Uniform circular motion

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The realm of physics consists of two types of circular motion: uniform circular motion and non-uniform circular motion.

Uniform circular motion describes motion in which an object moves with constant speed along a circular path.

[edit] Acceleration and velocity

Since the velocity is tangent to the circular path, no two velocities point in the same direction. Although the object has a constant speed, its direction is always changing. This change in velocity is caused by an acceleration, whose magnitude is (like that of the velocity) held constant, but whose direction is always changing. The acceleration points radially inwards (centripetally) and is perpendicular to its velocity. This acceleration is known as centripetal acceleration.

Image:Uniform circular motion diagram 0001.JPG

The magnitude of the acceleration is given by a = v2 / r, where v is the speed of the object and r is the radius of its path, or a = (4π2r) / t2, where r is the radius of the object and t is the time it takes the object to travel a distance.

[edit] How to derive these equations of acceleration

Image:uni_circular.jpg

The above picture shows a point of mass that is moving with a constant angular speed around a center. When the change in angle is Δθ , the change in displacement is Δs. Using the relationship of trignometric functions, we find that,
\Delta s=\frac{r sin \Delta \theta}{cos (\Delta \theta /2)} The equation is only valid when Δθ does not equal (2n + 1)π where n is integer.
Similarly, the magnitude of tangential speed is always the same. Let Δv be the change in velocity, v be the initial velocity or instantaneous velocity, and Δt be the change in time,
\Delta v=\frac{v sin \Delta \theta}{cos (\Delta \theta/2)}
\frac{\Delta s}{\Delta t}=\frac{r sin \Delta \theta}{\Delta t cos(\Delta \theta/2)}
When \Delta t \to 0, \Delta \theta \to 0,
\lim_{{\Delta t} \to 0}\frac{\Delta s}{\Delta t}=\lim_{{\Delta t \to 0}}\frac{r sin \Delta \theta}{\Delta t cos(\Delta \theta/2)}
v=\lim_{\Delta t \to 0}\frac{r \Delta \theta}{\Delta t}
v=r \frac{d \theta}{dt}=r \omega (ω is angular speed)
\frac{r}{\Delta s}=\frac{cos (\Delta \theta/2)}{sin \Delta \theta}=\frac{v}{\Delta v}
\Delta v=\frac{v \Delta s}{r}
\frac{\Delta v}{\Delta t}=\frac{v \Delta s}{r \Delta t}
\lim_{\Delta t \to 0}\frac{\Delta v}{\Delta t}= \lim_{\Delta t \to 0}\frac{v \Delta s}{r \Delta t}
a=\frac{v^2}{r}

[edit] Centripetal force

Image: Uniform_circular_motion_diagram_3.JPG

The acceleration is usually considered to be due to an inward acting force, which is known as the centripetal force. Centripetal force means “center seeking” force. It is the force that keeps an object in its uniform circular motion. We determine this force by using Newton's second law of motion, Fnet = ma, where Fnet is the net force acting on the object (this is the centripetal force, Fc, of an object in uniform circular motion), m is the mass of the object, and a is the acceleration of the object. Since the acceleration of the object in uniform circular motion is the centripetal acceleration, we can substitute v2 / r or (4π2r) / t2 for a. This gets us Fc = (mv2) / r or Fc = (4mπ2r) / t2

The centripetal force can be provided by many different things, such as tension (as in a string), and friction (as between a tire and the road).

An example of tension being the centripetal force is tying a mass onto a string and spinning it around in a horizontal circle above your head. The tension force is the centripetal force because it is the only force keeping the object in uniform circular motion.

Image: Uniform_circular_motion_tension.JPG

The m is the mass of the object, and the tension force is the centripetal force because it is keeping the object in uniform circular motion.

If a person were to cut the rope at one given point. The object would continue to move in the direction of the velocity.

Image: Uniform_circular_motion_tension_cut.JPG

As one can see, the string holding mass m is cut about ¾ of the way. After the string is cut, the tension force/centripetal force is no longer acting upon the object so there is no force holding the object in uniform circular motion. Therefore it continues going in the direction when it was last in contact with the force.

Similar to the tension force, the friction force between the tires of a car and the road is the centripetal force because it keeps the car moving in a circular path. If this were a frictionless plane, the car would not be able to move in uniform circular motion, and will instead travel in a straight line. Without the friction force acting upon it, no force is keeping the car in uniform circular, so it moves in a straight line.

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