Centripetal Force and Velocity

6 Key excerpts on "Centripetal Force and Velocity"

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  eBook - ePub

  Mechanical Design for the Stage

  • Alan Hendrickson(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  Figure 7.4c ). Mathematically, centripetal acceleration equals

  Figure 7.4 Illustration of the terms involved in: a. Angular and linear displacement, b. Angular and linear speed and acceleration, c. Tangential and centripetal acceleration. (In both b. and c. rotational speed and acceleration are indicated in an easily understood way. The actual vectors for rotational velocity and acceleration would be shown by arrows along the axis of rotation, straight out of the page.)

  a

  c e n t r i p e t a l

  =

  v 2

  r

  = r

  ω 2

  Newton’s second law states that force is required to accelerate mass, and so a centripetal force can be defined.

  F

  c e n t r i p e t a l

  = m

  a

  c e n t r i p e t a l

  = m r

  ω 2

  If you were to tie a weight to a string and spin it around above your head, the tension in the line would equal Fcentripetal . The cohesive forces within the steel of a shaft supply Fcentripetal , keeping the shaft from flying apart as it spins (though there are limits to the speed it can withstand). And a free-standing prop set on a turntable could slide outward, or away from the axis of rotation, if the turntable spins fast enough so that Fcentripetal exceeds Fstatic friction

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  Engineering Science

  • W. Bolton(Author)
  • 2015(Publication Date)
  • Routledge

    (Publisher)

  r gives the acceleration an object must experience, at right angles to its direction of motion, if it is to move in a circular path. The centripetal force necessary for this acceleration is thus:

  According to Newton’s third law, to every action there is an opposite and equal reaction. In this case the reaction to the centripetal force is called the centrifugal force and acts in an outwards direction on the pivot C around which the motion is occurring.

  Example An object of mass 0.5 kg is whirled round in a horizontal circle of radius 0.8 m on the end of a rope. What is the tension in the rope when the object rotates at 4 rev/s?

  F = mω 2 r = 0.5 X (2π X 4)2 × 0.8 = 253 N

  Example Calculate the force acting on a bearing which is carrying a crankshaft with an out- of-balance load of 0.10 kg at a radius of 100 mm and rotating at 50 rev/s.

  Centripetal force = mω 2 r = 0.10 X (2π X 50)2 X 0.100 = 987 N. The force acting on the bearing is the reaction force, i.e. the centrifugal force, and is thus 987 N radially outwards.

  Example

  An object of mass 3 kg is attached to the end of a rope and whirled round in a vertical circle of radius 1 m. What are the maximum and minimum values of the tension in the rope when it rotates at 3 rev/s?

  The centripetal force F = mω 2 r = 3 X (2π X 3)2 X 1 = 1066 N. At the top of the path, the centripetal force is provided by the tension in the rope plus the weight since they are acting in the same direction. Hence, T + 3 X 9.8 = 1066 and so tension T = 1037 N. At the bottom of the path, the tension and the weight act in opposite directions and so T - 3 X 9.8 = 1066 and the tension T = 1095 N. Thus the maximum tension is 1095 N and the minimum tension 1037 N.

  23.3 Cornering

  Figure 23.3 Cornering on the flat

  Consider a vehicle of mass m rounding a horizontal corner of radius r (Figure 23.3 ). Since the reactive force R is mg at right angles to the plane, the maximum frictional force F = μR = μmg . Hence, when the centripetal force exceeds this frictional force, skidding will occur. Thus the maximum speed is given when mv 2 /r = μmg

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  Biomechanics of Human Motion

  Applications in the Martial Arts, Second Edition

  • Emeric Arus, Ph.D.(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  centripetal force or center-seeking force.

  The radial acceleration involves change of direction continuing perpendicular to the direction of motion. Here are the equations for radial acceleration and radial (centripetal) force. Radial acceleration a rad = v 2 /r = ω2 r and radial force F rad = m ⋅ v 2 /r = m ⋅ r ⋅ ω2 . Here, v is the magnitude of the tangential circular velocity or speed and r is the length of the radius of rotation.

  Let us take an example and calculate the radial force. A karateka strikes toward the opponent’s face with a technique back-fist strike (Uraken-uchi) as shown in Figure 9.1 . The karateka weighs 70 kg, which we do not count at this time; his total arm mass is 3.45 kg; the velocity is 11 m/s; and the total arm length (with the fist closed) is 0.69 m. F rad = m ⋅ v 2 /r = 3.45 kg × 112 / 0.69 m = 605 N.

  FIGURE 9.1. Radial and tangential forces.

  Centripetal force coexists with another force named centrifugal force. The word comes from the Latin word “centrum” (center) and “fugere” to (flee). The centrifugal force has two slightly different manifestation forms:

  • Reactive centrifugal force that occurs in reaction to a centripetal acceleration on a mass. This force is equal in magnitude to the centripetal force and is directed from the center of rotation. We can observe this force when we are sitting in a car. When the car is turning to the left, our body moves to the right. This motion to the right is the centrifugal force. In martial arts, an example of the centrifugal force manifestation is the aikido technique named “entering throw—negative” execution (Irimi-nage). Here, the executor (attacker) (Shite) guides the opponent in a rotary fashion. The defender (Uke)

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  Science and Mathematics for Engineering

  • John Bird(Author)
  • 2019(Publication Date)
  • Routledge

    (Publisher)

  Chapter 19 Force, mass and acceleration

  Why it is important to understand: Force, mass and acceleration

  When an object is pushed or pulled, a force is applied to the object. The effects of pushing or pulling an object are to cause changes in the motion and shape of the object. If a change occurs in the motion of the object then the object accelerates. Thus, acceleration results from a force being applied to an object. If a force is applied to an object and it does not move, then the object changes shape. Usually the change in shape is so small that it cannot be detected by just watching the object. However, when very sensitive measuring instruments are used, very small changes in dimensions can be detected. A force of attraction exists between all objects. If a person is taken as one object and the Earth as a second object, a force of attraction exists between the person and the Earth. This force is called the gravitational force and is the force that gives a person a certain weight when standing on the Earth’s surface. It is also this force that gives freely falling objects a constant acceleration in the absence of other forces. This chapter defines force and acceleration, states Newton’s three laws of motion and defines moment of inertia, all demonstrated via practical everyday situations.

  At the end of this chapter, you should be able to:

  • define force and state its unit
  • appreciate ‘gravitational force’
  • state Newton’s three laws of motion
  • perform calculations involving force F = ma
  • define ‘centripetal acceleration’
  • perform calculations involving centripetal force =

    mv 2

    r
  • define ‘mass moment of inertia’

  Science and Mathematics for Engineering. 978-0-367-20475-4, © John Bird. Published by Taylor & Francis. All rights reserved.

  19.1   Introduction

  When an object is pushed or pulled, a force is applied to the object. This force is measured in

  newtons*

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  Basic Engineering Mechanics Explained, Volume 1

  Principles and Static Forces

  • Gregory Pastoll, Gregory Pastoll(Authors)
  • 2019(Publication Date)
  • Gregory Pastoll

    (Publisher)

  One might think that the displacement of the CG by so small an amount was insignificant. However, while rotating, the wheel is now behaving as if it were a point mass of 15.45 kg situated 0.233 mm from the axis of rotation. At a vehicle speed of 120 km/h, the centrifugal force on this wheel would be approximately 36 N, rotating at approximately 16 times per second, sufficient to cause a noticeable wobble. Centrifugal force is explained in a subsequent chapter, in volume 3 of this series.

  Centroids of areas and volumes

  The centroid of a plane area is a point on its surface that corresponds with the CG which that surface would have, if it were a thin plate, of uniform thickness and density. (This type of thin plate is often referred to as a ‘lamina’. This word comes directly from Latin, meaning a thin plate of marble or metal.)

  The centroid of a volume is that point within the volume that corresponds with the CG of a solid object with uniform density, and the same shape as that volume. Standard expressions for the location of the centroids of various regular geometric areas and volumes

  (derived by methods similar to that demonstrated at the end of this chapter. )

  Example

  The object illustrated here is turned from solid brass, density 8400 kg/m3 , and can be thought of as a hollow hemisphere adjoining a hollow cylinder.

  Determine the location of its CG above point ‘O’.

  Since all the constituent ‘parts’ of this assembly are of the same material, their masses are proportional to their volumes. It will thus not be necessary to determine their masses, as we can work with volumes instead. The equation for determining Ȳ becomes : Ȳ = (Σȳv)/(Σv)

  Consider the assembly to be made up of four parts:

  A. Solid hemisphere, r = 90 mm,

  B. Removed solid hemisphere, r = 50 mm, hence negative volume

  C. Solid cylinder, r = 90 mm and h = 150 mm

  D. Removed solid cylinder, r = 50 mm and h = 150 mm, hence negative volume

  Determine the volumes of the respective parts, and insert these values into the standard table. Note: the heights above the bases of the two solid hemispheres are added to the height of the cylinder, to give the ȳ dimensions relative to the base of the assembly.

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  Nursing HESI A2

  a QuickStudy Laminated Reference & Study Guide

  • April Michelle Davis(Author)
  • 2018(Publication Date)
  • QuickStudy Reference Guides

    (Publisher)

  Physics

  Mechanics

  • Mass: Amount of matter in an object
  • Displacement: Measure of how far an object has moved from its starting point
  • Velocity: Distance covered by an object over a given period of time
  • Acceleration: Change in velocity over time
  • Vectors have a magnitude and a direction, such as velocity or displacement
  • Scalars only have a magnitude, such as distance or speed
  • Momentum: Quantity made by multiplying an object’s mass by its velocity
  • Impulse: Change in momentum

  Forces

  • Force: The push that starts or stops an object’s motion, such as gravity, friction, tension, and electrical force
  • Centripetal force: Creates circular motion
  • Free body diagram: Vectors in the horizontal and vertical directions can be added to find the total force

  Circular & Rotational Motion

  • Torque: Form of work that is applied in a circular motion
  • Circular motion: The motion of an object around a central point
  • Centripetal acceleration: Points toward the center of the circle

  Energy, Work & Power

  • The capacity to do work or engage in physical activity; types of energy:
    • Kinetic: Energy of motion
    • Thermal (heat): Energy of a substance in relation to its temperature
    • Potential: Stored energy
    • Chemical:

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