The magnitude of the displacement is
$\left|\text{\Delta}\overrightarrow{r}\right|=\sqrt{{(4787)}^{2}+{(\mathrm{-11,557})}^{2}}=\mathrm{12,509}\phantom{\rule{0.2em}{0ex}}\text{km}.$ The angle the displacement makes with the
x- axis is
$\theta ={\text{tan}}^{\mathrm{-1}}\left(\frac{\mathrm{-11,557}}{4787}\right)=\mathrm{-67.5}\text{\xb0}.$
Significance
Plotting the displacement gives information and meaning to the unit vector solution to the problem. When plotting the displacement, we need to include its components as well as its magnitude and the angle it makes with a chosen axis—in this case, the
x -axis (
[link] ).
Note that the satellite took a curved path along its circular orbit to get from its initial position to its final position in this example. It also could have traveled 4787 km east, then 11,557 km south to arrive at the same location. Both of these paths are longer than the length of the displacement vector. In fact, the displacement vector gives the shortest path between two points in one, two, or three dimensions.
Many applications in physics can have a series of displacements, as discussed in the previous chapter. The total displacement is the sum of the individual displacements, only this time, we need to be careful, because we are adding vectors. We illustrate this concept with an example of Brownian motion.
Brownian motion
Brownian motion is a chaotic random motion of particles suspended in a fluid, resulting from collisions with the molecules of the fluid. This motion is three-dimensional. The displacements in numerical order of a particle undergoing Brownian motion could look like the following, in micrometers (
[link] ):
In the previous chapter we found the instantaneous velocity by calculating the derivative of the position function with respect to time. We can do the same operation in two and three dimensions, but we use vectors. The instantaneous
velocity vector is now
Let’s look at the relative orientation of the position vector and velocity vector graphically. In
[link] we show the vectors
$\overrightarrow{r}(t)$ and
$\overrightarrow{r}(t+\text{\Delta}t),$ which give the position of a particle moving along a path represented by the gray line. As
$\text{\Delta}t$ goes to zero, the velocity vector, given by
[link] , becomes tangent to the path of the particle at time
t .
Questions & Answers
A central force is given as F vector (r),where a=2NM².Assuming the potential energy at infinity to be zero,calculate the potential energy of a particle located at the point (3,4)
A vector is any physical quantity which has a magnitude as well as a direction associated to it. Which means a vector is some physical quantity which has magnitude and direction.
malayala
what is matter
Seth
nice
Faith
What is the equation illustrating Williamsons ether synthesis
examples: vibrating prongs of a tuning fork and a guittar string.
Salman
It is a repetitive periodic motion of a system about an equilibrium position
Felix
SHM is the repitition process of to and fro motion.
Younus
SHM is the motion in which the restoring force is directly proportional to the displacement of body from its mean position and is opposite in direction to the displacement.
From Hooke's law
F=-kx
Kushal
SHM is the motion in which the restoring force is directly proportional to the displacement of body from its mean position and is opposite in direction to the displacement.
From Hooke's law
F=-kx
a displacement vector has a magnitude of 1.62km and point due north . another displacement vector B has a magnitude of 2.48 km and points due east.determine the magnitude and direction of (a) a+ b and (b) a_ b
A student opens a 12kgs door by applying a constant force of 40N at a perpendicular distance of 0.9m from the hinges. if the door is 2.0m high and 1.0m wide determine the magnitude of the angular acceleration of the door. ( assume that the door rotates freely on its hinges.)
please assist me to d