The link A B has an angular velocity of 3 rad/s. Determine...

The $\operatorname{link} A B$ has an angular velocity of 3 rad/s. Determine the velocity of block $C$ and the angular velocity of $\operatorname{link} B C$ at the instant $\theta=45^{\circ} .$ Also, sketch the position of $\operatorname{link} B C$ when $\theta=60^{\circ}, 45^{\circ},$ and $30^{\circ}$ to show its general plane motion.

Key Concepts

Relative Motion in Rigid Body Mechanisms

Relative motion analysis involves studying how different parts of a mechanism move with respect to one another. This method is used to determine unknown velocities or angular velocities of interconnected components by employing vector relationships, kinematic constraints, and sometimes the concept of an instantaneous center of rotation.

Planar Kinematics and Linkage Motion

Planar kinematics deals with the motion of mechanisms in a two-dimensional plane. It involves analyzing the positions, velocities, and accelerations of various links. Sketching the positions of the links at different angles helps in visualizing the general path of motion, which is essential for understanding the behavior of mechanisms under different configurations.

Angular Velocity

Angular velocity is a measure of how quickly a body rotates around a fixed axis and is expressed in radians per second. It represents the rate of change of the angular position of the rigid body and is a fundamental concept in rotational dynamics and kinematics, essential for analyzing the motion of any rotating link in a mechanism.

Linear Velocity from Rotational Motion

The relationship between linear velocity and angular velocity is given by v = ?r, where v is the linear velocity, ? is the angular velocity, and r is the distance from the axis of rotation. This concept is crucial when determining the translational speed of a point on a rotating link, such as the velocity of block C attached to a rotating link.

Transcript

00:02 In this question, we are given this situation, so there's a link ab that rotates with an angular velocity of three radiance a second.

00:11 Then in this question we want to determine the velocity of block c and angular velocity of the link bc.

00:18 At the instance when data it goes to 45 degrees, data is this angle.

00:25 Okay, this data.

00:27 Then we need to sketch the position of link.

00:30 Bc when theta equals to 60 degrees, 45 degrees, and 30 degrees to show its general plane motion.

00:38 Okay, so let's just write down the given information.

00:43 So omega ab is 3k, radiance per second, and then we are also given that the position vector of from a to b.

00:59 It's actually 0 .5 times cosine.

01:03 45 degrees i plus sign 45 degrees j okay so using this tool we can determine vb either it's just omega ab cross rab 3k cross 0 .5 cosine 4 5 cosign 45 degrees i plus sign 45 degrees j this is 1 .5 root 2 over 2, k cross i gives j and then k cross j gives minus i.

01:53 So this is 0 .75 square root 2 minus i plus j.

02:04 Then we can use the relative velocity equation bc equals to bb plus omega bc cross r, the position of c relative to point b.

02:26 So ruby is 0 .75 root 2 minus i plus j.

02:44 Then omega bc will just be some generic omega omega and it's in the k direction then cross 1 .5 i okay so point b is here point c is here is 1 .5 meters to the right okay so it's 1 .5 i and then from the diagram vc we see has to be in the eye direction okay and it's going to be negative because as the as link ab rotates counterclockwise c is going to move to to the left.

03:27 Okay? so we just continue to do this part.

03:32 So 0 .75 square root 2 minus i plus j so 1 .5 omega bc k cross i gives j...

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