Electricity goes like many miles per second
Calculation of the angular velocity
In the everyday dicure "speed" and "speed" are often used synonymously. In physics, however, these terms have specific and different meanings. & qu
In everyday discourse, "speed" and "speed" are often used synonymously. However, in physics, these terms have specific and different meanings. "Speed" is the rate at which an object moves in space and is only indicated by a number with specific units (often in meters per second or miles per hour). The speed, on the other hand, is a speed that is coupled to a direction. The speed is called a scalar quantity, while the speed is a vector quantity.
When a car is driving on a freeway or a baseball is flying through the air, these objects are measured for speed relative to the ground, while speed is more information. For example, if you're in a car traveling 100 km / h on Interstate 95 on the east coast of the United States, it is also helpful to know whether you are heading northeast to Boston or south to Florida drive. In baseball, you may want to know if its y-coordinate changes faster than its x-coordinate (a fly ball) or if the opposite is the case (a line drive). But what about the spinning of the tires or the spin of the baseball as the car and ball move towards their final destination? For these kinds of questions, physics offers the concept of Angular velocity.
The basics of movement
Things move through three-dimensional physical space in two ways: translation and rotation. Translation is moving the entire object from one place to another, like a car going from New York City to Los Angeles. Rotation, on the other hand, is the cyclical movement of an object around a fixed point. Many objects, like the baseball in the example above, show both types of motion at the same time. When a fly ball moves through the air from the home plate to the outfield fence, it also rotates around its own center at a certain speed.
Describing these two types of motion is treated as separate physical problems. That said, if you calculate the distance the ball will travel in mid-air based on things like the initial launch angle and the speed at which it leaves the club, you can ignore its spin for current purposes.
The angular rate equation
First, when you talk about "angular" things, be it speed or some other physical quantity, you realize that because you are concerned with angles you are talking about traveling in circles or parts of them. Recall from geometry or trigonometry that the perimeter of a circle is the diameter multiplied by the constant pi, or πd. (The value of pi is approximately 3.14159.) This is more commonly expressed as the radius of a circle r, that's half the diameter, makes the circumference 2πr.
Also, you probably learned somewhere along the way that a circle is made up of 360 degrees (360 °). If you move a distance S along a circle, the angular displacement θ is equal to S / r. One full turn then gives 2πr / r, which leaves only 2π. This means that angles less than 360 ° can be expressed in pi or in other words as radians.
When you put all of this information together, you can express angles or parts of a circle in units other than degrees:
360 ° = (2π) radians or
1 radian = (360 ° / 2π) = 57.3 °,
While linear velocity is expressed in length per unit of time, angular velocity is measured in radians per unit of time, usually per second.
When you know that a particle is moving at one speed on a circular path v at some distance r from the center of the circle with the direction of v Always perpendicular to the radius of the circle, then the angular velocity can be written
ω = v / r,
Where ω is the Greek letter Omega. Units of angular velocity are radians per second; You can also treat this unit as a "reciprocal second" since v / r gives m / s divided by m or s-1This means that radians are technically a unitless quantity.
Equations of rotational motion
The angular acceleration formula is derived in the same way as the angular velocity formula: It is only the linear acceleration in a direction perpendicular to a radius of the circle (corresponding to its acceleration along a tangent to the circular path at any point) divided by the radius of the circle or a Section of a circle, which is:
α = at/ r
This is also given by:
α = ω / t
because for circular motion at = ωr / t = v / t.
αAs you probably know, the Greek letter is "alpha". The index "t" here means "tangent".
Curiously, however, the rotational motion exhibits a different type of acceleration called centripetal ("center-seeking") acceleration. This is given by the expression:
ac = v2/ r
This acceleration is directed towards the point around which the object in question rotates. This may seem strange since the object is not getting any closer to this center point since the radius r Is fixed. Think of centripetal acceleration as a free fall where there is no danger of the object hitting the ground, as the force pulling the object on itself (usually gravity) is exactly balanced by the tangential (linear) acceleration that is described by the first equation in this section. If ac weren't the same atthe object would either fly into space or soon crash into the center of the circle.
Related sets and expressions
Although angular velocity is commonly expressed in radians per second, there may be cases where it is preferable or necessary to use degrees per second instead, or conversely, convert degrees to radians before solving a problem.
Suppose you learned that a light source rotates 90 degrees every second at a constant speed. What is its angular velocity in radians?
First, remember that 2π radians = 360 ° and set up a proportion:
360 / 2π = 90 / x
360x = 180π
x = ω = π / 2
The answer is half a radian per second.
Further, if you knew that the beam of light had a range of 10 meters, what would be the peak of the beam's linear velocity? v, its angular acceleration α and its centripetal acceleration ac?
To solve for vfrom above, v = ωr, where ω = π / 2 and r = 10m:
(? / 2) (10) = 5? rad / s = 15.7 m / s
To solve for α, just add another unit of time to the denominator:
α = 5π rad / s2
(Note that this only works for problems where the angular velocity is constant.)
Finally also from above ac = v2/ r = (15.7)2/ 10 = 24.65 m / s2.
Angular velocity vs. linear velocity
Building on the previous problem, imagine a very large carousel with an improbable 10 kilometers (10,000 meters) radius. This carousel makes a complete revolution every 1 minute and 40 seconds or every 100 seconds.
A consequence of the difference between angular velocity, which is independent of distance from the axis of rotation, and circular linear velocity, which it is not, is that two people experience the same thing ω can suffer from very different physical experiences. If you happen to be 1 meter from the center on this suspected massive carousel, your linear (tangential) speed is:
ωr = (2π rad / 100 s) (1 m) = 0.0628 m / s or 6.29 cm (less than 3 inches) per second.
But when you are on the edge of this monster your linear speed is:
ωr = (2π rad / 100 s) (10,000 m) = 628 m / s, which is about 1,406 miles per hour, faster than a bullet. Wait!
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