A circular plate of 120-mm radius is supported by two bearings A and B as shown. The plate rotat

A circular plate of 120-mm radius is supported by two...

A circular plate of $120-\mathrm{mm}$ radius is supported by two bearings $A$ and $B$ as shown. The plate rotates about the rod joining $A$ and $B$ with a constant angular velocity of $26 \mathrm{rad} / \mathrm{s}$. Knowing that, at the instant considered, the velocity of point $C$ is directed to the right, determine the velocity and acceleration of point $E .$

A circular plate of $120-\mathrm{mm}$ radius is supported by two bearings $A$
and $B$ as shown. The plate rotates about the rod joining $A$ and $B$ with
a constant angular velocity of $26 \mathrm{rad} / \mathrm{s}$. Knowing that, at the instant
considered, the velocity of point $C$ is directed to the right, determine
the velocity and acceleration of point $E .$

Key Concepts

Rigid Body Rotation

Rigid body rotation refers to the motion of a solid body where all points retain fixed positions relative to one another. In such motion, every point in the body moves in a circle about a common axis, and the orientation of the body changes but its shape remains constant. This concept is critical in understanding how different points on a rotating object behave both in terms of velocity and acceleration.

Angular Velocity

Angular velocity is a measure of how fast an object rotates around a defined axis. It is a vector quantity with a magnitude representing the rate of rotation and a direction determined by the right-hand rule. In rigid body rotation, all points have the same angular velocity, which simplifies the analysis of motion for any point on the body.

Acceleration in Rotating Bodies

Acceleration in a rotating rigid body arises primarily from changes in the direction of the velocity vector, even when the angular velocity is constant. In such cases, the acceleration has components, notably the centripetal (or radial) acceleration, which points inward toward the center of rotation. The analysis of these accelerations is essential to understand the forces acting on different points of the body.

Centripetal Acceleration

Centripetal acceleration is the inward acceleration required to keep a point moving in a circular path. Its magnitude is given by ?²r, where r is the radius of the circular motion. This concept is crucial in rotational dynamics as it explains how the speed and curvature of the circular path combine to produce an acceleration directed towards the center of rotation.

Velocity in Rotational Motion

The velocity of any point on a rotating rigid body can be determined using the cross product of the angular velocity vector and the position vector from a known reference point to the location of interest. This relationship, given by v = ? × r, shows that the speed of a point depends on its distance from the rotation axis and that its direction is tangential to its circular path.

Transcript

00:01 Here we have a circular plate that is supported by two bearings as shown.

00:07 So again, i'm not going to try to reproduce the figure there, but you can look at it in the book.

00:12 And so the plate rotates about the rod joining a and b.

00:15 So he kind of rotates at this about this inclined axis.

00:24 And it rotates at the constant angular velocity of this value here.

00:30 And we know that the we're told that the radius is of the the disk is 120 millimeters so in this case we want to figure out what the the angular velocity and the angular acceleration of a point of point e which is along you know is at the at the edge of the of the disc knowing that is we're at the edge of the disc knowing that this is directed to the right.

01:05 Okay, so again, we need that that the velocity is directed to the right.

01:11 Basically, it says that, you know, tells us which direction that this is going to be positive.

01:18 If we point, if we get a vector from b to a, okay, so if you take a vector from b to a, we use the right hand rule.

01:27 This is a positive value.

01:30 Now, so here we write the vector from b to a.

01:35 Okay, so we can you know, either take it from the origin, um, and the two vectors from the origin and take the difference.

01:45 So this is a vector that points from b to a.

01:48 And it, it's not a unit vector.

01:49 It has a magnitude.

01:51 Um, and that's, this is given in meters.

01:55 And then we can figure out a vector from, from e from a to e or from e to a.

02:04 Um, and again, we're kind of using a as our reference here.