Consider Again the Turntable Described in the Last
Learning Objectives
By the end of this section, you will exist able to:
- Define arc length, rotation angle, radius of curvature and angular velocity.
- Calculate the angular velocity of a auto bicycle spin.
In Kinematics, nosotros studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. Ii-Dimensional Kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a altitude away. In this chapter, we consider situations where the object does not land but moves in a curve. Nosotros begin the study of uniform circular motion by defining two athwart quantities needed to describe rotational motion.
Rotation Bending
When objects rotate about some centrality—for example, when the CD (meaty disc) in Figure 1 rotates near its centre—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation bending is the amount of rotation and is analogous to linear altitude. We define the rotation angle Δθ to be the ratio of the arc length to the radius of curvature: [latex]\displaystyle\Delta\theta=\frac{\Delta{s}}{r}\\[/latex]
Effigy 1. All points on a CD travel in round arcs. The pits forth a line from the center to the edge all move through the same bending Δθ in a time Δt.
Figure ii. The radius of a circle is rotated through an bending Δθ. The arc length Δs is described on the circumference.
The arc lengthΔs is the distance traveled along a round path as shown in Figure 2 Note that r is the radius of curvature of the circular path.
We know that for one complete revolution, the arc length is the circumference of a circle of radius r. The circumference of a circumvolve is 2πr. Thus for one complete revolution the rotation bending is
[latex]\displaystyle\Delta\theta=\frac{2\pi{r}}{r}=two\pi\\[/latex].
This issue is the ground for defining the units used to measure rotation angles, Δθ to exist radians (rad), divers so that 2π rad = 1 revolution.
A comparison of some useful angles expressed in both degrees and radians is shown in Table i.
| Table ane. Comparison of Angular Units | |
|---|---|
| Degree Measures | Radian Mensurate |
| 30º | [latex]\displaystyle\frac{\pi}{6}\\[/latex] |
| 60º | [latex]\displaystyle\frac{\pi}{3}\\[/latex] |
| 90º | [latex]\displaystyle\frac{\pi}{2}\\[/latex] |
| 120º | [latex]\displaystyle\frac{2\pi}{iii}\\[/latex] |
| 135º | [latex]\displaystyle\frac{3\pi}{4}\\[/latex] |
| 180º | π |
Figure 3. Points 1 and 2 rotate through the same angle (Δθ), but point 2 moves through a greater arc length (Δs) because it is at a greater distance from the eye of rotation (r).
If Δθ= 2π rad, then the CD has made ane complete revolution, and every point on the CD is back at its original position. Because there are 360º in a circle or one revolution, the human relationship between radians and degrees is thus 2π rad = 360º so that
[latex]1\text{ rad}=\frac{360^{\circ}}{2\pi}\approx57.3^{\circ}\\[/latex].
Angular Velocity
How fast is an object rotating? Nosotros define angular velocity ω as the rate of change of an angle. In symbols, this is [latex]\omega=\frac{\Delta\theta}{\Delta{t}}\\[/latex], where an angular rotation Δθ takes identify in a time Δt. The greater the rotation angle in a given amount of fourth dimension, the greater the angular velocity. The units for angular velocity are radians per second (rad/s).
Angular velocity ω is analogous to linear velocity v. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length Δs in a time Δt, and so it has a linear velocity [latex]v=\frac{\Delta{s}}{\Delta{t}}\\[/latex].
From [latex]\Delta\theta=\frac{\Delta{due south}}{r}\\[/latex] we see that Δs= rΔθ. Substituting this into the expression for 5 gives [latex]v=\frac{r\Delta\theta}{\Delta{t}}=r\omega\\[/latex].
We write this human relationship in ii different means and proceeds 2 different insights:
[latex]v=r\omega\text{ or }\omega\frac{five}{r}\\[/latex].
The first relationship in [latex]v=r\omega\text{ or }\omega\frac{v}{r}\\[/latex] states that the linear velocity v is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest r), every bit y'all might expect. We can also telephone call this linear speed five of a point on the rim the tangential speed. The second relationship in[latex]v=r\omega\text{ or }\omega\frac{v}{r}\\[/latex] tin can be illustrated by considering the tire of a moving car. Notation that the speed of a bespeak on the rim of the tire is the same as the speed v of the car. See Figure 4. Then the faster the car moves, the faster the tire spins—large v means a large ω, because v=rω. Similarly, a larger-radius tire rotating at the aforementioned angular velocity (ω) will produce a greater linear speed (v) for the machine.
Figure 4. A car moving at a velocity 5 to the right has a tire rotating with an angular velocity ω.The speed of the tread of the tire relative to the axle is five, the aforementioned as if the car were jacked up. Thus the machine moves forwards at linear velocity v = rω, where r is the tire radius. A larger angular velocity for the tire ways a greater velocity for the automobile.
Example 1. How Fast Does a Car Tire Spin?
Calculate the angular velocity of a 0.300 m radius auto tire when the auto travels at 15.0 grand/south (about 54 km/h). Run into Figure four.
Strategy
Considering the linear speed of the tire rim is the same equally the speed of the car, we have 5 = 15.0 m/s . The radius of the tire is given to exist r = 0.300 m . Knowing v and r, we can apply the 2nd relationship in [latex]v=r\omega\text{ or }\omega\frac{v}{r}\\[/latex] to calculate the angular velocity.
Solution
To summate the angular velocity, nosotros will use the following relationship: [latex]\omega\frac{v}{r}\\[/latex].
Substituting the knowns,
[latex]\omega=\frac{15.0 \text{ m/south}}{0.300\text{ chiliad}}=50.0\text{ rad/s}\\[/latex].
Word
When we cancel units in the in a higher place calculation, nosotros get 50.0/s. Simply the athwart velocity must have units of rad/s. Because radians are really unitless (radians are defined every bit a ratio of distance), we can simply insert them into the reply for the angular velocity. Also notation that if an earth mover with much larger tires, say 1.twenty m in radius, were moving at the same speed of 15.0 m/s, its tires would rotate more slowly. They would take an angular velocity [latex]\omega=\frac{15.0\text{ m/s}}{1.20\text{ g}}=12.5\text{ rad/south}\\[/latex].
Both ω and five accept directions (hence they are angular and linear velocities, respectively). Angular velocity has simply two directions with respect to the axis of rotation—information technology is either clockwise or counterclockwise. Linear velocity is tangent to the path, every bit illustrated in Figure 5.
Take-Home Experiment
Tie an object to the end of a string and swing it around in a horizontal circle above your head (swing at your wrist). Maintain uniform speed as the object swings and measure the angular velocity of the move. What is the approximate speed of the object? Identify a betoken close to your paw and accept appropriate measurements to calculate the linear speed at this indicate. Identify other circular motions and measure their athwart velocities.
Figure 5. As an object moves in a circle, here a fly on the border of an erstwhile-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case.
PhET Explorations: Ladybug Revolution
Join the ladybug in an exploration of rotational move. Rotate the merry-go-round to change its bending, or cull a abiding athwart velocity or angular dispatch. Explore how circular motion relates to the bug'south ten,y position, velocity, and acceleration using vectors or graphs.
Click to download. Run using Java.
Section Summary
- Uniform circular motion is motion in a circle at constant speed. The rotation angle [latex]\Delta\theta\\[/latex] is defined as the ratio of the arc length to the radius of curvature: [latex]\Delta\theta=\frac{\Delta{s}}{r}\\[/latex], where arc length Δs is altitude traveled along a circular path and r is the radius of curvature of the circular path. The quantity [latex]\Delta\theta\\[/latex] is measured in units of radians (rad), for which [latex]2\pi\text{rad}=360^{\circ}= 1\text{ revolution}\\[/latex].
- The conversion betwixt radians and degrees is [latex]one\text{ rad}=57.iii^{\circ}\\[/latex].
- Athwart velocity ω is the rate of alter of an angle, [latex]\omega=\frac{\Delta\theta}{\Delta{t}}\\[/latex], where a rotation [latex]\Delta\theta\\[/latex] takes place in a fourth dimension [latex]\Delta{t}\\[/latex]. The units of athwart velocity are radians per second (rad/due south). Linear velocity v and angular velocity ω are related by [latex]5=\mathrm{r\omega }\text{ or }\omega =\frac{5}{r}\text{.}[/latex]
Conceptual Questions
- There is an analogy betwixt rotational and linear physical quantities. What rotational quantities are analogous to altitude and velocity?
Bug & Exercises
- Semi-trailer trucks take an odometer on one hub of a trailer cycle. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutions—it and so calculates the altitude traveled. If the wheel has a ane.fifteen m bore and goes through 200,000 rotations, how many kilometers should the odometer read?
- Microwave ovens rotate at a rate of almost 6 rev/min. What is this in revolutions per second? What is the angular velocity in radians per second?
- An automobile with 0.260 grand radius tires travels fourscore,000 km earlier wearing them out. How many revolutions exercise the tires make, neglecting whatever backing up and any alter in radius due to wear?
- (a) What is the menstruation of rotation of Earth in seconds? (b) What is the angular velocity of World? (c) Given that Earth has a radius of [latex]6.4\times{10}^6\text{ m}\\[/latex] at its equator, what is the linear velocity at World'south surface?
- A baseball game pitcher brings his arm frontwards during a pitch, rotating the forearm about the elbow. If the velocity of the brawl in the pitcher's mitt is 35.0 m/due south and the ball is 0.300 1000 from the elbow joint, what is the angular velocity of the forearm?
- In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 30.0 rad/s and the brawl is 1.30 m from the elbow joint, what is the velocity of the ball?
- A truck with 0.420-thou-radius tires travels at 32.0 m/due south. What is the angular velocity of the rotating tires in radians per second? What is this in rev/min?
- Integrated Concepts. When kick a football, the kicker rotates his leg virtually the hip articulation. (a) If the velocity of the tip of the kicker's shoe is 35.0 chiliad/s and the hip joint is one.05 m from the tip of the shoe, what is the shoe tip's athwart velocity? (b) The shoe is in contact with the initially stationary 0.500 kg football for xx.0 ms. What average force is exerted on the football to give it a velocity of 20.0 thou/south? (c) Detect the maximum range of the football, neglecting air resistance.
- Construct Your Ain Problem. Consider an entertainment park ride in which participants are rotated about a vertical axis in a cylinder with vertical walls. Once the angular velocity reaches its full value, the floor drops away and friction between the walls and the riders prevents them from sliding down. Construct a problem in which you calculate the necessary angular velocity that assures the riders will not slide down the wall. Include a costless body diagram of a single rider. Among the variables to consider are the radius of the cylinder and the coefficients of friction between the riders' wearable and the wall.
Glossary
arc length: Δs, the altitude traveled by an object along a round path
pit:a tiny indentation on the spiral track moulded into the height of the polycarbonate layer of CD
rotation bending:the ratio of the arc length to the radius of curvature on a circular path: [latex]\Delta\theta=\frac{\Delta{s}}{r}\\[/latex]
radius of curvature:radius of a circular path
radians:a unit of measurement of angle measurement
athwart velocity: ω, the charge per unit of alter of the angle with which an object moves on a round path
Selected to Solutions to Issues & Exercises
i. 723 km
three. 5 × 10seven rotations
5. 117 rad/due south
7. 76.2 rad/s; 728 rpm
viii. (a) 33.3 rad/southward; (b) 500 N; (c) twoscore.eight one thousand
Source: https://courses.lumenlearning.com/physics/chapter/6-1-rotation-angle-and-angular-velocity/
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