Question Video: Finding the Distance between the Centres of Two Bodies given the Gravitational Force between Them | Nagwa Question Video: Finding the Distance between the Centres of Two Bodies given the Gravitational Force between Them | Nagwa

Question Video: Finding the Distance between the Centres of Two Bodies given the Gravitational Force between Them Mathematics • Second Year of Secondary School

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Given that the gravitational force between two bodies of masses 4.6 kg and 2.9 kg was 3.2 × 10⁻¹⁰ N, find the distance between their centers. Take the universal gravitational constant 𝐺 = 6.67 × 10⁻¹¹ N ⋅ m²/kg².

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Video Transcript

Given that the gravitational force between two bodies of masses 4.6 kilograms and 2.9 kilograms was 3.2 times 10 to the power of negative 10 newtons, find the distance between their centres. Take the universal gravitational constant 𝐺 equals 6.67 times 10 to the power of negative 11 newton metres squared per kilogram squared.

Well, to solve this problem, what we have is a formula. And that formula is 𝐹 is equal to 𝐺𝑚 sub one 𝑚 sub two over 𝑆 squared, where 𝐹 is the gravitational force. Capital 𝐺 is the gravitational constant, not to be confused with small 𝑔, which is acceleration due to gravity. 𝑚 sub one and 𝑚 sub two are our masses. And 𝑆 is the distance between our centres. So, now, what we can do is use this formula to help us find the distance between the centres.

Now, the first thing we do is we write out the information we’ve been given. So, the first things that we have are the two masses. So, we got 𝑚 sub one, which is 4.6 kilograms, and 𝑚 sub two, which is 2.9 kilograms. So, then, next, what we have is the gravitational force. And this is 3.2 times 10 to the power of negative 10 newtons. Then, the last bit of information we have is that 𝐺, our gravitational constant, is equal to 6.67 times 10 to the power of negative 11 newton metres squared per kilogram squared. And then, 𝑆 is what we’re trying to find because that’s the distance between the centres.

Now, what I like to do is just check that we’ve got everything in the correct units. So, first of all, if we take a look at the masses. Well, these were in kilograms. And if we check out the gravitational constant, this also has kilograms because it’s newton metre squared per kilogram squared. So, yes, they’re in the correct units. So, then, if we look at the gravitational force, this is in newtons. That’s fine because again this appears in our gravitational constant.

So, then, finally, we’ve got metres. And that’s what our distance is gonna be in because if we look again at the gravitational constant, we’ve got newton metres squared per kilogram squared. Okay, so, we’re clear with all of our units. So, now, let’s substitute our values in to find our distance.

So, when we do that, we’re gonna get 3.2 multiplied by 10 to the power of negative 10 is equal to 6.67 times 10 to the power of negative 11 multiplied by 4.6 multiplied by 2.9 over 𝑆 squared. So, what we’re gonna do is rearrange it to make 𝑆 squared the subject. So, we multiply by 𝑆 squared and divide by 3.2 multiplied by 10 to the power of negative 10. So, then, we’ve got 𝑆 squared is equal to 6.67 multiplied by 10 to the power of negative 11 multiplied by 4.6 multiplied by 2.9 over 3.2 multiplied by 10 to the power of negative 10.

Well, we notice here that we’ve got 𝑆 squared on the left-hand side. And in fact, all we want is 𝑆. So, what we need to do is root both sides. And when we do that, we’re gonna get 𝑆 is equal to the square root of 6.67 multiplied by 10 to the power of negative 11 multiplied by 4.6 multiplied by 2.9 over 3.2 multiplied by 10 to the power of negative 10.

So, when we calculate this, we find that the answer is gonna be 𝑆 is equal to 1.6675. And as we stated before, that’s going to be in metres. And if we wanted to, we could convert it into centimetres, which would be 166.75 centimetres. So therefore, we can say that the distance between the centres of the two bodies of mass is 1.6675 metres, or 166.75 centimetres.

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