A 12-in.-radius cylinder of weight 16 lb rests on a 6-1 b...

A 12-in.-radius cylinder of weight 16 lb rests on a $6-1 b$ carriage. The system is at rest when a force $\mathbf{P}$ of magnitude 4 lb is applied. Knowing that the cylinder rolls without sliding on the carriage and neglecting the mass of the wheels of the carriage, determine (a) the acceleration of the carriage, (b) the acceleration of point $A$, (c) the distance the cylinder has rolled with respect to the carriage after $0.5 \mathrm{~s}$

A 12-in.-radius cylinder of weight 16 lb rests on a $6-1 b$ carriage. The system is at rest when a force $\mathbf{P}$ of magnitude 4 lb is applied. Knowing that the cylinder rolls without sliding on the carriage and neglecting the mass of the wheels of the carriage, determine
(a) the acceleration of the carriage,
(b) the acceleration of point $A$,
(c) the distance the cylinder has rolled with respect to the carriage after $0.5 \mathrm{~s}$

Key Concepts

Rolling Without Slipping

This principle states that when a body such as a cylinder rolls without slipping, the velocity at the point of contact with the surface is zero relative to that surface. This condition links linear and angular motion, as the linear speed of the center of mass is equal to the angular speed multiplied by the radius of the rolling object, and it is fundamental to analyzing problems involving combined rotational and translational motions.

Rotational Dynamics

Rotational dynamics involves studying how torques affect the rotation of a rigid body. Using the relationship that net torque equals the moment of inertia times angular acceleration (? = I?), this concept helps in understanding how forces cause rotational changes in objects. It is essential when evaluating the acceleration of points on a rolling object and the effect of applied forces on its rotation.

Rigid Body Kinematics

Rigid body kinematics addresses the motion of objects that do not deform, combining both translation and rotation. This concept allows the determination of the acceleration of any point on a moving object by considering the translation of the center of mass along with the rotation about that center. It is crucial in problems where one needs to analyze the motion of specific points on a rolling body relative to an external frame or another moving body.

Relative Motion Analysis

Relative motion analysis is the method of examining the movement of one object or point with respect to another moving reference frame. It is particularly important in systems where multiple bodies interact, such as a rolling object on an accelerating carriage. This concept helps in decomposing the overall motion into components, making it easier to analyze accelerations and displacements relative to different parts of the system.

Transcript

00:01 We have two different bodies here, so we will study them separately.

00:06 First we draw the fbd for the cylinder along.

00:10 The cylinder will be called a, like in the book, and the carriage b.

00:16 As you can see from the fvd, taking the sum of moments around the center of mass, that years mass moment of inertia, so it can be written as sigma -mg is a good.

00:35 To i alpha but sigma m g is equal to f f r is equal to one half m a r square alpha so we can write alpha is equal to 2 mu g upon r also taking newton's second law around the x -axis we can write sigma f xx is equal to m a a ax therefore a .a.

01:33 A is equal to mu g.

01:42 I changed the name of acceleration from ax to a since there only is acceleration in the x direction for the center of mass.

01:54 So it doesn't need to be denoted and it's easier to understand.

01:59 Now, if you have solved enough of these problems, you should feel that we need a relationship to relate the angular acceleration of the cylinder with the acceleration of its center of mass.

02:13 Now, if instead of the carriage, we would have solid heavy ground that isn't affected by the force of friction, the cylinder applies, we could drive that for the no sliding.

02:27 So accordingly and so accordingly i can write the equation as aa is equal to alpha r.

02:45 However, that is not the case here.

02:49 So let's think of how we can make that the case.

02:53 If we can make the carriage not move with respect to the cylinder, then we can use the above equation.

03:00 It turns out this is pretty easy to do.

03:05 Since for the relative acceleration, if we subtract the acceleration of the carriage from the acceleration of the cylinder, then to the cylinder the carriage appears to not be accelerating at all.

03:20 In math term, this is translated to alpha r is equal to alpha b minus...

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