The motor at C pulls in the cable with an and acceleration aC=(3 t^2) m / s^2, where t is in sec

The motor at C pulls in the cable with an and acceleration...

Key Concepts

Integration of Time?Dependent Acceleration

When acceleration is provided as a function of time, the process of integration is essential to determine the corresponding velocity and displacement. This sequential integration allows one to solve for motion parameters even when acceleration is non?constant, forming the backbone of solving kinematics problems involving variable rates of change.

Constraint Relationships in Pulley and Cable Networks

Pulley and cable systems involve interconnected motions where the lengths of the cables impose restrictions linking the movements of different objects. By establishing constraint equations that relate the displacements, velocities, or accelerations of these objects, one can effectively relate the individual motions to the overall motion of the system.

Relative Motion in Connected Systems

In systems with multiple connected objects, such as those using pulleys or cables, the analysis often requires understanding how the motion of one object is influenced by or related to the motion of another. This concept of relative motion aids in setting up the equations that express the interdependent kinematics of the system.

Application of Kinematics Equations

Kinematic equations provide the foundational relationships between displacement, velocity, acceleration, and time. Utilizing these equations, especially in situations combining constant and variable accelerations, allows one to solve for unknown quantities like time or speed by integrating the relationships and applying initial conditions.

Transcript

00:02 In this problem, we have a pulley system driven by motors d and c pulling blocks b and a.

00:12 We're given the accelerations caused by motors c and d.

00:18 A's accelerate or c's acceleration is defined by 3t squared meters per second squared, and d's acceleration is constant at 5 meters per second square.

00:29 The system is initially at rest, and what we're interested in finding is when the distance between the edge of blocks b and a, which we're calling d here, is zero.

00:40 And we're given that this is initially three.

00:44 So we'll go and find that.

00:46 I've gone ahead and defined some fixed points in the system.

00:52 So one coordinate i'm using is this dbc, or this db, which is the distance between b and the initial edge of block a.

01:07 And i've defined aw, which is the distance between a's pulley here and the wall, and ac, which is between a and the motor c.

01:21 So what we're interested in finding then is when d is equal to zero.

01:29 And this will occur when the change in db plus the change in ac or aw, it doesn't really matter, since these are going to be the same, is equal to our initial distance there.

01:48 So once the change of both of these blocks added up is equal to our initial distance, then the distance between them will be zero.

01:59 So to start with, we can begin by defining some equations for each of these deltas.

02:06 So delta db is pretty simple, uh, since it's just a block connected to a motor that's pulling its string.

02:16 It's just going to be the equation for constant acceleration motion since it's constantly accelerating.

02:24 So that's just v -0 -t -plus -half a t squared.

02:30 And we know that v -0 is zero since the system is initially at rest.

02:34 So we can discard that term.

02:36 And we'll find that if we plug in a here, this is just 5 over 2 t squared.

02:47 Now for delta ac, this one's a little trickier.

02:50 We'll begin by defining the changes, the instantaneous changes, the time derivatives in ac, aw, and l.

02:59 We know that the time derivative of l must be equal to negative of this.

03:16 And we know that the time derivative of l must also be equal to the sum of the time derivatives of aw and ac.

03:32 We also know that since aw and ac are on a taught string connected by a pulley, that their time derivatives must equal each other.

03:47 So since we were defining this in terms of ac up here, we can go ahead and substitute in aw for ac here to get that l.

03:56 Dot must just be to ac.

04:02 Now that we have that, we can plug it.

04:05 In our equation for l dot here that we defined as minus 3 t squared excuse me this should be l double dot because this is the acceleration.

04:18 So before we before we get l dot we should before we actually get l dot we should figure out what that is.

04:26 So l dot is just going to be the integral of l double dot with respect to time which if we take the integral of minus 3 t squared with respect to time.

04:42 This is just going to be negative t cubed, and then plus some constant for the initial velocity, but we know that the system is initially at rest, so this is zero.

05:00 So we have l.

05:02 Dot, and it's just minus t cubed, and so therefore a c dot is just minus one half.

05:12 T cubed.

05:18 Now, we have this time derivative, and so to get the change in ac, we'll just integrate this again...

Please give Ace some feedback