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4. Kinetics of Particles

4.1 Introduction

4.2 Rotation

4.3 Absolute motion

4.4 Relative velocity

4.5 Instantaneous center of zero velocity

4.6 Relative acceleration

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4.1 Introduction

• Rigid body kinematics involves both linear and angular displacements, velocities, and accelerations.

Rigid Body Assumption

• A rigid body is a system of particles for which the distances between the particles remain unchanged.

Plane Motion

• A rigid body executes plane motion when all parts of the body move in parallel planes.

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4.1 Introduction Plane Motion (contd.)

• The plane motion of a rigid body may be divided into several categories, as represented in the Figure.

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4.1 Introduction Plane Motion (contd.)

• Translation is defined as any motion in which every line in the body remains parallel to its original position at all times. In translation there is no rotation of any line in the body. In rectilinear translation, all points in the body move in parallel straight lines. In curvilinear translation, part b, all points move on congruent curves.

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4.1 Introduction Plane Motion (contd.)

• Rotation about a fixed axis, is the angular motion about the axis. It follows that all particles in a rigid body move in circular paths about the axis of rotation, and all lines in the body which are perpendicular to the axis of rotation (including those which do not pass through the axis) rotate through the same angle in the same time.

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4.1 Introduction Plane Motion (contd.)

• General plane motion of a rigid body is a combination of translation and rotation.

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4.2 Rotation

• The angular positions of any two lines 1 and 2 attached to the body are specified by 𝜃1 and 𝜃2 measured from any convenient fixed reference direction. Because the angle 𝛽 is invariant, the relation 𝜃2 = 𝜃1 + 𝛽 upon differentiation with

respect to time gives 𝜃 2 = 𝜃 1 and 𝜃 2 = 𝜃 1 or, during a finite interval, ∆𝜃2 = ∆𝜃1. Thus, all lines on a rigid body in its plane of motion have the same angular displacement, the same angular velocity, and the same angular acceleration.

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4.2 Rotation Angular-Motion Relations

𝜔 =𝑑𝜃

𝑑𝑡= 𝜃

𝛼 =𝑑𝜔

𝑑𝑡= 𝜔 𝑜𝑟 𝛼 =

𝑑2𝜃

𝑑𝑡2= 𝜃

𝜔 𝑑𝜔 = 𝑎 𝑑𝜃 𝑜𝑟 𝜃 𝑑𝜃 = 𝜃 𝑑𝜃

• For rotation with constant angular acceleration, the integrals of Equations become

𝜔 = 𝜔0 + 𝛼𝑡

𝜔2 = 𝜔0

2 + 2𝛼 𝜃 − 𝜃0

𝜃 = 𝜃0+ 𝜔0𝑡 +1

2𝛼𝑡2

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4.2 Rotation Rotation about a Fixed Axis

𝑣 = 𝑟𝜔

𝑎𝑛 = 𝑟𝜔2 = 𝑣2/𝑟 = 𝑣𝜔

𝑎𝑡 = 𝑟𝛼

𝐯 = 𝐫 = 𝛚 × 𝐫

𝐚 = 𝐯 = 𝛚 × 𝐫 + 𝛚 × 𝐫

= 𝛚× 𝛚× 𝐫 +𝛚 × 𝐫

= 𝛚×𝐯 + 𝛂× 𝐫

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4.2 Rotation Rotation about a Fixed Axis (contd.)

𝐯 = 𝛚 × 𝐫 𝐚𝐧 = 𝛚× 𝛚× 𝐫

𝐚𝐭 = 𝛂 × 𝐫

• For three-dimensional motion of a rigid body, the angular-velocity vector 𝛚 may change direction as well as magnitude, and in this case, the angular acceleration, which is the time derivative of angular velocity, 𝛂 = 𝛚 , will no longer be in the same direction as 𝛚.

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4.2 Rotation Sample Problem (1)

The pinion A of the hoist motor drives gear B, which is attached to the hoisting drum. The load L is lifted from its rest position and acquires an upward velocity of 2 m/s in a vertical rise of 0.8 m with constant acceleration. As the load passes this position, compute

(a) the acceleration of point C on the cable in contact with the drum and

(b) the angular velocity and angular acceleration of the pinion A.

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Solution

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4.2 Rotation Sample Problem (2)

The right-angle bar rotates clockwise with an angular velocity which is decreasing at the rate of 4 rad/s2. Write the vector expressions for the velocity and acceleration of point A when 𝜔 = 2 rad/s.

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Solution

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4.3 Absolute Motion

• We now develop the approach of absolute-motion analysis to describe the plane kinematics of rigid bodies. In this approach, we make use of the geometric relations which define the configuration of the body involved and then proceed to take the time derivatives of the defining geometric relations to obtain velocities and accelerations.

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4.3 Absolute Motion Sample Problem (3)

A wheel of radius r rolls on a flat surface without slipping. Determine the angular motion of the wheel in terms of the linear motion of its center O. Also determine the acceleration of a point on the rim of the wheel as the point comes into contact with the surface on which the wheel rolls.

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Solution

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Solution (contd)

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4.3 Absolute Motion Sample Problem (4)

The linear actuator is designed for rapid horizontal velocity 𝑣 of jaw C for a slow change in the distance between A and B. If the hydraulic cylinder decreases this distance at the rate 𝑢 , determine the horizontal velocity of jaw C In terms of the angle 𝜃.

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Solution

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4.3 Absolute Motion Sample Problem (5)

The circular cam is mounted eccentrically about its fixed bearing at O and turns counterclockwise at the constant angular velocity 𝜔. The cam causes the fork A and attached control rod to oscillate in the horizontal x-direction. Write expressions for the velocity 𝑣𝑥 and acceleration 𝑎𝑥 of the control rod in terms of the angle measured from the vertical. The contact surfaces of the fork are vertical.

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Solution

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4.4 Relative Velocity Relative Velocity due to Rotation

• With 𝐵 as the reference point, we see from Figure that the total displacement of 𝐴 is

∆𝐫𝐴 = ∆𝐫𝐵 + ∆𝐫𝐴/𝐵

where ∆𝐫𝐴/𝐵 has the magnitude 𝑟∆𝜃 as ∆𝜃 approaches zero. We note that the relative linear motion ∆𝐫𝐴/𝐵 is accompanied by the absolute angular motion , as seen from the translating axes 𝑥′ − 𝑦′. Dividing the expression for ∆𝐫𝐴 by the corresponding time interval ∆𝑡 and passing to the limit, we obtain the relative-velocity equation

𝐯𝐴 = 𝐯𝐵 + 𝐯𝐴/𝐵

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4.4 Relative Velocity Relative Velocity due to Rotation (contd.)

𝐯𝐴 = 𝐯𝐵 + 𝐯𝐴/𝐵

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4.4 Relative Velocity Relative Velocity due to Rotation (contd.)

• The magnitude of the relative velocity is thus seen to be

𝑣𝐴/𝐵 = lim∆𝑡→0

∆𝐫𝐴/𝐵 /∆𝑡 = lim∆𝑡→0

𝑟∆𝜃/∆𝑡 which, with 𝜔 = 𝜃

becomes

𝑣𝐴/𝐵 = 𝑟𝜔

Using 𝐫 to represent the vector 𝐫𝐴/𝐵, we may write the relative

velocity as the vector

𝐯𝐴/𝐵 = 𝛚× 𝐫

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4.4 Relative Velocity Interpretation of the Relative-Velocity Equation

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4.4 Relative Velocity Sample Problem (6)

The wheel of radius r = 300mm rolls to the right without slipping and has a velocity v0 = 3m/s of its center 𝑂. Calculate the velocity of point 𝐴 on the wheel for the instant represented.

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Solution

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4.4 Relative Velocity Sample Problem (7)

Crank CB oscillates about C through a limited arc, causing crank OA to oscillate about O. When the linkage passes the position shown with CB horizontal and OA vertical, the angular velocity of CB is 2 rad/s counterclockwise. For this instant, determine the angular velocities of OA and AB.

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Solution

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4.4 Relative Velocity Sample Problem (8)

The elements of a switching device are shown. If the vertical control rod has a downward velocity 𝑣 = 0.9 m/s of when 𝜃 = 60° and if roller 𝐴 is in continuous contact with the horizontal surface, determine the magnitude of the velocity of 𝐶 for this instant.

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Solution

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4.5 Instantaneous Center of Zero Velocity

• As far as velocities are concerned, a body may be considered to be in pure rotation about an axis, normal to the plane of motion, passing through this point. This axis is called the instantaneous axis of zero velocity, and the intersection of this axis with the plane of motion is known as the instantaneous center of zero velocity.

𝜔 =𝑣𝐴𝑟𝐴

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4.5 Instantaneous Center of Zero Velocity Motion of the Instantaneous Center

• As the body changes its position, the instantaneous center C also changes its position both in space and on the body.

• Although the instantaneous center of zero velocity is momentarily at rest, its acceleration generally is not zero. Thus, this point may not be used as an instantaneous center of zero acceleration in a manner analogous to its use for finding velocity.

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4.5 Instantaneous Center of Zero Velocity Sample Problem (9)

The wheel rolls to the right without slipping, with its center O having a velocity 𝑣𝑂 = 3 m/s. Locate the instantaneous center of zero velocity and use it to find the velocity of point A for the position indicated.

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Solution

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4.5 Instantaneous Center of Zero Velocity Sample Problem (10)

Arm OB of the linkage has a clockwise angular velocity of 10 rad/s in the position shown where = 45. Determine the velocity of A, the velocity of D, and the angular velocity of link AB for the position shown.

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Solution

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4.6 Relative Acceleration

𝒂𝐴 = 𝒂𝐵 + 𝒂𝐴/𝐵

• The above equation states that the acceleration of point A equals the vector sum of the acceleration of point B and the acceleration which A appears to have to a nonrotating observer moving with B.

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4.6 Relative Acceleration Relative Acceleration Due to Rotation

Alternatively we may write

𝐚𝐴 = 𝐚𝐵 + 𝐚𝐴/𝐵 𝑛+ 𝐚𝐴/𝐵 𝑡

where the magnitudes of the relative-acceleration components are

𝑎𝐴/𝐵 𝑛= 𝑣𝐴/𝐵

2/𝑟 = 𝑟𝜔2

𝑎𝐴/𝐵 𝑡= 𝑣 𝐴/𝐵 = 𝑟𝛼

In vector notation the

acceleration components are

𝐚𝐴/𝐵 𝑛= 𝛚× (𝛚 × 𝐫)

𝐚𝐴/𝐵 𝑡= 𝛂× 𝐫

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4.6 Relative Acceleration Solution of the Relative-Acceleration Equation

• Before attempting a solution, identify the knowns and unknowns, keeping in mind that a solution to a vector equation in two dimensions can be carried out when the unknowns have been reduced to two scalar quantities. These quantities may be the magnitude or direction of any of the terms of the equation.

• Because the normal acceleration components depend on velocities, it is generally necessary to solve for the velocities before the acceleration calculations can be made.

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4.6 Relative Acceleration Sample Problem (12)

The wheel of radius r rolls to the left without slipping and, at the instant considered, the center O has a velocity 𝐯𝑂 and an acceleration 𝐚𝑂 to the left. Determine the acceleration of points A and C on the wheel for the instant considered.

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Solution

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Solution (Contd.)

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4.6 Relative Acceleration Sample Problem (13)

Crank CB has a constant counterclockwise angular velocity of 2 rad/s in the position shown during a short interval of its motion. Determine the angular acceleration of links AB and OA for this position. Solve by using vector algebra.

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Solution

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Solution (Cntd)

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4.6 Relative Acceleration Sample Problem (14)

If the velocity of point A is 3 m/s to the right and is constant for an interval including the position shown, determine the tangential acceleration of point B along its path and the angular acceleration of the bar.

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Solution

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4.6 Relative Acceleration Sample Problem (15)

The hydraulic cylinder imparts motion to point B which causes link OA to rotate. For the instant shown where OA is vertical and AB is horizontal, the velocity 𝑣𝐵 of pin B is and is 4 m/s increasing at the rate of 20 m/s2 . For this position determine the angular acceleration of OA.

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Solution

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