Angular Velocity

5 Key excerpts on "Angular Velocity"

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  eBook - ePub

  Classical Mechanics

  A Computational Approach with Examples Using Mathematica and Python

  • Christopher W. Kulp, Vasilis Pagonis(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  Figure 10.2 represents a wheel rotating in a counter clockwise direction. As you curl your fingers in the direction of the arrows on the wheel, your thumb should point up, away from the page (or screen if you are reading this on a device). Your thumb is pointing in the direction of the wheel’s Angular Velocity.

  Figure 10.2: The right-hand rule. The circle represents a wheel and the arrows on the circle represent a counter clockwise rotation of the wheel. Curl the fingers of your right hand in the direction of the arrows on the circle. Your thumb should point in the direction of the vector

  ω .

  If an object’s Angular Velocity is changing with time, then both the rate of rotation and/or the orientation of the rotation axis is changing in time. As you might imagine, situations that involve changes in the orientation of the rotation axis can be difficult to describe mathematically. In this chapter, we will focus primarily on systems with fixed angular velocities, i.e., both the rate of rotation and the direction of the rotation axis will remain constant.

  In addition to a formal definition of Angular Velocity, we will also need a relationship between the velocity of a particle and its Angular Velocity. You may recall from your introductory physics course that the tangential velocity v of a particle moving along a circle of radius r with Angular Velocity ω is v = ωr. In this particular case, the axis of rotation went through the center of the circle and was perpendicular to the plane of the circle. This simple relationship between v and ω results from the fact that all of the motion is restricted to a plane. That is not generally true.

  Consider a particle, represented by the black dot in Figure 10.3 fixed to a location on the Earth’s surface in the Northern Hemisphere. The Angular Velocity is essentially constant and points from the South Pole to the North Pole. While every point on the Earth has the same Angular Velocity, the particle’s latitude will affect the particle’s tangential velocity.

  Figure 10.3: The northern hemisphere of the Earth showing the location of a point particle (black dot) and its velocity, v.

  In Figure 10.3 , the particle travels in a circle of radius

  ρ =

  | ρ |

  as the Earth rotates. The latitude of the particle is determined by the angle θ (the latitude is π/2 − θ). As θ increases, the particle needs to travel a greater distance in the same amount of time (the Angular Velocity is the same at all latitudes). Therefore, v = |v | must increase with an increasing θ. We know that the tangential velocity of the particle is v = ρω from introductory physics. However, ρ = rsinθ. Therefore, v = rωsinθ

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  eBook - ePub

  Foundations of Mechanical Engineering

  • A. D. Johnson(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  fraction of circumference  × total circumference

  =

  1 .5 × 560 π

  2 π

  = 420mm

  2.6.3 Angular Velocity

  A car travelling along the road at a steady velocity possesses linear velocity measured in m/s. The wheels are also rotating at a steady velocity and their ‘Angular Velocity’ is measured in rad/s and is defined as ‘the rate of change of angular displacement compared to elapsed time’ and may be denoted by the symbol ω (omega).

  Angular Velocity parallels linear velocity in that it shares the same concepts of average velocity and constant velocity, but it must be noted that these now refer to rotational velocities and are defined in appropriate units.

  If a chalk mark on a bicycle wheel takes a time t to rotate through an angle θ , then the average Angular Velocity is

  ω =

  θ t

  =

  angular displacement time

  ⁢ ( 2.6 )

  Constant Angular Velocity means that equal, consecutive angular displacements (rad) are covered in equal intervals of time. The bicycle wheel would, therefore, rotate at a non-varying velocity.

  Angular Velocity is related to the ‘frequency of rotation’, or the number of revolutions made per second. Since there are 2π radians in one revolution, then n revolutions per second will have an Angular Velocity of 2πn rad/s, as follows:

  1 rev/s = 2 π   rad/s

  2 rev/s = 2 π × 2   rad/s

  3 rev/s = 2 π × 3   rad/s

  n     rev/s = 2 π × n   rad/s

  Hence

  angular velocity  = ω = 2 π n   rad/s

  ⁢ ( 2.7 )

  In many cases Angular Velocity is quoted in the form of revolutions per minute, which requires a modification of equation (2.7) , as follows:

  ω =

  2 π n

  60

  rad/s

  ⁢ ( 2.8 )

  where n is expressed in rev/min.

  Example 2.13 A flywheel rotates at 3000 rev/min. Find its Angular Velocity in rad/s. Solution

  The Angular Velocity can be found by using equation (2.8) :

  ω =

  2 π n

  60

  =

  2 π × 3000

  60

  = 314rad/s

  2.6.4 Angular acceleration

  When traffic lights turn green a car steadily builds up its velocity after being at rest. The car experiences linear acceleration. The wheels are also stationary when the lights change to green and they also steadily build up their velocity. Their Angular Velocity increases second by second and can be said to possess ‘angular acceleration’, denoted by α (alpha). The units of angular acceleration are radians/second/second (rad/s2

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  eBook - ePub

  Doing Physics with Scientific Notebook

  A Problem Solving Approach

  • Joseph Gallant(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Evaluate Numerically to find the number of times the record turns while it’s slowing down.

  The record turns about 2.8 times while it’s slowing down.

  When the angular acceleration is not constant, you must use calculus to describe the object’s rotational motion. The Angular Velocity is the rate of change of the angular displacement.

  (7.9)

  If you know the angular position as a function of time, you can calculate the angular speed.

  The angular acceleration is the rate of change of angular speed.

  (7.10)

  If you know the angular speed as a function of time, you can calculate the angular acceleration. From these definitions we can derive four basic equations which describe the angular position and speed of objects rotating with a varying acceleration.

  (7.11a)

  (7.11b)

  (7.11c)

  (7.11d)

  For varying angular acceleration, the average Angular Velocity does not equal the arithmetic average of the initial and final angular velocities.

  The Compact Disk

  The rotational motion of a CD is an excellent example of nonuniform rotational motion with a time-dependent acceleration. To read the information on a record, a needle starts at the outer edge and follows a single spiral track while the disk rotates at a constant angular speed. To read the information on a CD, a laser starts at the inner radius and follows a single spiral track while the disk rotates at a decreasing angular speed. This ensures the data-read rate, which is proportional to the linear speed, is constant. [31]

  The data on a typical CD is stored between R0 = 2.3 cm and Rf = 5.8 cm, and the spacing between tracks is Δr = 1.6 μm/rev. We can use Evaluate Numerically

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  eBook - ePub

  Classical Mechanics

  • Tom W B Kibble, Frank H Berkshire(Authors)
  • 2004(Publication Date)
  • ICP

    (Publisher)

  Chapter 5

  Rotating Frames

  Hitherto, we have always used inertial frames, in which the laws of motion take on the simple form expressed in Newton’s laws. There are, however, a number of problems that can most easily be solved by using a non-inertial frame. For example, when discussing the motion of a particle near the Earth’s surface, it is often convenient to use a frame which is rigidly fixed to the Earth, and rotates with it. In this chapter, we shall find the equations of motion with respect to such a frame, and discuss some applications of them.

  5.1Angular Velocity; Rate of Change of a Vector

  Let us consider a solid body which is rotating with constant Angular Velocity ω about a fixed axis. Let n be a unit vector along the axis, whose direction is defined by the right-hand rule: it is the direction in which a right-hand-thread screw would move when turned in the direction of the rotation. Then we define the vector Angular Velocity ω to be a vector of magnitude ω in the direction of n: ω = ωn. Clearly, Angular Velocity, like angular momentum, is an axial vector (see §3.3 ).

  For example, for the Earth, ω is a vector pointing along the polar axis, towards the north pole. Its magnitude is equal to 2π divided by the length of the sidereal day (the rotation period with respect to the fixed stars, which is less than that with respect to the Sun by one part in 365), that is

  If we take the origin to lie on the axis of rotation, then the velocity of a point of the body at position r is given by the simple formula

  To prove this, we note that the point moves with Angular Velocity ω around a circle of radius ρ = r sin θ (see Fig. 5.1 ). Thus its speed is

  Fig. 5.1

  Moreover, the direction of v is that of ω ∧ r; for clearly, v is perpendicular to the plane containing ω and r

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  eBook - ePub

  Higher Engineering Science

  • William Bolton(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  The instantaneous Angular Velocity ω is the change in angular displacement with time when the time interval tends to zero. It can be expressed as:

  [20]

  3 Angular accelerationThe average angular acceleration over some time interval is the change in Angular Velocity during that time divided by the time:

  [21]

  The unit is rad/s2 . The instantaneous angular acceleration a is the change in Angular Velocity with time when the time interval tends to zero. It can be expressed as:

  [22]

  4.4.1 Motion with constant angular acceleration

  For a body rotating with a constant angular acceleration α, when the Angular Velocity changes uniformly from ω0 to co in time t, as in Figure 4.19 , equation [21 ] gives:

  Figure 4.19 Uniformly accelerated motion

  and hence:

  ω = ω0 + at

  [23]

  The average Angular Velocity during this time is ½(ω + ω0 ) and thus if the angular displacement during the time is θ:

  Substituting for co using equation [23 ]:

  Hence:

  θ = ω0 t + ½at2

  [24]

  Squaring equation [23 ] gives:

  Hence, using equation [24 ]:

  [25]

  Example

  An object which was rotating with an Angular Velocity of 4 rad/s is uniformly accelerated at 2 rad/s. What will be the Angular Velocity after 3 s?

  Using equation [23 ]:

  ω = ω0 + at = 4 + 2 × 3 = 10 rad/s

  Example

  The blades of a fan are uniformly accelerated and increase in frequency of rotation from 500 to 700 rev/s in 3.0 s. What is the angular acceleration?

  Since ω = 2πf, equation [23 ] gives:

  2π × 700 = 2π × 500 + a × 3.0

  Hence a = 419 rad/s2 .

  Example

  A flywheel, starting from rest, is uniformly accelerated from rest and rotates through 5 revolutions in 8 s. What is the angular acceleration?

  The angular displacement in 8 s is 2π × 5 rad. Hence, using equation [24 ], i.e. θ = ω0 t + ½at2 :

  2π × 5 = 0 + ½a × 82

  Hence the angular acceleration is 0.98 rad/s2 .

  Revision

  13 A flywheel rotating at 3.5 rev/s is accelerated uniformly for 4 s until it is rotating at 9 rev/s. Determine the angular acceleration and the number of revolutions made by the flywheel in the 4 s.

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