A body is at rest under the action of four forces. Which of the following can be correctly inferred? I) The vector sum of the four forces is zero. II) None of the forces act in the same direction as each other. III) All the forces are equal in magnitude. a) I and III only; b) I and II only; c) I, II, and III; d) I only; e) II only.
The question asks us about a body at rest. If a body is at rest, it means that body is not moving. In other words, its velocity is zero. We’re also told that the body is being acted upon by four forces. Let’s recall that Newton’s second law tells us that force is equal to mass times acceleration. And acceleration is a change in velocity. So force acting on an object tends to change the object’s velocity.
But we’re told that our object is at rest, even though there are forces acting on it, which means the total acceleration must be zero, since the velocity remains zero and doesn’t change. But 𝑎 total, the net acceleration, is just the acceleration associated with the net force on the object. And using Newton’s second law, the net force is the mass times that acceleration. But then the acceleration is zero. So the net force is also zero.
We recall that force is a vector, which means it has both size and direction. So when we talk about the sum of the four forces making up force total, it’s really the vector sum. And we just figured out that this vector sum has to be equal to zero. Well, this is just Roman numeral I. So Roman numeral I can be correctly inferred from the problem statement.
To think about Roman numerals II and III, we’ll have to understand how to actually perform a vector sum. And for our purposes, we’ll perform the sum geometrically. Recall that a vector has both size and direction. Geometrically, we represent this with an arrow, where the length of the arrow is the size of the vector and the direction of the arrow is the direction of the vector. Thus, this vector and this vector point in the same direction. But the second vector has a size that’s twice as large because it’s twice as long. On the other hand, this vector and this vector have the same size but point in different directions.
To add two vectors geometrically, we simply line the tail, the blank end of one vector, up to the head, the arrow end of the other vector. It looks something like this. Here’s the first vector and here’s the second vector. The resulting sum is the vector that points from the tail of the first vector to the head of the second vector. So there is the resulting vector sum of the two vectors on the left-hand side.
For more than two vectors, simply align the tail and head of each vector successively. For example, here’s a sum of three vectors. But let’s add a fourth vector to this sum. It’ll be this vector here. Now the resulting vector goes from the tail of the first vector here to the head of the final vector, also here. Well, this is just a point. It’s zero. So the vectors have wrapped all the way back around to where they started. And the resulting sum is zero. This is what it means for a vector sum to be zero. When the head of the final vector falls on the tail of the initial vector, the total size is zero. And so the vector is zero.
Let’s turn to Roman numerals II and III and try to find a counter example. Here’s our body, with two of the four forces acting on it, labeled 𝐹 one and 𝐹 two for force one and two. And they act in the same direction. Remember from Roman numeral I that, for an object at rest, the total force is zero. So if II can be correctly inferred, then with 𝐹 one and 𝐹 two acting in the same direction, we shouldn’t be able to put two more forces on the object and get a total of zero. On the other hand, if it can’t be correctly inferred, we could get a total force of zero.
There’s one more thing we need to know to make this counter example work. For every nonzero vector, there’s another nonzero factor that we can add to it to give a result of zero. And that vector has the same size but opposite direction. Connecting the tail of the second vector to the head of the first vector, we see that the second vector exactly retraces the first vector. So its head lies at the first vector’s tail, which means that the sum is zero.
Using the same idea, we’ll cancel 𝐹 one and 𝐹 two on our object. 𝐹 three has the same size as force one but opposite direction. And 𝐹 four has the same size as force two but opposite direction. Let’s double check that the net force really is zero. Here we have 𝐹 one plus 𝐹 two. And now we’ll put in 𝐹 three and 𝐹 four. So this is the sum 𝐹 one plus 𝐹 two plus 𝐹 four plus 𝐹 three.
But remember that 𝐹 four and 𝐹 two have the same magnitude but opposite direction. So the head of 𝐹 four is at the tail of 𝐹 two. But 𝐹 three and 𝐹 one also have the same magnitude but opposite direction. So the head of 𝐹 three winds up with the tail of 𝐹 one. Since the head of the final vector coincides with the tail of the initial vector, the overall sum is zero. And a vector sum of the forces equaling zero is the condition for a body at rest. This means that the body is at rest even though two of the forces point in the same direction, which means that II cannot be correctly inferred. Because two of the four forces can point in the same direction and the body can still be at rest.
We approach number III the same way. Here we again have 𝐹 one and 𝐹 two, this time pointing in different directions and with different magnitudes. Now we add 𝐹 three with the same magnitude but opposite direction to 𝐹 one and 𝐹 four with the same magnitude but opposite direction to 𝐹 two, just like before. So the total sum is 𝐹 one plus 𝐹 two plus 𝐹 three plus 𝐹 four. But just like before, since 𝐹 one and 𝐹 three point in opposite directions but have the same magnitude, 𝐹 one plus 𝐹 three is zero. Similarly, 𝐹 two and 𝐹 four point in opposite directions but have the same magnitude. So their sum is also zero. And zero plus zero is zero.
So in perfect analogy with II, we found a situation where there are two forces of different magnitudes. And nevertheless, the vector sum of the four forces acting on the object is zero. This means that III is not correctly inferred because there is a situation where the forces are not all the same magnitude. But nevertheless, the body is at rest. So I can be correctly inferred. II and III cannot be correctly inferred. And the correct choice is d I only.