Video: Minimizing the Resultant Force on an Object

Two forces, 𝐴 and 𝐵, act on an object. If 𝜃 is the angle between the directions of the two forces, for what value of 𝜃 is the resultant force minimum? [A] 180° [B] 45° [C] 0° [D] 60° [D] 90°

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Video Transcript

Two forces, 𝐴 and 𝐵, act on an object. If 𝜃 is the angle between the directions of the two forces, for what value of 𝜃 is the resultant force minimum? a) 180 degrees, b) 45 degrees, c) zero degrees, d) 60 degrees, or e) 90 degrees.

To begin answering the question, we can visualise each of our answers by drawing the object and the two corresponding forces acting on it. For choice a), our object has two forces acting on it that are 180 degrees to each other. This means that the forces are pulling in opposite directions. Because the problem did not state the magnitude of the forces, we can draw them as different sizes to imagine the general case.

The problem also does not state the direction of our vectors. For this visualisation, we’ve chosen to draw vector 𝐀 to the right and vector 𝐁 to the left. Making sure that the angle between the two is 180 degrees. We have also chosen to make the vectors different colours to help later when we add them together.

For choice b), our vectors are at 45 degrees to each other. We have chosen to draw vector 𝐀 to the right and vector 𝐁 at 45 degrees above vector 𝐀. For choice c), our vectors are at zero degrees to each other. This means that both vectors, 𝐀 and 𝐁, are pulling in the same direction. We have chosen to draw vector 𝐀 and 𝐁 both to the right. For choice d), the forces are at 60 degrees to one another. For this diagram, we chose to draw vector 𝐀 to the right and vector 𝐁 at 60 degrees above vector 𝐀. For choice e), our vectors are at 90 degrees to one another. We chose to draw vector 𝐀 pulling to the right and vector 𝐁 at 90 degrees above vector 𝐀.

To determine which of the angles has the minimum resultant force, we must add our vectors together. A quick way to add vectors without having to do any math is the tip-to-tail method. The tip-to-tail method is when we slide one vector until the tail of the vector is on the tip of the other vector. For each of our answer choices, we will slide vector 𝐁 over such that the tail of vector 𝐁 will be on the tip of vector 𝐀.

For answer choice a), we start by drawing vector 𝐀 to the right. We draw vector 𝐁 starting at the tip of vector 𝐀 and pointing to the left. The resultant force will be pointing in the direction from where we began to where we ended. In this case, we began at the tail of 𝐀 and ended at the tip of 𝐁. We can see from our diagram that our resultant force is a small arrow pointing to the right.

For answer choice b), we once again begin by choosing to draw vector 𝐀 to the right. We then slide vector 𝐁 over until the tail of vector 𝐁 is on the tip of vector 𝐀. Our resultant vector starts at the tail of vector 𝐀 and ends at the tip of vector 𝐁. As we can see in the diagram we just drew, the size of resultant vector 𝐑 is larger than vector 𝐀 and vector 𝐁. This is because both vector 𝐀 and vector 𝐁 have components pointing to the right. Vector 𝐀 is completely to the right. And vector 𝐁 has a component both to the right and up, based on the orientation we chose. The components of both vectors add together to make a larger resultant. Whereas in answer choice a), vector 𝐀 was pointing to the right and vector 𝐁 was pointing to the left. This had a resultant that was smaller because they ended up subtracting from each other.

For answer choice c), we will once again begin by drawing vector 𝐀 to the right. We slide vector 𝐁 over until the tail of vector 𝐁 is on the tip of vector 𝐀. Our resultant vector will begin at the tail of vector 𝐀 and end at the tip of vector 𝐁. We can see that since vector 𝐀 and vector 𝐁 are pointing in the same direction, we end up with a resultant that is the largest so far.

For answer choice d), we once again begin by drawing vector 𝐀 to the right. We slide vector 𝐁 over so that the tail of vector 𝐁 is on the tip of vector 𝐀. The resultant vector begins at the tail of vector 𝐀 and ends at the tip of vector 𝐁. From our diagram, we can see that the resultant vector for answer choice d) is smaller than the resultant vector for answer choice b). However, the angle between the resultant vector 𝐑 and vector 𝐀 in answer choice d) is bigger than the angle between the resultant vector 𝐑 and vector 𝐀 in answer choice b). This is because the vector component of vector 𝐁 pointing to the right for answer choice d) is not as big as the vector component pointing to the right for vector 𝐁 in answer choice b).

For the final answer choice e), we begin by drawing vector 𝐀 to the right. We slide vector 𝐁 over so that the tail of vector 𝐁 is on the tip of vector 𝐀. The resultant vector begins at the tail of vector 𝐀 and ends at the tip of vector 𝐁. Because vector 𝐁 does not have a component pointing to the right, we can see that our resultant vector is not as large as answer choices b), c), or d).

Returning to the problem, to determine which of our answer choices has the minimum resultant force, we can see that that would be choice a). Looking back at the diagrams we drew for our problem, we can see that answer choice a) is the only one that had vector 𝐁 being subtracted from vector 𝐀. This was due to the orientation of the two vectors compared to one another. Vectors who are oriented 180 degrees from each other will always produce a minimum resultant. Whereas vectors that are oriented zero degrees to each other will always produce a maximum resultant. As can be seen in answer choice c), where the vectors add completely together.

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