The 3 -lb uniform rod B D is connected to crank A B and to a...

The 3 -lb uniform rod $B D$ is connected to crank $A B$ and to a collar of negligible weight. A couple (not shown) is applied to crank $A B$ causing it to rotate. At the instant shown, crank $A B$ has an angular velocity of $12 \mathrm{rad} / \mathrm{s}$ and an angular acceleration of $80 \mathrm{rad} / \mathrm{s}^{2} ;$ both are counterclockwise. Neglecting the effect of friction, determine the reaction at $D .$

Key Concepts

Distributed Mass and Center of Mass

The concept of distributed mass refers to the way mass is evenly or unevenly distributed along a body. For uniform rods, the center of mass is located at the geometric midpoint, and this property is fundamental when determining the effects of weight in dynamic analysis, as it influences both the translational and rotational behavior of the system.

Free Body Diagram Analysis

A free body diagram is a graphical representation that isolates a body from its environment and depicts all external forces and moments acting on it. In problems involving mechanical linkages and rotating bodies, constructing a free body diagram is crucial for applying the equilibrium equations and Newton’s laws to solve for unknown forces and reaction components.

Reaction Forces in Mechanical Systems

Reaction forces are the forces exerted by supports, joints, or connections to maintain the equilibrium or motion of a structure. In systems with moving components, these forces ensure compatibility and continuity of motion between interconnected parts, and they can be determined by analyzing the net effects of inertial forces, applied loads, and moments acting on the system.

Dynamics of Rigid Bodies

The dynamics of rigid bodies encompasses the analysis of forces and torques acting on objects that do not deform under loading. This concept is essential in determining how applied forces, including couple moments and gravitational forces, impact the motion and acceleration of each part of a connected mechanical system such as a rod rotating about a pivot.

Rotational Kinematics

Rotational kinematics involves the study of angular motion without regard to its causes, focusing on parameters such as angular displacement, angular velocity, and angular acceleration. In applications where a crank rotates and imparts motion to a connected component, these kinematic quantities allow one to determine linear and tangential accelerations of points along the rotating body.

Transcript

00:01 Hello friends as shown in the figure rod vd of weight 3 pound connected to the crank av and caller d.

00:11 If moment is given to the crank av so that it having angular velocity of 12 radian per second and angular acceleration of 80 radian per second square both in counter clock by direction at that instant calculate the reaction at d.

00:34 Let discuss the kinematics of crank av this is a this is the alpha a v 80 radial per second is square a v normal acceleration at the tension i think the value of normal and tangential acceleration of v point the normal acceleration of v point will be omega square are this length is given 3 inch omega is given 12 so 12 square into 3 by 12 so it would be 36 fit per second square and tangential acceleration is alpha into r 80 into 3 by 12 so it would be 20 fit per second square now kinematics of the rod vd, velocity of v, velocity of d, instantaneous center, omega vd, this will be of 6 inch.

04:25 The velocity of v will be omega av into av.

04:36 So velocity of v you will get 3 by 12 into 12, 3 fit per second.

04:44 So angular velocity of vd, velocity of b upon bc, 3 upon 0 .5, 6 radian per second.

05:08 Now acceleration, if we discuss at b point, there will be normal acceleration, there will be a tangential acceleration.

05:27 At d point there will be a tangential acceleration.

05:38 There will be a normal acceleration.

05:55 This is the center of mass, alpha v .d and omega v .d.

06:16 So we can write ad is equal to ab plus acceleration of d with respect to b.

06:30 So ad is in downward direction or upper direction...