When dealing with a tilted coordinate system, finding the components of a vector can be a bit challenging. However, it's a problem that can be easily solved with some basic trigonometry and vector operations. In this article, we will discuss how to find the components Nx and Ny of N? in the tilted coordinate system of part B.
Understanding the Tilted Coordinate System
A tilted coordinate system is simply a coordinate system that is rotated at an angle with respect to the standard coordinate system. In this system, the x and y-axes are no longer perpendicular to each other, which means that the components of vectors in this system are different from the components in the standard system.
In the diagram above, you can see that the tilted coordinate system is rotated at an angle of ? with respect to the standard system. This means that the x-axis makes an angle of ? with the horizontal, while the y-axis makes an angle of (? + 90?) with the horizontal.
Finding the Components Nx and Ny of N?
Now that we understand the tilted coordinate system, let's move on to finding the components Nx and Ny of N?. To do this, we need to use some basic trigonometry and vector operations.
First, we need to resolve the vector N? into its components in the standard coordinate system. We can do this by using the following equations:
Nx = N? cos ?
Ny = N? sin ?
Where Nx and Ny are the components of N? in the standard system, and ? is the angle that N? makes with the horizontal in the standard system.
Next, we need to find the components of N? in the tilted coordinate system. To do this, we can use the following equations:
Nx' = Nx cos ? + Ny sin ?
Ny' = -Nx sin ? + Ny cos ?
Where Nx' and Ny' are the components of N? in the tilted coordinate system.
Let's take an example to understand this better. Suppose we have a vector N? with magnitude 10 N and angle 30? with the horizontal in the standard coordinate system. If the tilted coordinate system is rotated at an angle of 45? with respect to the standard system, then we can find the components Nx and Ny as follows:
Nx = 10 cos 30? = 8.66 N
Ny = 10 sin 30? = 5 N
Now, using the equations for finding the components in the tilted system, we get:
Nx' = 8.66 cos 45? + 5 sin 45? = 9.9 N
Ny' = -8.66 sin 45? + 5 cos 45? = 0.83 N
So, the components Nx and Ny of N? in the tilted coordinate system are 9.9 N and 0.83 N, respectively.
Conclusion
Finding the components Nx and Ny of N? in a tilted coordinate system may seem daunting at first, but it's a problem that can be easily solved with some basic trigonometry and vector operations. By following the steps outlined in this article, you should be able to find the components of any vector in a tilted coordinate system.
