Curvilinear Motion

4 Key excerpts on "Curvilinear Motion"

• 

  eBook - PDF

  Engineering Mechanics

  Problems and Solutions

  • Arshad Noor Siddiquee, Zahid A. Khan, Pankul Goel(Authors)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  11.1 Introduction When the motion of a body takes place along a curved path, it is called Curvilinear Motion. The Curvilinear Motion is known as a two-dimensional motion as it takes place in the X–Y plane. The basic difference between rectilinear and Curvilinear Motion is that in a rectilinear motion, all particles of a body move parallel to each other along a straight line but in a Curvilinear Motion all particles move parallel to each other along a curved path in a common plane. Some common examples of Curvilinear Motion are as follows: a. A vehicle passing over a speed breaker. b. Turning of a vehicle along a curved path. c. Motion of a ball during lofted six in cricket. d. Motion of missile fired from a fighter plane. e. Oscillatory motion of a pendulum. 11.2 Rectangular Coordinates This system is used to analyse the Curvilinear Motion of a particle moving in X–Y plane. Here the velocity and acceleration are resolved in two perpendicular components along X and Y axis. The resultant value of velocity and acceleration can be determined by using vector approach or combining its components as discussed below: Velocity: Consider a ball during lofted six, it consists two types of motions, one rotary motion and another Curvilinear Motion. Here the rotary motion is neglected and assumed that all particles of ball will travel with same displacement, velocity and acceleration. Consider a displacement-time graph where a particle is moving along a curved path in X–Y plane. Let at any instant t , particle has position A. If particle moves from point A to B by displacement Δ s during time interval Δ t as shown in Fig. 11.1. Chapter 11 Kinematics: Curvilinear Motion of Particles 530 Engineering Mechanics where Δ s = Δ s x + Δ s y We know that the instantaneous velocity is given by v s t v s s t v s t inst velocity t t x y t x .

  Sign up to read

  Learn more about book
• 

  No longer available |Learn more

  Engineering Mechanics

  • Ping YI, Jun LIU, Feng JIANG(Authors)
  • 2022(Publication Date)
  • EDP Sciences

    (Publisher)

  DOI: 10.1051/978-2-7598-2901-9.c006 © Science Press, EDP Sciences, 2022 6.1 General Curvilinear Motion When a particle moves along a straight line, it undergoes rectilinear motion; whereas a particle moving along a curved path undergoes Curvilinear Motion. Rectilinear motion has been considered extensively in physics and it will be treated as a special case of Curvilinear Motion. The kinematics of a particle includes specifying the particle’s position, dis- placement, velocity, and acceleration. Position. A particle is moving along a space curve, figure 6.3. The position of the particle at any instant can be defined by position vector r measured from fixed point O. Apparently, both the magnitude and direction of the vector r will change as the particle moves along its curved path. Then the position vector is a single-valued function of time as follows: r ¼ rðt Þ ð6:1Þ FIG. 6.2 – Plane’s flight attitude. FIG. 6.1 – Plane’s trajectory. 194 Engineering Mechanics Displacement. The displacement of a particle represents the change in its position. Suppose that at a given instant t, the particle is at point P on the curve and position vector is r, as shown in figure 6.3. After a small time interval Δt, the particle moves to new position P 0 and the position vector becomes r 0 . The displacement is then determined by Dr ¼ r 0  r . Velocity. The velocity of a particle is defined as the time rate of change in its position. During the time interval Δt, the average velocity of the particle is v avg ¼ Dr Dt Then the instantaneous velocity is determined by letting Dt ! 0, i.e., v ¼ lim Dt !0 v avg ¼ lim Dt !0 Dr Dt ¼ dr dt ð6:2Þ Since the direction of Dr gradually approaches the tangent to the curved path as Dt ! 0, the direction of dr, also the direction of v, is tangent to the path, as shown in figure 6.3. The magnitude of v, also called the speed, is usually expressed in units of m/s or km/h (kilometer per hour).

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Dynamics of Mechanical Systems

  • Carl T. F. Ross(Author)
  • 1997(Publication Date)
  • Woodhead Publishing

    (Publisher)

  CHAPTER TWO Kinematics of Particles 2.1 INTRODUCTION This chapter is divided into two sections, one section is on rectilinear motion and the other section is on plane Curvilinear Motion. Now rectilinear motion can be described as one dimensional motion or motion in a straight line. The section on rectilinear motion is sub-divided into two sub-sections, namely motion when the acceleration is constant and motion when the acceleration varies with time or distance or velocity. The section on plane Curvilinear Motion is divided into three sub-sections. One of these sub-sections is on plane motion with reference to the rectangular axes, namely x and y, while another sub-section is on plane motion with reference to the normal and tangential axes of the path of the motion. These axes are often called the not axes, where 'n' is perpendicular to the path of the motion and 't' is tangential to the path of the motion. Thus, if the path of the motion is curvilinear, the directions of 'n' and 't' will vary. The third sub-section on plane motion is on motion in terms of the polar co-ordinates 'r' and '8', where 'r' is radially outwards and '8' is tangential to 'r', in an angular direction. The x-y rectangular co-ordinate system is popular when the motion or its direction is known in terms of the x-y co-ordinates. The not co-ordinate system is popular when the direction of the path that the motion follows is known. This may be in the case of a car or a train travelling around a curve, whose equation is known. Polar co-ordinates are popular for analysing mechanisms and other artefacts, which are rotating about a fixed point. Also considered in this chapter is relative motion of translating axes and the motion of connected particles. 2.2 RECTILINEAR MOTION Rectilinear motion of a particle can be described as motion in a straight line, as shown by Figure 2.1, where the particle 'P' moves to the position 'ph in time'!1t'.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introduction to Fluid Dynamics

  Understanding Fundamental Physics

  • Young J. Moon(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  105 4 Curved Motions 4.1 Centrifugal Force Curved motion is a generic term but its implicated meaning goes beyond definition. The most fundamental physics of curved motion is the centrifugal effect, which is coupled with viscosity to make physics more complex and diverse. Curved motion is often associated with the flows of rotat- ing machineries (e.g. fans, compressors, and turbines) and vortex flows in nature (e.g. tornados, low-pressure cyclones, and fire whirl). It is also related to the complex dynamics of wall-bounded shear flows on curved walls (e.g. Coanda effect, origin of lift). 4.1.1 Radial Force Balance When a fluid undergoes curved motion, it is brought into a state of radial compression by two forces in action and reaction: the centrifugal force of the fluid (i.e. inertia force) versus the force exerted in the centripetal direction by reaction (Figure 4.1). The latter can be the reaction force from curved boundaries, or the fields under two different gravitational body forces (e.g. sink and surroundings). Therefore, the local pressure is determined by these two forces in action and reaction and also increases outward due to the mass accumulated in the radial direction. For a fluid in curved motion, the equation of motion can be set in the radial direction as follows: 𝜌 (dn dA) ⋅ a n = −dp n ⋅ dA (4.1) where dm = 𝜌 (dn dA), a n is the centripetal acceleration, dp n = (𝜕p∕𝜕n)dn is the pressure differ- ence in the radial direction, and n denotes the local coordinates normal (outward) to the streamline. The centripetal acceleration a n equals the product of the tangential flow speed v (= ds∕dt) and the rate of turn of the flow direction ̇ 𝜃 (= d𝜃∕dt) a n = −v ⋅ ̇ 𝜃 = −v ⋅ ds∕r c dt = − v 2 r c (4.2) where the negative sign indicates that the direction of action is in the centripetal direction, and r c is the local radius of curvature of the streamline.

  Sign up to read

  Learn more about book