The system shown starts from rest, and the length of the...

The system shown starts from rest, and the length of the upper cord is adjusted so that $A, B$, and $C$ are initially at the same level. Each component moves with a constant acceleration, and after 2 s the relative change in position of block $C$ with respect to block $A$ is $280 \mathrm{mm}$ upward. Knowing that when the relative velocity of collar $B$ with respect to block $A$ is $80 \mathrm{mm} / \mathrm{s}$ downward, the displacements of $A$ and $B$ are $160 \mathrm{mm}$ downward and $320 \mathrm{mm}$ downward, respectively, determine $(a)$ the accelerations of $A$ and $B$ if $a_{B}>10 \mathrm{mm} / \mathrm{s}^{2},(b)$ the change in position of block $D$ when the velocity of block $C$ is $600 \mathrm{mm} / \mathrm{s}$ upward.

Key Concepts

Constant Acceleration Kinematics

This concept involves the equations that describe the relationship between displacement, velocity, acceleration, and time when an object's acceleration is constant. It allows for the prediction of an object's position and speed at any instance given initial conditions and is crucial in problems where motion starts from rest or when accelerations are specified.

Relative Motion Analysis

Relative motion analysis addresses the movement of one object as observed from another moving object. This concept is vital in systems where several components move in relation to one another, requiring the calculation of relative displacements or velocities between parts of a mechanical system.

Constraint Relationships in Inextensible Systems

In systems with inextensible (non-stretchable) cords or ropes, the total length remains constant. This leads to constraint relationships that connect the displacements, velocities, and accelerations of various parts of the system. By analyzing these constraints, one can determine the motion of interconnected bodies based on the motion of others.

Pulley System Mechanics

Pulley system mechanics involves understanding how ropes and pulleys modify force and motion in a system. Such systems often use pulleys to change the direction of forces or to create relationships between the movements of different bodies, making it essential to consider how the configuration affects mechanical advantages and motion constraints.

Transcript

00:01 In this problem we have the same contraption we had in the previous problem.

00:05 And we're told that the blocks a, b, and c are at the same initial height.

00:16 We're told that after two seconds, block c relative to a is at minus 280 millimeters.

00:27 So that means it was c is above a by 280 millimeters.

00:36 We're told that at some time the velocity of b with respect to a is 80 millimeters per second.

00:45 And at that same time t1, the position of block a has moved down 160 millimeters and block b has moved down.

00:57 320 millimeters.

01:00 And so our constraint from before we had 2.

01:04 Y .a plus 2 yb plus y c is a constant and then 2 y b minus y b as a constant and then taking derivatives to get the constraints on the velocity and acceleration.

01:18 We're told that the system starts from rest and we'll basically measure the position of each of the blocks from zero when t equals zero so each of the blocks has a position that's related to one -half times the acceleration of block times t squared the first thing they ask is for the acceleration of blocks a and b well we can use the information given here to get that so we know that the position of block a at t1 is a 160 millimeters so we have that position of black b is a hundred and three hundred and thirty two millimeters at t1 and then the relative velocity the relative velocity is the velocity of b with respect to a as a velocity of b which is the acceleration of b times t1 minus the velocity of a minus the velocity of a which is the acceleration of a times t1 and we know that's 80 millimeters per second so this gives us three equations and three unknowns and we can solve that and we get that t1 is four seconds.

02:33 The acceleration of block a is 20 millimeters per second squared and that of block b is 40 millimeters per second squared.

02:43 So those are constants, so that's true for all time.

02:49 And so now they want us to figure out the change in position of d when the velocity of block c is 600 millimeters per second.

02:59 So we can then, if we know the acceleration of a and b, we can get the acceleration of c and d and those are minus 120 millimeters per second squared and 30 millimeters per second squared respectively...

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