Velocity and Acceleration Analysis MCQ Quiz - Objective Question with Answer for Velocity and Acceleration Analysis - Download Free PDF

Explanation:

Angular Velocity and Linear Velocity Relationship

• In the context of rotational motion, angular velocity (ω) is a measure of how quickly an object rotates or revolves relative to a point or axis.
• It is typically measured in radians per second (rad/s).
• Linear velocity (v), on the other hand, is the rate at which an object moves along a path and is measured in meters per second (m/s).
• The radius (r) is the distance from the center of the rotating object to the point where the linear velocity is measured.

The angular velocity (ω) of a rotating object is directly related to its linear velocity (v) at a point on the object's circumference and the radius (r) of the rotation. The relationship can be described by the formula:

\(\omega~=~\frac{ v}{r}\)

Explanation:

At the instant when AP is horizontal and OP is vertical, V_P will come in the line of AP, then the perpendicular component of V_P will be zero.

⇒ V_P = (OP) × ω_OP = (OP) × 7 m/s; (direction is along AP)

Hence, ω_ABC = 0;

No perpendicular component of velocity

Explanation:

Kennedy’s theorem states that if three bodies have plane motion relative to one another, then there I-centres i.e. instantaneous centre must lie on a straight line.

Explanation:

Double Slider Crank Chain:

It consists of four pairs out of which two are turning pairs and two others are sliding pairs.

• Link 1 and link 4 is sliding pair
• link 1 and link 2 is turning pair
• link 2 and link 3 is the second turning pair
• link 3 and link 4 is the second sliding pair

• When link 4 is fixed, the first inversion is obtained. This inversion is used in the Elliptical trammel which is an instrument used for drawing ellipses. 
• Each and every point on the link connecting the sliders traces an ellipse.
• But, the midpoint of connecting rod of sliders will trace the circle.

Explanation:

Analyzing Acceleration in a Rotating Link:

• When analyzing the acceleration of a point in a rotating link, it is essential to consider the different components of acceleration that contribute to the overall motion.
• The relative acceleration equation is a fundamental tool used in this analysis, breaking down the acceleration into various components that act on the point in question.

Components of Acceleration: The relative acceleration equation typically considers the following components:

• Tangential Acceleration: This component is due to the change in the magnitude of the velocity of the point. It acts tangentially to the path of the point and is given by the product of the angular acceleration and the radius of rotation.
• Centripetal (or Radial) Acceleration: This component is due to the change in direction of the velocity of the point. It acts radially inward towards the center of rotation and is given by the square of the angular velocity times the radius of rotation.
• Coriolis Acceleration: This component arises when a point moves in a rotating reference frame. It acts perpendicular to the velocity of the point and the axis of rotation. It is given by twice the product of the angular velocity and the relative velocity of the point in the rotating frame.

Gravitational Acceleration

• When analyzing the acceleration of a point in a rotating link using the relative acceleration equation, gravitational acceleration is NOT directly considered. The relative acceleration equation focuses on the components of acceleration that arise due to the rotational motion and the relative motion of the point within the rotating link. Gravitational acceleration, while acting on the point, is a constant acceleration that affects all points equally and is not specific to the rotational motion being analyzed.

Concept:

Velocity at point of contact = Rω

(R = Radius of the point from Instantaneous centre I)

∴ The instantaneous centre is at the intersection of object and floor, hence radius R = 0

Velocity at the point is zero.

Concept:

The instantaneous centre method of analysing the motion in a mechanism is based upon the concept that any displacement of a body having motion in one plane, can be considered as a pure rotational motion of a rigid link as a whole about some centre, and known as an instantaneous centre.

The number of instantaneous centres in a considered kinematic chain is equal to the number of combinations of two links.

If N is the number of instantaneous centres and n is the number of links, then the relation is given by

Explanation:

Klein's Construction:

• It is used to draw the velocity and acceleration diagrams for a single slider crank mechanism.
• The velocity and acceleration of piston of a reciprocating engine mechanism can be determined by the figure given below.

Velocity diagram:

• Draw the configuration diagram OAB for the slider-crank mechanism.
• Draw OI perpendicular to OB and produce BA to meet OI at C. Then the triangle formed, is known as Klein's velocity diagram.
  • Velocity of connecting rod AB, V_BA = ω × AC.
  • Velocity of piston B,V_P = V_BO = ω × OC.
  • Velocity of crankpin A, V_AO = ω × OA.

Acceleration diagram:

• With A as centre and AC as radius, draw a circle.
• Locate D as the midpoint of AB.
• With D as centre, and DA as radius, draw a circle to intersect the previously drawn circle at points H and E.
• Join HE intersecting AB at F.
• Produce HE to meet OB at G.
• Then OAFG is Klein's acceleration diagram.
  • Acceleration of piston (Slider B), f_BO = ω² × OG.
  • Tangential acceleration of connecting rod, f^t_BA = ω² × FG.
  • Normal acceleration of connecting rod, fⁿ_BA = ω² × AF.
  • Total acceleration of connecting rod, f_BA = ω2 × OG.
• Angular acceleration of connecting rod, AB \(\alpha_{AB}=f^{t}_{BA}/AB=ω^{2}\frac{FG}{AB}\;\;\;\;(ccw)\)
• To find the acceleration of any point x in AB, draw a line X-X parallel to OB to intersect at X. Join OX.
• Then, Acceleration of point x, f_x = ω² × OX.

Though, we can calculate both velocity and acceleration through Klein's construction but mainly it is used in calculating the linear acceleration of the piston.

The instantaneous centre of a rigid body is located by the point where the lines perpendicular to the direction of the velocity of two different points on the body meets. If two points have velocity parallel to each other then the line drawn perpendicular to those points will meet at infinity, therefore their instantaneous centre will lie at infinity.

Location of Instantaneous Centres for common configurations:

a) When the two links are connected by a pin joint (or pivot joint), the instantaneous centre lies on the centre of the pin.

b) When the two links have a pure rolling contact the instantaneous centre lies on their point of contact.

c) When the two links have a sliding contact, the instantaneous centre lies on the common normal at the point of contact.

• When the link 2 (slider) moves on fixed link 1 having straight surface the instantaneous centre lies at infinity
• When the link 2 (slider) moves on fixed link 1 having curved surface, the instantaneous centre lies on the centre of curvature of the curvilinear path

Explanation:

Because PQ is a rigid link so the distance between P and Q can not changes. Therefore, the velocity of Q relative to P is VQP, which has an only component perpendicular to PQ.

∴ V_Pcosβ = V_Qcosθ

∴ V_QP = V_Qsinθ – V_Psinβ

Hence only one component is present and it is perpendicular to PQ.

Concept:

Relative velocity method

The relative velocity method is based upon the relative velocity of the various points of the link. The velocity of any point on a link with respect to another point on the same link is always perpendicular to the line joining these points on the configuration (or space) diagram.

ω = Angular velocity of the link AB about A

VBA = ω.(AB)

Calculation:

Given:

V_B = 3u, V_A = u, AB = r

VBA =VB - VA  = ω.(AB)

3u - u = ω × r

ω = 2u/r

Concept:

VQ = ω(PQ)

Calculation:

Given:

N = 100 rpm, PQ = 30 cm = 0.3 m

Now,

\(\omega = \frac{{2\pi N}}{{60}} = \frac{{2\pi \times 100}}{{60}} = 3.34\;\pi \;rad/s\)

V_Q = ω(PQ)

VQ = 3.34 π × 0.3

∴ VQ = 3.14 m/s

Explanation:

Coriolis component of acceleration: 

• When a point on one link is sliding along another rotating link such as in quick return motion mechanism, then the Coriolis component of acceleration comes into account.
• It is the tangential component of the acceleration of the slider with respect to the coincident point on the link.
• Mathematically, Coriolis component of acceleration, ac = 2vω
• The direction of the Coriolis component of acceleration is given by rotating the velocity of the slider by 90° in the direction of the angular velocity of the rotating link.

Explanation:

• In general, the motion of a link in a mechanism is neither a pure translation nor pure rotation. It is a combination of translation and rotation, which we normally say the link is in general motion. But any link at any instant can be assumed to be in pure rotation w.r.t the point in the space known as Instantaneous Centre of Rotation.
• As the link changes its position, In general, its I-centre keeps on changing, therefore the Locus of Instantaneous centre for a particular link during its whole motion is known as Centrode.of the link.
• The locus of Instantaneous Axis of rotation of a particular link during its whole motion is known as Axode of the link.

Concept:

Coriolis component of acceleration – When a point on one link is sliding along another rotating link such as in quick return motion mechanism, then the Coriolis component of acceleration comes into account. It is the tangential component of the acceleration of the slider with respect to the coincident point on the link.

Mathematically,

Coriolis component of acceleration, ac = 2vω

The direction of the Coriolis component of acceleration is given by rotating the velocity of the slider by 90° in the direction of the angular velocity of the rotating link.