Angular Acceleration

5 Key excerpts on "Angular Acceleration"

• 

  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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  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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  BIOS Instant Notes in Sport and Exercise Biomechanics

  • Paul Grimshaw, Neil Fowler, Adrian Lees, Adrian Burden(Authors)
  • 2007(Publication Date)
  • Routledge

    (Publisher)

  direction). Usually, we are normally concerned with rotation about one axis of rotation and it is applicable therefore to refer to angular momentum about a single origin or a single axis of rotation. In this manner we can consider it as a scalar quantity where we refer to its direction as either positive (anti-clockwise rotation) or negative (clockwise rotation). In addition, it is worth repeating that the total angular momentum of a body about any axis of rotation is made up by adding all the angular momenta of the various parts or segments of the body which are rotating about that axis. Within biomechanics this has important implications for understanding human movement and in more complex analyses the study of angular momenta about multiple axes of rotation is required.

  As we have seen, a net torque (that is not zero) that acts on an object will cause an Angular Acceleration of the object in the direction of the net torque. The amount of net external torque will equal the rate of change of angular momentum (i.e., from the angular analog of Newton’s second law).

  The change in angular momentum of an object can be determined by examining the initial and final angular momentum possessed by the object: Change in angular momentum = Angular momentum (final) − Angular momentum (initial) Thus we can now include this in the equation for torque: Mathematically this can be expressed as: Rearranging this equation produces the following: T(net) × (t2 − t1 ) = L(f) − L(i) This can now be expressed as the equation for angular impulse: This equation has important implications for the effective execution of rotational movements with human motion.

  Application

  Considering the diver in

  Fig. C2.3

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  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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  Engineering Science

  For Foundation Degree and Higher National

  • Mike Tooley, Lloyd Dingle(Authors)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  We considered angular motion in section 8.3. We now extend the notion of angular motion to include the accelerations and forces that act on bodies that move in a circular path, for example, motor vehicles that negotiate bends, aircraft that bank and turn or governors that have rotating bob weights to control the rate of opening of valves and the like. We start by considering the acceleration and forces that occur as a result of circular motion.

  8.8.1 Centripetal acceleration and force

  You will be aware of the relationship between linear and angular velocity through use of the transformation equations, where we know that s = rθ , v = rω and a = rα . We can use these formulae to form relationships for the accelerations and forces that act on a body during circular motion. Consider Figure 8.22 , where a point mass is subject to a rotational velocity.

  Figure 8.22 A point mass subject to a rotational velocity

  From the figure it can be seen that the direction of motion of the mass must be continually changing, in order to produce the circular motion. Therefore, it is being subjected to an acceleration that is acting towards the centre of rotation ; this is known as centripetal acceleration .

  Now, from the transformation equations we know that the tangential velocity v = r ω and that the acceleration of the mass is equal to its change in velocity divided by the time taken for the mass to move through some angular distance θ. Thus the change in the tangential velocity of the mass divided by the time, that is its acceleration , is given as and since the angular velocity is equal to the angle turned through in radian divided by time, Then, from above, the centripetal acceleration of the mass = ω2 r. Also, we know that v = rω or so that centripetal therefore we may write that:

  Now when this acceleration acts on a mass, as in this case, it produces a force known as centripetal force, thus: centripetal force (

  Fc

  ) = mass × centripetal acceleration . Then:

  Now, remembering Newton’s third law, that to every action there is an equal and opposite reaction, the centripetal force acting on the mass pushing it towards the centre of rotation must be balanced by the radial force created by the mass maintaining the orbit. This force opposing the centripetal force is known as the centrifugal force

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