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How fast is an object rotating? We can answer this question by using the concept of angular velocity. Consider first the angular speed (ω)(ω) is the rate at which the angle of rotation changes. In equation form, the angular speed is which means that an angular rotation (Δθ)(Δθ) occurs in a time, ΔtΔt. If an object rotates through a greater angle of rotation in a given time, it has a greater angular speed. The units for angular speed are radians per second (rad/s). Now let’s consider the direction of the angular speed, which means we now must call it the angular velocity. The direction of the angular velocity is along the axis of rotation. For an object rotating clockwise, the angular velocity points away from you along the axis of rotation. For an object rotating counterclockwise, the angular velocity points toward you along the axis of rotation. Angular velocity (ω) is the angular version of linear velocity v. Tangential velocity is the instantaneous linear velocity of an object in rotational motion. To get the precise relationship between angular velocity and tangential velocity, consider again a pit on the rotating CD. This pit moves through an arc length (Δs)(Δs) in a short time (Δt)(Δt) so its tangential speed is From the definition of the angle of rotation, Δθ=ΔsrΔθ=Δsr, we see that Δs=rΔθΔs=rΔθ. Substituting this into the expression for v gives The equation v=rωv=rω says that the tangential speed v is proportional to the distance r from the center of rotation. Consequently, tangential speed is greater for a point on the outer edge of the CD (with larger r) than for a point closer to the center of the CD (with smaller r). This makes sense because a point farther out from the center has to cover a longer arc length in the same amount of time as a point closer to the center. Note that both points will still have the same angular speed, regardless of their distance from the center of rotation. See Figure 6.4.
Now, consider another example: the tire of a moving car (see Figure 6.5). The faster the tire spins, the faster the car moves—large ωω means large v because v=rωv=rω. Similarly, a larger-radius tire rotating at the same angular velocity, ωω, will produce a greater linear (tangential) velocity, v, for the car. This is because a larger radius means a longer arc length must contact the road, so the car must move farther in the same amount of time.
However, there are cases where linear velocity and tangential velocity are not equivalent, such as a car spinning its tires on ice. In this case, the linear velocity will be less than the tangential velocity. Due to the lack of friction under the tires of a car on ice, the arc length through which the tire treads move is greater than the linear distance through which the car moves. It’s similar to running on a treadmill or pedaling a stationary bike; you are literally going nowhere fast.
Angular velocity ω and tangential velocity v are vectors, so we must include magnitude and direction. The direction of the angular velocity is along the axis of rotation, and points away from you for an object rotating clockwise, and toward you for an object rotating counterclockwise. In mathematics this is described by the right-hand rule. Tangential velocity is usually described as up, down, left, right, north, south, east, or west, as shown in Figure 6.6.
This video reviews the definition and units of angular velocity and relates it to linear speed. It also shows how to convert between revolutions and radians. Click to view content
For an object traveling in a circular path at a constant speed, would the linear speed of the object change if the radius of the path increases?
In a circle, there are various angles that can be formed joining the endpoints of arcs and those angles are termed subtended angles. There are different categories of angles formed by these arcs, for example, angles in the same segment, angles in a semi-circle, angles at the circumference, etc. Let's learn about it in detail in this lesson. How do you Find the Angle Subtended by an Arc in Circle?An arc of a circle is any part of the circumference. The angle subtended by an arc at any point is the angle formed between the two line segments joining that point to the end-points of the arc. In the following figure, an arc of the circle shown subtends an angle α at a point on the circumference, and an angle β at the center O. We are interested in the relation between the angle which a given arc subtends anywhere on the circumference and the angle which that arc subtends at the circle’s center. The relation between them is simple and extremely important. Angle Subtended by an Arc at the CenterAt the center of a circle, if there are two line segments originated from the end-points of the arc of the circle intersect, that angle is called an angle subtended by the arc at the center. Let us understand a theorem and its proof based on it. Theorem:The angle subtended by an arc of a circle at its center is twice the angle it subtends anywhere on the circle’s circumference. The proof of this theorem is quite simple, and uses the exterior angle theorem – an exterior angle of a triangle is equal to the sum of the opposite interior angles. If the two opposite interior angles happen to be equal, then the exterior angle will be twice any of the opposite interior angles. Proof:Consider the following figure, in which an arc (or segment) AB subtends ∠AOB at the center O and ∠ACB at a point C on the circumference. We have to prove that ∠AOB = 2 × ∠ACB. Draw the line through O and C, and let it intersect the circle again at D, as shown. There are two triangles formed ΔOAC and ΔOBC. So, we make the following observations. 1. In ΔOAC, ∠OAC = ∠OCA, because OA = OC 2. In ΔOBC, ∠OBC = ∠OCB, because OB = OC. Hence, using the exterior angle theorem, we get, ∠AOD= 2×∠ACO ⋯ (1) ∠DOB= 2×∠OCB ⋯ (2) Add equations (1) and (2): ∠AOD+ ∠DOB= 2× (∠ACO+ ∠OCB) ⇒ ∠AOB= 2× ∠ACB This completes the proof of the theorem. You can experiment with the simulation below to understand the relationship between the angles subtended by an arc at the center and at a point on the circle. What if arc AB is such that it subtends a reflex angle at the center? Well, our proof doesn’t change, as the following figure shows. You can now apply exactly the same steps as we did earlier and arrive at the same result: ∠AOB= 2× ∠ACB. Another configuration would be when O does not lie within ∠ACB. Will the result still hold, as in the following figure? Yes, it will. Join C to O again and extend it to D on the circumference of the circle. Use the Exterior Angle Theorem again and note that, 1. ∠BOD= 2× ∠BCD ⋯ (1) 2. ∠AOD= 2× ∠ACD ⋯ (2) To avoid cluttering, these angles have not been highlighted in the figure, but make sure you observe them carefully. Subtract Equation (2) from (1) ∠BOD− ∠AOD= 2× (∠BCD− ∠ACD) ⇒ ∠AOB= 2× ∠ACB This theorem leads to an interesting corollary, which is discussed next. Corollary:The angle in a semi-circle is a right angle. Consider the figure below, where AB is the diameter of the circle. We need to prove that ∠ACB= 90∘ Proof:Since ∠AOB= 180º, ∠ACB= 1/2× ∠AOB= 90º This theorem also leads to the following result. Angles in the Same Segment of a Circle are EqualIn a circle, there are two segments- major segments and minor segments. Angles formed by the endpoints of the chord in either major or minor segments are always equal. Let's understand this theorem and its proof in detail. Theorem:Angles in the same segment of a circle are equal. In other words, an arc in a circle will subtend equal angles anywhere on the circumference. Consider the following figure, which shows an arc AB subtending angles ACB and ADB at two arbitrary points C and D on the circumference. O is the center of the circle. We need to prove that ∠ACB= ∠ADB. Proof:Using our previous theorem, we have that ∠ACB= 1/2× ∠AOB ⋯ (1) ∠ADB= 1/2× ∠AOB ⋯ (2) From equations (1) and (2), we get ∠ACB= ∠ADB. Important Notes:
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FAQs on Arcs and Subtended AnglesIn Geometry, subtended means to extend under. A subtended angle is formed by the intersection of more than 1 line, or the rays joining the endpoints of a line or arc. What is a Subtended Central Angle?A subtended central angle is the angle formed at the center of the circle and the legs of the angle are the radii of the circle intersecting the circle at two different points. How do you Find the Angle Subtended by an Arc at the Centre?To find the value of angle subtended by an arc at the center we have to multiply the angle formed through the same end-points of the arc on the circumference by two. For example, if the angle subtended at any point on the circumference is 60º, that means the angle subtended by the same arc at the center is 120º. What is the Central Angle of a Circle?The central angle of a circle is that angle whose vertex is the same as the center point of the circle. The other two arms are the radius which meets the endpoints of any chord or arc of the circle. What is Arc in Circle?In a circle, the arc can be defined as any portion of the boundary. It is a curve that forms the circumference of a circle. In other words, we can say that circumference is a complete or closed arc of the circle. What is the Measure of Angle in a Semi-circle?The angle in a semi-circle is always 90º. Arc drawn from the endpoints of the diameter of a circle always form a 90º angle on its circumference. |