Equations of Motion for Uniform Circular Motion
The object moving in a uniform circular motion will have a certain position and speed at each point of time and hence, can be denoted by a position vector. Let us consider that the particle has a position vector [Tex]\vec{r}(t) [/Tex] with an amplitude of P. As circular motion is a 2-dimensional motion, the vector is traveling in both x and y coordinates hence, it will have components of both the x and y-axis. In general, if the object is moving with a uniform angular frequency of ‘w’, at a time ‘t’, it will have the following values,
Position vector (P) ⇢ [Tex] \vec{r}(t) [/Tex]
X-Component of position vector ⇢ [Tex]|\vec{r}(t)|Coswt [/Tex]
Y- Component of position vector ⇢ [Tex]|\vec{r}(t)|sinwt [/Tex]
Where, w is given as, [Tex]w=\frac{2\pi}{T} [/Tex]
T= Time Period
Equation for Position Vector, [Tex]\vec{r}(t)=iP|\vec{r}(t)|coswt+jP|\vec{r}(t)|sinwt [/Tex]
The velocity vector can easily be obtained by differentiating the position vector w.r.t time,
[Tex]\vec{v}(t)=-iwP|\vec{r}(t)|sinwt+jwP|\vec{r}(t)|coswt [/Tex]
Similarly, the acceleration vector can be easily obtained by differentiating the velocity vector w.r.t. time,
[Tex]\vec{a}(t)=-iw^2P|\vec{r}(t)|coswt-jw^2P|\vec{r}(t)|sinwt [/Tex]
From the above equation, the relationship between the acceleration vector and position vector can be easily calculated,
[Tex]\vec{a}(t)=-w^2|\vec{r}(t)| [/Tex]
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Uniform Circular Motion
Uniform Circular Motion as the name suggests, is the motion of a moving object with constant speed in a circular path. As we know, motion in a plane only has two coordinates, either x, and y, y and z, or z and x. Except for Projectile motion, circular motion is also an example of motion in a 2-D plane.
In a uniform circular motion, the object moves with constant speed but not with constant velocity as the direction of the motion is due to the circular path always changing. From the motion of electrons in Bohr’s Atomic model to the motion of the hands of an analog clock, we can see Uniform Circular Motion around us.
In this article, we will learn about the details of Uniform Circular Motion i.e., formulas related to uniform circular motion, examples, and the equation of motion of the uniform circular motion.
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