Physics
Circular Motion and Free-Body Diagrams
Circular motion involves an object moving in a circular path at a constant speed. Free-body diagrams are used to analyze the forces acting on an object in circular motion, including centripetal force, which is directed towards the center of the circle. These diagrams help to understand the forces involved and how they contribute to the object's motion.
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5 Key excerpts on "Circular Motion and Free-Body Diagrams"
eBook - PDF- Philip Dyke, Roger Whitworth(Authors)
- 2017(Publication Date)
- Red Globe Press(Publisher)
CHAPTER 6 Circular Motion 6.1 Introduction We may not realise it yet, but the effects of circular motion are a common daily experience. The motion of record players, the act of cornering in a car or on a bicycle, and the behaviour of washing in a spin dryer are common examples, and there are plenty more. Circular motion is an example of two-dimensional motion. Previously, it has helped to analyse motion in two perpendicular directions, and a Cartesian coordinate system has proved ideal for this purpose. In Cartesians, the position of a body is represented parametrically, in terms of the time t , in the form x x t , y y t ± see for example the case of projectile motion in Chapter 5. For the analysis of motion in a circle, it is more convenient to use polar coordinates ( r , ). The advantages of using such a coordinate system will become obvious in the next section. 6.2 Polar coordinates The particle P shown in Figure 6.1 is describing a circle with centre O in a plane. The radius of the circle is a metres. At a particular instant, the position of P is given in polar coordinates by ( r a , 0 radians) and at some subsequent time, its position has moved to ( r a , = 3 radians). The dis-tance between the two positions is the most direct distance, which using trigonometry, is 2 a sin = 6 . However, a more significant measure of the motion is the distance that P has travelled (that is, along the arc of the circle between the two points). This is given by the expression a = 3 metres. More generally, if the angular coordinates of two points on a circle, of radius r metres, differ by an angle of radians, then the distance along the circular arc between them is r metres. In Figure 6.2, a particle has travelled from a point A to point B , on a circle of radius r . The angle subtended at the centre of the circle by the arc AB is radians. The angle radians is defined as the angular displacement experienced by the particle moving from A to B .
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
5.1 | Uniform Circular Motion There are many examples of motion on a circular path. Of the many possibilities, we single out those that satisfy the following definition: Definition of Uniform Circular Motion Uniform circular motion is the motion of an object traveling at a constant (uniform) speed on a circular path. As an example of uniform circular motion, Figure 5.1 shows a model airplane on a guideline. The speed of the plane is the magnitude of the velocity vector v B , and since the speed is constant, the vectors in the drawing have the same magnitude at all points on the circle. Sometimes it is more convenient to describe uniform circular motion by specifying the period of the motion, rather than the speed. The period T is the time required to travel once around the circle—that is, to make one complete revolution. There is a relationship between period and speed, since speed v is the distance traveled (here, the circumference of the circle 5 2pr) divided by the time T: v 5 2pr T (5.1) If the radius is known, as in Example 1, the speed can be calculated from the period, or vice versa. While performing a circular loop-the-loop stunt, the pilot of this small airplane experiences a net force and acceleration that points toward the center of the circle. We will now see how and why this net force and acceleration arise. 5 | Dynamics of Uniform Circular Motion 121 EXAMPLE 1 | A Tire-Balancing Machine The wheel of a car has a radius of r 5 0.29 m and is being rotated at 830 revolutions per minute (rpm) on a tire-balancing machine. Determine the speed (in m/s) at which the outer edge of the wheel is moving. Reasoning The speed v can be obtained directly from v 5 2pr/T, but first the period T is needed. The period is the time for one revolution, and it must be expressed in seconds, because the problem asks for the speed in meters per second.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
CONCEPTUAL EX AMPLE 2 Which Way Will the Object Go? In Figure 5.4 an object, such as a model airplane on a guideline, is in uniform circular motion. The object is symbolized by a dot (•), and at point O it is released suddenly from its circular path. For instance, suppose that the guideline for a model plane is cut suddenly. Does the object move (a) along the straight tangent line between points O and A or (b) along the circular arc between points O and P? Reasoning Newton’s first law of motion (see Section 4.2) guides our reasoning. This law states that an object continues in a state of rest or in a state of motion at a constant velocity (i.e., at a constant speed along a straight line) unless compelled to change that state by a net force. When an object is suddenly released from its circular path, there is no longer a net force being applied to the object. In the case of the model airplane, the guideline cannot apply a force, since it is cut. Gravity certainly acts on the plane, but the wings provide a lift force that balances the weight of the plane. Answer (b) is incorrect. An object such as a model airplane will remain on a circular path only if a net force keeps it there. Since there is no net force, it cannot travel on the circular arc. Answer (a) is correct. In the absence of a net force, the plane or any object would continue to move at a constant speed along a straight line in the direction it had at the time of release, consis- tent with Newton’s first law. This speed and direction are given in Figure 5.4 by the velocity vector → v. C O P A v θ FIGURE 5.4 If an object (•) moving on a circular path were released from its path at point O, it would move along the straight tangent line OA in the absence of a net force. As Example 2 discusses, the object in Figure 5.4 would travel on a tangent line if it were released from its circular path suddenly at point O.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
5.1 | Uniform Circular Motion There are many examples of motion on a circular path. Of the many possibilities, we single out those that satisfy the following definition: Definition of Uniform Circular Motion Uniform circular motion is the motion of an object traveling at a constant (uniform) speed on a circular path. As an example of uniform circular motion, Figure 5.1 shows a model airplane on a guideline. The speed of the plane is the magnitude of the velocity vector v B , and since the speed is constant, the vectors in the drawing have the same magnitude at all points on the circle. Sometimes it is more convenient to describe uniform circular motion by specifying the period of the motion, rather than the speed. The period T is the time required to travel once around the circle—that is, to make one complete revolution. There is a relationship between period and speed, since speed v is the distance traveled (here, the circumference of the circle 5 2pr) divided by the time T: v 5 2pr T (5.1) If the radius is known, as in Example 1, the speed can be calculated from the period, or vice versa. 5 | Dynamics of Uniform Circular Motion 108 EXAMPLE 1 | A Tire-Balancing Machine The wheel of a car has a radius of r 5 0.29 m and is being rotated at 830 revolutions per minute (rpm) on a tire-balancing machine. Determine the speed (in m/s) at which the outer edge of the wheel is moving. Reasoning The speed v can be obtained directly from v 5 2pr/T, but first the period T is needed. The period is the time for one revolution, and it must be expressed in seconds, because the problem asks for the speed in meters per second. Figure 5.1 The motion of a model airplane flying at a constant speed on a horizontal circular path is an example of uniform circular motion. © Gari Wyn Williams/Age Fotostock Chapter | 5 LEARNING OBJECTIVES After reading this module, you should be able to... 5.1 | Define uniform circular motion. 5.2 | Solve uniform circular motion kinematic problems.
- Michael Clifford, Kathy Simmons, Philip Shipway(Authors)
- 2009(Publication Date)
- CRC Press(Publisher)
In this case it is clear that the tension in the string acts towards O the centre of rotation. In other cases the direction of the forces Figure 6.27 Free body diagram of a particle in circular motion-tension force (solid line); radial acceleration (dashed line) O R a R = R ω 2 . θ = ω P T An Introduction to Mechanical Engineering: Part 1 420 may not be so clear. In these cases it is worth noting that if the actual force is in the opposite direction to that indicated on the free body diagram, the calculation will yield a negative value. In contrast, the direction of the acceleration here is specified from kinematic considerations, and must be positive towards the centre of rotation (see ‘Circular motion of a particle’). Applying Newton’s second law to the particle along the string gives T m ( R 2 ) This equation defines the tension in the string, as required. This problem can also be solved using d’Alemberts Principle, in which the free body diagram is redrawn as shown in Figure 6.28. In this figure an imaginary inertia force of magnitude ma R has been included, where a R R 2 . This force is in the opposite direction to the actual acceleration of the particle. The tension force is obtained by considering the particle to be in static equilibrium, i.e. T mR 2 0 This equation defines the tension force T mR 2 , as required. Figure 6.28 Free body diagram of a particle in circular motion using d’Alemberts Principle Point mass in a slot A particle A of mass m fits loosely in a smooth slot with vertical sidewalls cut into a disc mounted in the horizontal plane, as shown in Figure 6.29.The disc is rotating at constant angular speed about its fixed centre O as shown.The particle is held in position by an inextensible cord having one end secured at B . Determine an expression for the tension in the cord. As the disc rotates, the particle remains at A and as a consequence rotates at a constant radius R about O .
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