If a set of displacement vectors laid head to tail forms a closed polygon, the resultant is zero. Is this statement true? Explain your reasoning.
Well, to answer your question, we have to "dis-place" some misconceptions! Haha, get it?
In physics, the resultant vector of a closed polygon formed by head-to-tail addition of displacement vectors is indeed zero. Why? Well, imagine two vectors pointing in opposite directions. When you add them together head to tail, they perfectly cancel each other out. It's like having a superhero and a supervillain in a tug of war – they balance each other perfectly in the middle! So, when all the vectors in a closed polygon balance out, the resultant becomes zero.
Think of it like a group of clowns trying to push a giant balloon in different directions – they might create a lot of chaos and funny faces, but at the end of the day, the balloon won't really move. It's all about finding that "sweet spot" of balance!
I hope that clowny explanation clears things up for you!
Yes, the statement is true.
The resultant of a set of displacement vectors is the vector obtained by adding all the individual vectors together. If these displacement vectors form a closed polygon when laid head to tail, it means that the initial and final points coincide, forming a closed loop.
Since the initial and final points are the same, it implies that the net displacement is zero. In other words, the sum of all the displacement vectors cancels out and there is no overall displacement.
To put it mathematically, if we label the displacement vectors as A, B, C, ..., with A being the initial displacement and C being the final displacement, the relationship A + B + C + ... = 0 holds true when a closed polygon is formed.
Yes, the statement is true. This can be explained using vector addition and the concept of vector components.
When displacement vectors are laid head to tail to form a closed polygon, the final displacement from the starting point to the ending point is the resultant displacement. If the resultant displacement is zero, it means that the vector sum of all the displacement vectors is zero.
To understand why the resultant is zero for a closed polygon, we can break down each displacement vector into its components. Each vector can be represented by two perpendicular components: one in the horizontal direction (x-component) and one in the vertical direction (y-component).
Now, let's consider a closed polygon formed by laying the displacement vectors head to tail. Starting from the initial point, we can go through each displacement vector in the polygon and add its components to the previous components. As we move along the polygon, some vectors will add positive values to the components, while others will add negative values, depending on their directions.
Since the polygon is closed, the last vector will bring us back to the initial point. Therefore, the resultant components in the horizontal and vertical directions should add up to zero, indicating that the displacement in all directions cancel each other out.
Mathematically, if we denote the x-component of each displacement vector as Dx and the y-component as Dy, then the resultant x-component (Rx) and y-component (Ry) can be calculated as the sum of all the individual components:
Rx = ∑(Dx)
Ry = ∑(Dy)
For the resultant to be zero, Rx = Ry = 0.
So, when the set of displacement vectors laid head to tail forms a closed polygon, the resultant is zero, indicating that the displacement in all directions cancels each other out, leading back to the initial point.