A 9-in.-radius cylinder of weight 18 lb rests on a 6-lb...

Name: Problem 72, Chapter 17: Vector Mechanics for Engineers: Statics and Dynamics
Uploaded: 2020-08-14T02:09:07-08:00
Duration: 6 min 17 s
Channel: Aatish Gupta
Description: A 9-in.-radius cylinder of weight 18 lb rests on a 6-lb carriage. The system is at rest when a force P of magnitude 2.5 lb is applied as shown for 1.2 s. Knowing that the cylinder rolls without sliding on the carriage and neglecting the mass of the wheels of the carriage, determine the resulting velocity of (a) the carriage, (b) the center of the cylinder.

A 9-in.-radius cylinder of weight 18 lb rests on a 6-lb carriage. The system is at rest when a force P of magnitude 2.5 lb is applied as shown for 1.2 s. Knowing that the cylinder rolls without sliding on the carriage and neglecting the mass of the wheels of the carriage, determine the resulting velocity of (a) the carriage, (b) the center of the cylinder.

Key Concepts

Rolling Without Slipping

This concept describes the condition where the point of contact between a rolling object and the surface does not slide. It creates a kinematic constraint linking the translational and rotational motion of the object, such that the linear velocity of the center of mass is proportional to the angular velocity times the radius. This is essential when analyzing problems where rotational motion converts directly into linear displacement, as it allows the derivation of relationships between linear and angular quantities.

Impulse and Momentum Principles

Impulse, defined as the product of force and the time interval over which it is applied, is directly related to the change in momentum of a system according to the impulse-momentum theorem. In dynamics problems involving a time-varying force, this concept aids in determining the resultant velocities after the force is applied by equating the impulse with the changes in linear momentum of the involved bodies.

Newton’s Laws of Motion

Newton's laws, particularly the second law (F = ma), are fundamental in analyzing both the translational and rotational aspects of a system. They provide a framework for relating the applied forces and torques to the linear and angular accelerations, respectively. This principle is crucial for setting up equations of motion for coupled systems like the rolling cylinder and the carriage.

Rotational Dynamics

Rotational dynamics involves the study of torques and moments of inertia, which are key to understanding how forces affect the rotational motion of bodies. When dealing with rolling objects, it is necessary to consider how an applied force results not only in translation but also in rotation about the center of mass, thereby influencing the overall kinetic energy and momentum distribution of the system.

Transcript

00:01 So we have a cylinder rolling on a cart that is being pulled with a force t.

00:07 So the first thing we're going to do is we're going to, i've listed all our givens here and converted the weights into masses.

00:15 So these are, this is mass a, this is mass b, this is the radius and feet, this is the pulling force, and we have a t of 1 .2 seconds.

00:27 So let's examine this rolling motion of this cylinder.

00:35 So we have a normal force being exerted.

00:41 We have a frictional force being exerted.

00:44 And we have a m .g that's being exerted directly downwards along the center.

00:51 And this is all around a point c.

00:54 So if we have moments about c, you'll notice that there are no, there is no impulse being imparted upon this cylinder because none of these forces create a moment.

01:19 So we have 0 is equal to i omega, or i'm just going to say 1 half m r squared omega minus m -a -v -a -r.

01:31 And this is the linear component of the momentum, and this is the rotation.

01:37 So that means that we have 1⁄2 .559 times 0 .75 squared omega minus 0 .559 times 0 .75va.

01:58 So we get 0 .157 omega minus 0 .419 va.

02:27 So now we have that.

02:32 We are going to be, we're going to use this as one of these simultaneous equations that we will use to solve this problem.

02:46 So we're going to move this up here and continue.

02:53 So next we have, we're going to look at the cart, the cart pulley system as a whole.

03:06 And note that there is an impulse being created called pt, that the cart is also moving with some mbvb...

Please give Ace some feedback