How To Find Linear Velocity Of A Rotating Object

Section Learning Objectives

By the finish of this section, you volition be able to practice the post-obit:

• Describe the angle of rotation and relate information technology to its linear counterpart
• Describe angular velocity and relate information technology to its linear counterpart
• Solve issues involving angle of rotation and athwart velocity

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The learning objectives in this section volition aid your students master the following standards:

• (4) Science concepts. The educatee knows and applies the laws governing motion in a variety of situations. The student is expected to:
  • (C) clarify and depict accelerated motion in two dimensions using equations, including projectile and circular examples.

Section Key Terms

angle of rotation	angular velocity	arc length	round motion
radius of curvature	rotational motion	spin	tangential velocity

Angle of Rotation

What exactly do nosotros mean past circular motion or rotation? Rotational movement is the circular move of an object about an centrality of rotation. We will discuss specifically round motion and spin. Circular move is when an object moves in a circular path. Examples of round motion include a race car speeding around a circular curve, a toy attached to a string swinging in a circle around your head, or the circular loop-the-loop on a roller coaster. Spin is rotation virtually an axis that goes through the center of mass of the object, such every bit Earth rotating on its axis, a wheel turning on its beam, the spin of a tornado on its path of destruction, or a figure skater spinning during a functioning at the Olympics. Sometimes, objects volition be spinning while in circular motion, like the Earth spinning on its axis while revolving around the Dominicus, just we will focus on these two motions separately.

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[BL] [OL] Explain the difference between circular and rotational motions by using the Earth's rotation about its axis and its revolution nigh the Sunday. Explain that Earth's rotation is slightly elliptical, although information technology is very nearly circular.

[OL] [AL] Enquire students to come up upwards with examples of circular motion.

When solving problems involving rotational motion, nosotros employ variables that are similar to linear variables (altitude, velocity, acceleration, and forcefulness) but have into account the curvature or rotation of the move. Here, we define the angle of rotation, which is the angular equivalence of distance; and angular velocity, which is the angular equivalence of linear velocity.

When objects rotate nigh some axis—for case, when the CD in Effigy 6.2 rotates about its eye—each point in the object follows a round path.

Figure vi.2 All points on a CD travel in round paths. The pits (dots) forth a line from the center to the border all move through the same angle $\text{Δ}\theta$ in fourth dimension $\text{Δ}t$ .

The arc length , , is the distance traveled along a circular path. The radius of curvature, r, is the radius of the circular path. Both are shown in Effigy vi.iii.

Effigy 6.3 The radius (r) of a circle is rotated through an angle $\text{Δ}\theta$ . The arc length, $\text{Δ}s$, is the distance covered along the circumference.

Consider a line from the middle of the CD to its edge. In a given fourth dimension, each pit (used to record data) on this line moves through the aforementioned angle. The bending of rotation is the amount of rotation and is the athwart analog of altitude. The angle of rotation $\text{Δ}\theta$ is the arc length divided by the radius of curvature.

$$\text{Δ}\theta =\frac{\text{Δ}s}{r}$$

The angle of rotation is ofttimes measured by using a unit called the radian. (Radians are really dimensionless, because a radian is defined every bit the ratio of 2 distances, radius and arc length.) A revolution is one complete rotation, where every bespeak on the circumvolve returns to its original position. One revolution covers $\mathrm{two}\pi$ radians (or 360 degrees), and therefore has an angle of rotation of $2\pi$ radians, and an arc length that is the aforementioned as the circumference of the circumvolve. We tin can convert between radians, revolutions, and degrees using the relationship

1 revolution = $2\pi$ rad = 360°. See Table 6.1 for the conversion of degrees to radians for some common angles.

$$\begin{array}{ccc}\hfill \mathrm{two}\pi \text{ rad}& \text{=}& \text{360}°\hfill \\ \hfill 1\mathrm{rad}& \text{=}& \frac{360°}{2\text{π}}\approx 57.3°\hfill \end{array}$$

6.1

Caste Measures	Radian Measures
${30}^{\circ }$	$\frac{\pi }{\mathrm{six}}$
${60}^{\circ }$	$\frac{\pi }{\mathrm{iii}}$
${90}^{\circ }$	$\frac{\pi }{2}$
${120}^{\circ }$	$\frac{2\pi }{3}$
${135}^{\circ }$	$\frac{3\pi }{\mathrm{four}}$
${180}^{\circ }$	$\pi$

Table 6.i Commonly Used Angles in Terms of Degrees and Radians

Angular Velocity

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[BL] Review displacement, speed, velocity, dispatch.

[AL] Ask students whether or not velocity changes in compatible circular motion. What nigh speed? What nearly acceleration?

How fast is an object rotating? We can respond this question by using the concept of athwart velocity. Consider first the angular speed $(\omega )$ is the charge per unit at which the angle of rotation changes. In equation course, the angular speed is

$$\omega =\frac{\text{Δ}\theta }{\text{Δ}t}\text{,}$$

6.2

which ways that an angular rotation $(\Delta \theta )$ occurs in a time, $\text{Δ}t$ . 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/southward).

Now allow's consider the direction of the angular speed, which means we now must telephone call it the athwart velocity. The direction of the angular velocity is forth the axis of rotation. For an object rotating clockwise, the angular velocity points away from you along the centrality of rotation. For an object rotating counterclockwise, the athwart velocity points toward you along the axis of rotation.

Angular velocity (ω) is the athwart version of linear velocity v. Tangential velocity is the instantaneous linear velocity of an object in rotational motion. To go the precise relationship between angular velocity and tangential velocity, consider again a pit on the rotating CD. This pit moves through an arc length $$(\Delta \mathrm{due\; south})$$ in a curt time $$(\Delta t)$$ so its tangential speed is

$$v=\frac{\text{Δ}\mathrm{due\; south}}{\text{Δ}t}\text{.}$$

half dozen.3

From the definition of the angle of rotation, $\text{Δ}\theta =\frac{\text{Δ}s}{r}$ , we run into that $\text{Δ}s=r\text{Δ}\theta$ . Substituting this into the expression for five gives

$$v=\frac{\text{r}\text{Δ}\theta }{\text{Δ}t}=r\omega \text{.}$$

The equation $5=r\omega$ says that the tangential speed v is proportional to the altitude r from the centre of rotation. Consequently, tangential speed is greater for a point on the outer edge of the CD (with larger r) than for a bespeak closer to the center of the CD (with smaller r). This makes sense because a indicate further out from the center has to embrace a longer arc length in the aforementioned corporeality of time as a point closer to the heart. Note that both points will however have the same angular speed, regardless of their altitude from the eye of rotation. See Figure half dozen.4.

Figure six.four Points ane and 2 rotate through the aforementioned bending ( $\text{Δ}\theta$ ), but indicate two moves through a greater arc length ( $\text{Δ}{\mathrm{south}}_2$ ) because it is farther from the heart of rotation.

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[AL] Explain that the time menses $\text{Δ}t$ in the equation that defines tangential velocity ( $\mathrm{five}=\frac{\text{Δ}s}{\text{Δ}t}$ ) must be short so that the arc described past the moving object tin can be approximated every bit a direct line. This allows us to define the direction of the tangential velocity as existence tangent to the circle. This approximation becomes increasingly accurate as $\text{Δ}t$ becomes increasingly pocket-sized.

Now, consider another instance: the tire of a moving automobile (come across Effigy 6.5). The faster the tire spins, the faster the car moves—large $\omega$ means large v because $v=r\omega$ . Similarly, a larger-radius tire rotating at the aforementioned angular velocity, $\omega$ , will produce a greater linear (tangential) velocity, 5, for the car. This is considering a larger radius means a longer arc length must contact the route, so the automobile must movement farther in the aforementioned amount of time.

Figure 6.five A car moving at a velocity, v, to the correct has a tire rotating with athwart velocity $\omega$ . The speed of the tread of the tire relative to the axle is v, the aforementioned as if the motorcar were jacked upwards and the wheels spinning without touching the road. Directly below the axle, where the tire touches the road, the tire tread moves backward with respect to the axle with tangential velocity $v=r\omega$, where r is the tire radius. Because the road is stationary with respect to this signal of the tire, the automobile must move forward at the linear velocity 5. A larger angular velocity for the tire means a greater linear velocity for the car.

However, in that location are cases where linear velocity and tangential velocity are not equivalent, such as a car spinning its tires on water 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 automobile on ice, the arc length through which the tire treads move is greater than the linear distance through which the machine moves. Information technology's similar to running on a treadmill or pedaling a stationary bike; you are literally going nowhere fast.

Tips For Success

Athwart velocity ω and tangential velocity v are vectors, so we must include magnitude and direction. The direction of the angular velocity is forth the axis of rotation, and points abroad from y'all for an object rotating clockwise, and toward you for an object rotating counterclockwise. In mathematics this is described by the right-paw rule. Tangential velocity is commonly described equally up, downwards, left, right, n, south, east, or w, equally shown in Effigy 6.half dozen.

Effigy six.6 As the fly on the edge of an old-fashioned vinyl record moves in a circle, its instantaneous velocity is always at a tangent to the circle. The management of the athwart velocity is into the folio this case.

Watch Physics

Relationship between Angular Velocity and Speed

This video reviews the definition and units of angular velocity and relates it to linear speed. It as well shows how to convert between revolutions and radians.

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For an object traveling in a circular path at a constant angular speed, would the linear speed of the object change if the radius of the path increases?

1. Yep, because tangential speed is independent of the radius.

2. Yes, because tangential speed depends on the radius.

3. No, considering tangential speed is independent of the radius.

4. No, because tangential speed depends on the radius.

Solving Problems Involving Angle of Rotation and Angular Velocity

Snap Lab

Measuring Angular Speed

In this action, you will create and measure out compatible circular motion then contrast it with circular motions with unlike radii.

• One string (i m long)
• One object (two-pigsty rubber stopper) to tie to the end
• One timer

Procedure

1. Tie an object to the cease of a cord.
2. Swing the object around in a horizontal circumvolve to a higher place your caput (swing from your wrist). It is of import that the circle be horizontal!
3. Maintain the object at uniform speed as it swings.
4. Measure out the athwart speed of the object in this manner. Measure the fourth dimension it takes in seconds for the object to travel ten revolutions. Divide that time past 10 to get the angular speed in revolutions per second, which you can convert to radians per 2nd.
5. What is the guess linear speed of the object?
6. Motion your hand upward the cord so that the length of the cord is 90 cm. Echo steps two–v.
7. Motility your hand upwards the string then that its length is 80 cm. Repeat steps ii–5.
8. Movement your hand up the cord and so that its length is lxx cm. Repeat steps 2–5.
9. Move your mitt up the string so that its length is threescore cm. Repeat steps 2–5
10. Motility your mitt up the string and then that its length is l cm. Repeat steps 2–5
11. Brand graphs of athwart speed vs. radius (i.east. string length) and linear speed vs. radius. Draw what each graph looks like.

If yous swing an object slowly, information technology may rotate at less than 1 revolution per 2d. What would be the revolutions per 2nd for an object that makes 1 revolution in five seconds? What would be its angular speed in radians per second?

1. The object would spin at \frac{i}{5}\,\text{rev/s}. The angular speed of the object would be \frac{2\pi}{v}\,\text{rad/southward}.

2. The object would spin at \frac{1}{five}\,\text{rev/south}. The angular speed of the object would be \frac{\pi}{5}\,\text{rad/s}.

3. The object would spin at five\,\text{rev/south}. The angular speed of the object would be x\pi\,\text{rad/s}.

4. The object would spin at v\,\text{rev/s}. The angular speed of the object would be v\pi\,\text{rad/southward}.

At present that we have an understanding of the concepts of angle of rotation and angular velocity, we'll employ them to the existent-world situations of a clock tower and a spinning tire.

Worked Case

Angle of rotation at a Clock Tower

The clock on a clock tower has a radius of ane.0 m. (a) What angle of rotation does the hour mitt of the clock travel through when information technology moves from 12 p.thousand. to 3 p.m.? (b) What's the arc length forth the outermost edge of the clock between the hr paw at these two times?

Strategy

We tin effigy out the angle of rotation past multiplying a full revolution ( $\mathrm{ii}\pi$ radians) by the fraction of the 12 hours covered past the hour manus in going from 12 to 3. Once nosotros have the angle of rotation, nosotros can solve for the arc length by rearranging the equation $\text{Δ}\theta =\frac{\text{Δ}\mathrm{southward}}{r}$ since the radius is given.

Discussion

We were able to drop the radians from the final solution to part (b) because radians are actually dimensionless. This is because the radian is defined as the ratio of two distances (radius and arc length). Thus, the formula gives an answer in units of meters, every bit expected for an arc length.

Worked Instance

How Fast Does a Car Tire Spin?

Summate the athwart speed of a 0.300 yard radius car tire when the car travels at fifteen.0 m/s (about 54 km/h). Come across Figure half dozen.5.

Strategy

In this case, the speed of the tire tread with respect to the tire axle is the same as the speed of the car with respect to the road, so we accept 5 = xv.0 m/southward. The radius of the tire is r = 0.300 g. Since we know 5 and r, nosotros can rearrange the equation $\mathrm{five}=r\omega$ , to get $\omega =\frac{5}{r}$ and discover the athwart speed.

Discussion

When nosotros abolish units in the to a higher place adding, we get l.0/southward (i.e., 50.0 per second, which is normally written equally l.0 southward^−ane). Only the athwart speed must have units of rad/due south. Because radians are dimensionless, we can insert them into the answer for the angular speed because we know that the movement is circular. Also note that, if an globe mover with much larger tires, say 1.twenty m in radius, were moving at the same speed of 15.0 one thousand/s, its tires would rotate more slowly. They would have an angular speed of

$$\begin{array}{ccc}\hfill \omega & =& \frac{\mathrm{fifteen.0}\text{grand/south}}{1.20\text{m}}\hfill \\ & =& \mathrm{12.five}\text{rad/southward}\hfill \end{array}$$

6.8

Practice Issues

1 .

What is the angle in degrees between the hour hand and the minute hand of a clock showing 9:00 a.1000.?

1. 0°
2. 90°
3. 180°
4. 360°

2 .

What is the approximate value of the arc length between the hour hand and the minute hand of a clock showing 10:00 a.chiliad if the radius of the clock is 0.2 grand?

1. 0.ane 1000
2. 0.two k
3. 0.3 m
4. 0.half dozen chiliad

Check Your Understanding

three .

What is circular motion?

1. Circular motion is the movement of an object when it follows a linear path.

2. Circular move is the motion of an object when it follows a zigzag path.

3. Round motion is the motion of an object when information technology follows a circular path.

4. Option D is disruptive as a distractor

four .

What is meant past radius of curvature when describing rotational movement?

1. The radius of curvature is the radius of a circular path.
2. The radius of curvature is the diameter of a circular path.
3. The radius of curvature is the circumference of a circular path.
4. The radius of curvature is the surface area of a circular path.

5 .

What is angular velocity?

1. Angular velocity is the rate of change of the diameter of the circular path.

2. Angular velocity is the rate of change of the angle subtended past the circular path.

3. Angular velocity is the rate of modify of the surface area of the circular path.

4. Athwart velocity is the charge per unit of change of the radius of the circular path.

6 .

What equation defines angular velocity, ω when r is the radius of curvature, θ is the angle, and t is the time?

1. \omega = \frac{\Delta\theta}{\Delta{t}}

2. \omega = \frac{\Delta{t}}{\Delta\theta}

3. \omega = \frac{\Delta{r}}{\Delta{t}}

4. \omega = \frac{\Delta{t}}{\Delta{r}}

7 .

Place three examples of an object in circular motion.

1. an artificial satellite orbiting the Globe, a race car moving in the circular race track, and a pinnacle spinning on its axis

2. an artificial satellite orbiting the Earth, a race machine moving in the circular race rails, and a brawl tied to a string beingness swung in a circle around a person's caput

3. Earth spinning on its ain axis, a race car moving in the circular race rails, and a ball tied to a cord being swung in a circle around a person'due south caput

4. Earth spinning on its own axis, blades of a working ceiling fan, and a height spinning on its ain axis

8 .

What is the relative orientation of the radius and tangential velocity vectors of an object in uniform circular motion?

1. Tangential velocity vector is always parallel to the radius of the round path along which the object moves.

2. Tangential velocity vector is always perpendicular to the radius of the round path forth which the object moves.

3. Tangential velocity vector is always at an acute angle to the radius of the circular path along which the object moves.

4. Tangential velocity vector is always at an obtuse angle to the radius of the circular path forth which the object moves.

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Apply the Cheque Your Agreement questions to assess whether students master the learning objectives of this section. If students are struggling with a specific objective, the formative cess volition aid place which objective is causing the trouble and straight students to the relevant content.

Source: https://openstax.org/books/physics/pages/6-1-angle-of-rotation-and-angular-velocity

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