The bowling ball shown rolls without slipping on the...

The bowling ball shown rolls without slipping on the horizontal $x z$ plane with an angular velocity $\omega=\omega_{x} \mathbf{i}+\omega_{y} \mathbf{j}+\omega_{z} \mathbf{k} .$ Knowing that $\mathbf{v}_{A}=(4.8 \mathrm{m} / \mathrm{s}) \mathrm{i}-(4.8 \mathrm{m} / \mathrm{s}) \mathrm{j}+(3.6 \mathrm{m} / \mathrm{s}) \mathrm{k}$ and $\mathrm{v}_{D}=(9.6 \mathrm{m} / \mathrm{s}) \mathrm{i}$ $+(7.2 \mathrm{m} / \mathrm{s}) \mathrm{k},$ determine $(a)$ the angular velocity of the bowling ball, $(b)$ the velocity of its center $C .$

The bowling ball shown rolls without slipping on the horizontal
$x z$ plane with an angular velocity $\omega=\omega_{x} \mathbf{i}+\omega_{y} \mathbf{j}+\omega_{z} \mathbf{k} .$ Knowing
that $\mathbf{v}_{A}=(4.8 \mathrm{m} / \mathrm{s}) \mathrm{i}-(4.8 \mathrm{m} / \mathrm{s}) \mathrm{j}+(3.6 \mathrm{m} / \mathrm{s}) \mathrm{k}$ and $\mathrm{v}_{D}=(9.6 \mathrm{m} / \mathrm{s}) \mathrm{i}$
$+(7.2 \mathrm{m} / \mathrm{s}) \mathrm{k},$ determine $(a)$ the angular velocity of the bowling ball,
$(b)$ the velocity of its center $C .$

Key Concepts

Center of Mass Velocity

The velocity of a rigid body's center of mass represents its translational motion. It is especially important in rolling motion where it, in combination with the angular velocity, defines the complete kinematics of the body. Understanding this concept allows for the analysis of how external points on the body move relative to a fixed frame of reference.

Rigid Body Kinematics

Rigid body kinematics deals with the motion of solid bodies in which the distances between any two points remain constant. It includes the study of simultaneous translation and rotation, with applications in determining velocities of different points on the body based on the angular velocity and the velocity of the center of mass.

Rolling Without Slipping

Rolling without slipping is a condition where a rolling object (like a ball or cylinder) has no relative motion at the contact point with the surface. This condition provides a direct relationship between the translational velocity of the object's center of mass and its angular velocity, allowing for the derivation of kinematic equations that combine both types of motion.

Angular Velocity

Angular velocity is a vector quantity that describes the rate of rotation of a rigid body about a fixed axis, indicating both how fast and in which direction the rotation occurs according to the right-hand rule. It is central to understanding and analyzing rotational motion in rigid body dynamics.

Transcript

00:01 So for problem 15 .184, we've been given the velocity, the vector velocities for points a and d.

00:12 We've been asked to find the angular velocity of the rolling ball and the velocity of point c, which is the center of the ball.

00:22 In order to do this, we have to relate the vector velocities for point a and d.

00:31 With the angular velocity and the velocity of the center.

00:39 This can be time using equation 15 .43 from the textbook.

00:59 You will see right away why i'm writing the one for point d first.

01:13 This notation here is the relative position vector for point d relative to c.

01:22 And now the same for point a is the first product of the angular momentum excuse me of the angular velocity with the relative position vector of a with respect to c now in order to get the angular velocity we can do this little trick and subtract the two equations and we get the vector difference between velocities vd minus va.

02:02 And this expression here, for which if we use the linearity property of the process product, we can write us as this vector difference.

02:21 For which though, if we use the definition of this relative position vectors, it can be simplified.

02:45 This is clear.

02:47 And we get the vector difference of velocities from point d to point a equals the cross product of the angular velocity to this difference, the position vectors of points d and a.

03:21 So now we have to calculate explicitly the values for these two.

03:29 Differences in order to in order than to solve for the angular velocity.

03:39 Let's begin with the velocities.

03:45 This must be quite straightforward...

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