Slider A moves in the horizontal slot with a constant speed v for a short interval of motion. De

Slider A moves in the horizontal slot with a constant speed...

Key Concepts

Angular Velocity

Angular velocity is the rate at which a rigid body rotates about an axis. It is defined as the change in the angular displacement with time and is typically measured in radians per second. In mechanics, knowing the angular velocity of a component is crucial for analyzing its rotational motion and for linking rotational dynamics with other aspects of system behavior.

Kinematic Constraints in Rigid Body Mechanisms

Kinematic constraints refer to the geometrical and motion restrictions that are imposed by the connections between the different parts of a mechanical system. In systems where a sliding motion is connected to a rotating bar, the geometry of the mechanism constrains the motion such that the displacement of one component directly affects the rotational position of another. Understanding these constraints is key to establishing the relationships between linear and angular motion.

Geometric Relationships in Linkages

Geometric relationships in linkages involve the use of trigonometry and geometry to relate the positions, angles, and displacements of components in a mechanism. For instance, knowing the geometry of a bar connected to a slider allows one to derive expressions that relate the slider’s displacement to the angle of the bar. This forms the basis for further kinematic analysis.

Differentiation and the Chain Rule in Kinematics

Differentiation, particularly using the chain rule, plays an essential role in converting static geometric relationships into dynamic relationships. By differentiating the geometric constraints with respect to time, one can relate the rate of linear displacement to the angular velocity. This process is key for analyzing the motion of systems where different types of motion are interdependent.

Transcript

00:01 Okay, in this question we have this bar with sliders at point b and a here.

00:09 They're moving along tracks that are in these directions that i'm going to highlight in red here.

00:18 This bar of length l is a fixed length, and this bar will move, these objects will move in given slots here, one of them at this angle of 40 degrees with the horizontal, one of them horizontally.

00:38 And we want to know what the angular velocity of the bar will be in terms of the displacement.

00:48 So to do that, we'll have to work out a little bit of geometry first.

00:53 So first we're going to use the law of signs to compare that horizontal position xa over the sign of theta.

01:05 We'll have to equal that length l over the sign of 40 degrees.

01:17 So next we can differentiate, well, we can solve for xa and then we can differentiate with respect to time to get the velocity of a.

01:31 So the velocity of a will equal l is not changing.

01:37 So that's just a constant.

01:39 I'm going to pull that out of our derivative here.

01:51 So we have the derivative respect to time, sine of theta, d theta, not a d theta there yet, just a sine of theta.

02:04 So when we take that derivative with respect to time, we're left with l over the sign of 40 degrees.

02:12 Times the cosine of theta, and then theta dot, which is just the time derivative of theta.

02:21 So this then gives us the rate at which that angle theta is changing, and that's just that theta dot.

02:31 So we can solve for that.

02:36 We'll call that omega here, and that will just equal the velocity of that bar, or velocity a times the sign of 40 degrees, divided by l times the cosine of theta.

03:02 However, we're not done yet.

03:04 We still need to find the value of the cosine of theta.

03:11 So we can do a few things to get there...

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