The disk A rotates about the vertical z -axis with a constant speed ω=θ̇=π/ 3 rad / s . Simultan

The disk A rotates about the vertical z -axis with a...

The disk $A$ rotates about the vertical $z$ -axis with a constant speed $\omega=\dot{\theta}=\pi / 3 \mathrm{rad} / \mathrm{s} .$ Simultaneously, the hinged arm $O B$ is elevated at the constant rate $\dot{\phi}=2 \pi / 3$ rad/s. At time $t=0,$ both $\theta=0$ and $\phi=0$ The angle $\theta$ is measured from the fixed reference $x$ -axis. The small sphere $P$ slides out along the rod according to $R=50+200 t^{2}$, where $R$ is in millimeters and $t$ is in seconds. Determine the magnitude of the total acceleration a of $P$ when $t=\frac{1}{2}$ s.

Key Concepts

Rotational Kinematics

Rotational kinematics involves the study of the motion of objects rotating about an axis. It deals with quantities like angular displacement, angular velocity, and angular acceleration, and is essential for understanding how rotational motion contributes to overall acceleration, notably via centripetal and tangential components.

Spherical Coordinate System Kinematics

The spherical coordinate system expresses the position and motion of points in space using a radial distance and two angles (typically the elevation and azimuth angles). This framework is crucial for decomposing the acceleration of an object moving with both radial and angular components, as it naturally incorporates changes in both distance and direction.

Time-Dependent Nonlinear Motion

This concept deals with motions where the kinematic variables such as displacement, velocity, and acceleration are functions of time in a nonlinear manner. When a motion parameter, like the radial distance given by R = 50+200t², is nonlinearly dependent on time, differentiation processes such as the chain rule are used to obtain the acceleration, highlighting the interplay between the linear expansion and the angular motions.

Transcript

00:01 In my very high quality diagram here, so i'd reference, i'd suggest you go look at the one in the book.

00:06 We've got a disk gate that's rotating vertically on its, about the z axis, with a constant angular acceleration of pi over three rad radians per second.

00:19 Hinged arm ob is elevated at constant rate, as shown, 2 pi over 3 rads per second.

00:29 At time equals zero, both theta and phi equals zero.

00:35 The angle theta is measured from the fixed reference on the x -axis, and the small sphere, p, slides out along the ride abroad, according to r equals 50 plus 200 t squared, r's in millimeters, and t's in seconds.

00:51 We're asked to find the total magnitude of a, or total acceleration, a of p when time equals.

00:59 Half a second.

01:01 Okay, so i'm going to first start with the most general expression for acceleration, acceleration in spherical coordinates.

01:28 So let me start by just writing this down.

02:00 Okay, add to that.

02:37 Okay, plus getting close here.

02:40 Not getting close.

02:41 I'm getting close to the first part.

03:15 Okay.

03:16 Derivatives of both theta and five are constant.

03:29 So there's some equal terms, some terms equal to zero.

03:33 So let's do this.