Rotational Equilibrium

6 Key excerpts on "Rotational Equilibrium"

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  eBook - ePub

  Introductory Physics for the Life Sciences: Mechanics (Volume One)

  • David V. Guerra(Author)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  No matter how many equal parts a uniform bar is divided up into the center of mass will always be at the center. For objects that are not uniform, the center of mass will need to be measured or if the function that described the object is known an integral needs to be used in a process like the one described for the uniform bar, but with different values for each part of the object. It cannot be assumed that the center of mass is at the center of the object that is not uniform.

  3.5 Rotational Equilibrium

  In Chapter 2 , translational equilibrium was defined as the condition in which all the forces in each coordinate direction add to zero. In some situations, translational equilibrium is not enough to explain what is happening. For example, if the bar in Figure 3.12 is 10 m long and pinned at its center, it is free to rotate in the x-y plane about its center, and 10 N forces are applied to each end as shown in the figure, the bar would be in translational equilibrium, but it would not be at rest.

  FIGURE 3.12 Bar in translational equilibrium but not in Rotational Equilibrium.

  It is obvious that the bar would rotate counterclockwise about the center. Thus, to understand this situation, another condition is needed. This additional condition for this system is Rotational Equilibrium.

  Similar to translational equilibrium of forces, Rotational Equilibrium is the sum of all torques adding up to zero, this can be written mathematically in equation (3.6) as

  τ ⇀

  Net

  = 0 , which means that ∑  

  τ ⇀

  = 0

  (3.6)

  which states that the sum of the torques acting on an object in equilibrium must equal zero. For the problems in this text, all the forces will lie in the x-y plane, so all the torques will be parallel to the z-axis so the relevant torque component equation for Rotational Equilibrium will be given by equation (3.7) as:

  τ Net, z

  = 0  which means that

  ∑

  τ z

  = 0

  (3.7)

  which states that the sum of the z-components of the torques acting on an object in equilibrium must equal zero.

  3.5.1 Rotational Equilibrium Concept Map

  In Figure 3.13

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  eBook - ePub

  Statics For Dummies

  • James H. Allen(Author)
  • 2010(Publication Date)
  • For Dummies

    (Publisher)

  Rotation is always caused by the application of an applied moment, couple, or eccentric load (see Chapter 12 if you need more information). If an object is subjected to unbalanced moments, couples, or eccentric loads, that object experiences a rotation. Thus, if all the rotational behaviors are balanced, an object is in

  Rotational Equilibrium, and the net effect of the moments (or the resultant ) of all the rotational behaviors must also be zero.

  In the real world, you can observe an endless combination of translational and rotational effects. Objects can be displacing (translating) and, at the same time, spinning (rotating). Rolling

  is actually a combination of translation and rotation. For example, a tire on a moving car experiences rolling. The tire itself is rotating about its center point (the axle of the car), but at the same time, the center of the tire is moving in a straight line in the direction of the traveling car (assuming, of course, that the car is traveling on a straight and level stretch of road).

  In order for an object to be in total equilibrium, it must be balanced for all translational behaviors at the same time that it’s balanced for the rotational behaviors. If either one of these behaviors isn’t balanced, the object can’t be considered to be in equilibrium.

  Looking for Equilibrium with Newton’s Laws

  The basic equations of statics and equilibrium are founded in the principles of Newtonian mechanics, developed by Sir Isaac Newton in the 17th century. Newton’s three laws of motion help describe the way forces and objects interact, and ultimately provide the basic equations of equilibrium.

  Reviewing Newton’s laws of motion

  Newton’s laws provide a solid foundation for dealing with objects subjected to forces (also known as the study of mechanics). Perhaps the most famous of his explanations are contained in his three fundamental laws of motion:

  Newton’s first law: Newton’s first law of motion, sometimes referred to as the law of inertia,

  states that an object at rest tends to stay at rest until acted upon by an unbalanced force. Likewise, an object in motion stays in motion with the same speed and in the same direction until acted upon by an unbalanced force.

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  eBook - ePub

  Predicting Motion

  • Robert Lambourne(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  Rotational Equilibrium:

  ∑ i

  Γ i

  = 0

  (4.13)

  where the Γ i are the external torques about the centre of mass. Note though that in this case there are still situations (as pointed out in Section 2.4 ) where the absence of a resultant external torque is not sufficient to ensure that the angular velocity will remain constant. Nonetheless, this is, in effect, the best we can do when it comes to specifying a condition for Rotational Equilibrium.

  To complete the terminology of equilibrium conditions, we can combine Equations 4.12 and 4.13 to provide the following condition for mechanical equilibrium:

  ∑ i

  F i

  = 0         and      

  ∑ i

  Γ i

  = 0 .

  (4.14)

  A static system is a special case of a system in equilibrium, so it must necessarily satisfy this condition. Remember, however, that the mechanical equilibrium condition is not sufficient to guarantee that a system will be static.

  Question 4.7 A pair of oppositely directed forces of equal magnitude, with different lines of action is said to comprise a couple. Such a couple is applied to the uniform wheel of radius of 0.345 m in Figure 4.25 . If the magnitude of each force is |F | = 20.2 N, and no other forces act on the wheel, what is the total force on the wheel? What is the torque Γ about the centre of the wheel, and will the wheel be in any sort of equilibrium?

  Question 4.8 A disc of diameter 1.50 m is mounted on an axle of diameter 5.50 × 10−2 m, that passes through the centre of the disc and is perpendicular to its surface. The axle is supported horizontally by a worn bearing. When rotating at a constant angular speed ω , the frictional force experienced by the rotating axle is of magnitude 0.168 N. What force must be applied to the rim of the disc if it is to maintain a constant angular speed? ■

  Figure 4.25 Two forces applied to the rim of a wheel. The forces are of equal magnitude but act in different directions along different lines of action. Such a pair of forces tends to cause rotational acceleration but not translational acceleration, and is sometimes referred to as a couple.

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  eBook - ePub

  Understanding Physics

  • Michael M. Mansfield, Colm O'Sullivan(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Equation (7.7) which in this case becomes

  (7.8)

  The significance of torque in relation to the rotation of rigid bodies can be seen from the following example. Consider a horizontal wheel of radius a which is constrained to rotate about an axle through its centre C, as illustrated in Figure 7.8 .

  If we apply an external force to the rim of the wheel at any point there will be an equal and opposite reaction (a constraint force) applied to the wheel by the axle. How do we apply a force to the wheel to achieve maximum rotational effect, that is to produce the maximum rate of change of angular momentum? If a radial force is applied, that is a force

  Fr

  in Figure 7.8 such that the line of action of the force passes through the centre of the wheel, there will be no torque about the axis (that is,

  |Fr |d = 0

  since

  d = 0

  in this case) and the force will be ineffective in providing a rotational effect. On the other hand, if a force of the same magnitude is applied at a tangent to the rim of the wheel, that is

  Ft

  in Figure 7.8 , this will result in the maximum rate of change of angular momentum since, in this case, the torque

  |Fr |d = |Ft |a

  is maximum.

  Figure 7.7

  Motion of a rigid body projectile is seen as the superposition of two independent motions. The centre of mass of the body follows a parabolic path under the influence of gravity. The body rotates with constant angular velocity about the centre of mass (since there is no net moment relative to the centre of mass, in this case).

  7.5 Measurement of torque: the torsion balance

  Consider what happens when one end of a piece of wire or a metal rod is twisted about its axis while the other end is held fixed. This can be studied experimentally by fixing one end of a horizontal metal rod and attaching a solid disc to the other end as shown in Figure 7.9

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  eBook - ePub

  Evolution

  Classical Philosophy Meets Quantum Science

  • Somnath Bhattacharyya(Author)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  Why is angular or rotational motion so important? It is the basis of our civilization. Wheels, pulleys, and gears give stability to a vehicle. It starts from a point and returns to the same point. Linear motion is not associated with an external force that is always present since the world is curved and no motion is free from force. The freefall of a particle from a height follows a straight line since gravitational force acts toward the center of the Earth. Any motion in a straight line not exactly normal to the surface of the Earth is curvilinear as a result of the applied force and the gravitational force. The criterion for the rotation of a particle is that the tangential velocity of the particle is always normal to the line joining the gap, i.e., the axis of rotation to the rotating particle. The rotation of an object is described by the centripetal and the centrifugal accelerations that act toward the center of rotation and outward, respectively. The criterion for the rotational motion is that the distance vector and the force are perpendicular to each other, forming a moment. The tendency for rotation is called the moment of a force that works in the perpendicular direction of the applied force. Rotation without translation can be achieved by the application of a pair of equal and opposite forces at two different points of an object. This combination of two moments around a fixed point is called a ‘couple’. The operation of the wheel was invented based on the principle of rotational motion. The inertia (or mass) in linear motion can be replaced by the moment of inertia. The equivalent force in rotational motion is called ‘torque’, which is proportional to the centripetal acceleration pointing to the center of the motion and the moment of inertia. In the final part of this chapter, we unify the linear (along the axis) as well as the rotational motion in a vortex.

  Vortex

  In fluid dynamics, a vortex is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids and may be observed in smoke rings, whirlpools in the wake of a boat, and the winds surrounding a tropical cyclone, tornado, or dust devil. Vortices are a major component of a turbulent flow. The distribution of velocity, vorticity (the curl of flow velocity), as well as circulation is used to characterize vortices. In most vortices, the fluid flow velocity is greatest next to its axis and decreases in inverse proportion to the distance from the axis. In the absence of external forces, viscous friction within the fluid tends to organize the flow into a collection of irrotational vortices, possibly superimposed to larger-scale flows, including larger-scale vortices. Once formed, vortices can move, stretch, twist, and interact in complex ways. A moving vortex carries some angular and linear momentum, energy, and mass with it.

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  No longer available |Learn more

  AP Physics C Premium, 2024: 4 Practice Tests + Comprehensive Review + Online Practice

  • Robert A. Pelcovits, Joshua Farkas(Authors)
  • 2023(Publication Date)
  • Barrons Educational Services

    (Publisher)

  . This is indeed the case:

  Relationship between net torque and angular momentum

  This equation was derived for one rotating object with one angular acceleration. However, we can generalize to a system of objects simply by summing both sides of the equation:

  Conservation of Angular Momentum

  Recall how the equation Fnet,system = dpsystem /dt implies that linear momentum is conserved if Fnet = 0. (The latter condition is satisfied whenever the net external force is zero; internal forces always cancel each other out and do not contribute to Fnet .) The analogous statement, τnet,system of particles = dLsystem /dt, requires angular momentum to be conserved when τnet = 0 (which is satisfied whenever the net external torque is zero and the internal forces act along the line joining one particle with another; in this case the internal torques cancel and do not contribute to τnet ).

  If the net external torque acting on an object or system of objects is zero, the total angular momentum is constant.

  Example 7.6

  A person is sitting on a rotating lab stool holding a top that is free to rotate, as shown in Figure 7.6 . The top is initially pointing vertically and rotating counterclockwise (as viewed from above) so that its angular momentum vector of magnitude L points in the +z-direction. The person turns the top (which continues to rotate about its axis with a constant speed) to a final angle θ as shown. If the person is initially at rest, what is the person’s final angular velocity with respect to the laboratory? (The person-top system has a rotational inertia of I about the z-axis.)

  Figure 7.6

  Solution

  We start by defining the system as consisting of the person and the top. Neglecting any friction in the axle of the stool, there are no external torques exerted in the z-direction. Therefore, Lz is conserved. Conservation of Lz

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