What is the gravitational field strength on a planet, where an object with a mass of 25 kilograms has a weight of 300 newtons?
Let us begin by underlining the important information in the question. So we’re being asked to find the gravitational field strength on a planet. And on this planet, we’ve got an object with a mass of 25 kilograms. It weighs 300 newtons on this planet. So to reiterate, we are being asked to find the gravitational field strength. Let’s call that 𝑔 sub 𝑝.
The reason we use something like 𝑔 sub 𝑝 rather than just calling it 𝑔 is because conventionally we often use 𝑔 to mean the gravitational field strength of the Earth. However, in this case, we don’t know if we were on the Earth. We just know that we’re on a planet. And actually once we work out the value of 𝑔 sub 𝑝, we we’ll be able to compare it with the value of 𝑔 on Earth and see if we actually could possibly be on the Earth.
Anyway, so we’ve been asked to find 𝑔 sub 𝑝, the gravitational field strength. As well as this, we know a couple of other pieces of information. For example, the mass of the object that we’re looking at is 25 kilograms. And we’ll label the mass of the object with lowercase 𝑚. Finally, we know that the weight, which we will label capital 𝑊, of the object on that planet is 300 newtons.
So we need to find a relationship between these three terms: 𝑔𝑝, the gravitational field strength of the planet, 𝑚, the mass of the object on the planet, and 𝑊, the weight of the object on the planet. We can find this relationship by looking at the definition of weight. Weight is defined as a force exerted on a body or an object by gravity. And that’s an important definition to remember because weight is a force.
There’s a common misconception, where people confuse weight and mass as being the same thing. They’re not. One reason for this misconception is because people often say the weight of an object is 25 kilograms for example, whereas kilograms is a measure of mass not weight.
Anyway, getting back to the definition of weight, we can actually write this in symbols. It’s written as 𝑊, the weight, is equal to 𝑚, the mass, multiplied by the gravitational field strength, 𝑔𝑝. As a quick aside by the way, notice the parallels with Newton’s second law of motion. That’s 𝐹 is equal to 𝑚𝑎.
Now, we’ve already mentioned that weight is a force. So on the left-hand side, we’ve got two identical things just labelled differently. On the right-hand side, we’ve got the masses of the objects, 𝑚. And then what’s left over is the gravitational field strength and the acceleration. Since we’ve said the left-hand side is a force and on the right-hand side we’ve got a mass multiplying something else, in both cases then that must mean that gravitational field strength and acceleration are somehow identical or at least intricately linked. And they are. In fact, in Einstein’s theory of relativity, that’s exactly what he says. Acceleration and gravitational field strength are intricately linked.
However, let’s get back to the question. We can use this definition of weight to solve our problem. First of all, we need to rearrange the equation to solve for the gravitational field strength. If we divide both sides of the equation by the mass 𝑚, then we’ll have what we’re looking for. We’ve just got the gravitational field strength left on the right-hand side of the equation. So all that’s left to do is just substitute in our values for the weight and the mass which simplifies to 12 newtons per kilogram. And that’s the answer to our question.
Now, we did say earlier that we could compare this gravitational field strength that we got in our solution with that of the Earth to see if the planet in the question is actually Earth. So the answer we’ve got here is 12 newtons per kilogram. But we know that the Earth’s gravitational field strength is about 9.81 newtons per kilogram, which means that the planet in this question cannot be the Earth.