Earth, which we’ve labeled here as 𝐹 subscript e, and the gravitational force on the stone, which we’ve labeled as 𝐹 subscript s, have the same magnitude as each other but act in opposite directions. Now for this third part of the question, we need to relate this gravitational force acting on Earth to Earth’s acceleration.

To do this, we can make use of another of Newton’s laws of motion. Specifically, we need Newton’s second law, which says that the force 𝐹 acting on an object is equal to the object’s mass 𝑚 multiplied by its acceleration 𝑎. If we then divide both sides of this equation by the mass 𝑚, then we can see that on the right-hand side, the 𝑚 in the numerator cancels with the 𝑚 in the denominator. This means that we can rewrite the Newton’s second law equation as saying that acceleration 𝑎 is equal to force 𝐹 divided by mass 𝑚. Looking at the statement given in option (A), we can see that this claims that Earth is not accelerated at all by the gravitational field of the stone.

Now, common sense and experience tells us that Earth likely doesn’t experience some huge acceleration every time somebody throws a stone in the air. So it might seem then the statement in option (A) seems plausible. However, we’ve already worked out that there is a force acting on Earth as a result of the gravitational field of the stone. So in this equation from Newton’s second law, we have a nonzero value of 𝐹, which means that we must have a nonzero value of the acceleration 𝑎. This means that we know that Earth does experience an acceleration as a result of the stone’s gravitational field. So the answer given in option (A) cannot be correct.

The remaining three answer options all claim that Earth does experience an acceleration. However, they each make different claims about the magnitude of this acceleration. We can use this equation from Newton’s second law in order to work out which of these three remaining options is the correct one. On the right-hand side of the equation, 𝑚 is the mass of the object that’s being accelerated, which in this case is Earth. Meanwhile, 𝐹 is the force acting on Earth in order to accelerate it. So that’s this force that we’ve labeled as 𝐹 subscript e. If we label the mass of Earth as 𝑚 subscript e and the acceleration it experiences due to the gravitational field of the stone as 𝑎 subscript e, then from Newton’s second law, we have that 𝑎 subscript e is equal to 𝐹 subscript e divided by 𝑚 subscript e.

Now technically, this equation relates the vector quantities 𝑎 subscript e and 𝐹 subscript e. Since mass is a directionless scalar quantity, then in order for the acceleration and the force to be related like this, they must both be in the same direction as each other, and the magnitude of 𝑎 subscript e must be equal to the magnitude of 𝐹 subscript e divided by 𝑚 subscript e. We can express this equation in words by saying that the magnitude of the acceleration of Earth due to the stone’s gravitational field is equal to the force applied by the gravitational field of the stone divided by Earth’s mass. This matches the statement that’s given here in option (C).

So our answer for this third part is that the statement that most correctly describes the motion of Earth due to the gravitational field of the stone is statement (C). The magnitude of the acceleration of Earth due to the stone’s gravitational field is equal to the force applied by the gravitational field of the stone divided by Earth’s mass.

Let’s now clear some space one last time to look at the fourth and final part of the question.

What magnitude force does the stone exert on the person who throws it?

Let’s recall that we’ve got a person throwing a stone. And we’re told that they throw it vertically upward with a force of 45 newtons. We can say then that the person throwing the stone is exerting a force on it in the upward direction with a magnitude of 45 newtons. However, we’re being asked for the magnitude of the force that the stone exerts on the person. And for this, we need to once more recall Newton’s third law of motion.

Remember that this law tells us that every action has an equal and opposite reaction. So if the person exerts an upward force on the stone, then the stone must also exert a downward force on the person. This directional information is covered by the word “opposite.” And then we’ve also got this word “equal” that tells us that these two oppositely directed forces must have equal magnitudes. This means that since the force exerted by the person on the stone has a magnitude of 45 newtons, then the force exerted by the stone on the person must also have a magnitude of 45 newtons. Our answer to this final part of the question then is that the magnitude of the force that the stone exerts on the person who throws it is equal to 45 newtons.