Differentiating Constraints for Velocity and Acceleration

1. From Position to Motion: The Role of Differentiation

The bridge between the constraint equations we derived previously and the kinematics we want to study is the time derivative. Recall the fundamental definitions from calculus in physics:

• Velocity is the time rate of change of position: $\vec{v}=\frac{d\vec{r}}{dt}$.
• Acceleration is the time rate of change of velocity: $\vec{a}=\frac{d\vec{v}}{dt}=\frac{d^2\vec{r}}{d{t}^2}$.

Our constraint equations are equations of position. By differentiating them with respect to time, we can generate new equations that relate the velocities and accelerations of the system's components.

The key insight is that the total length of an inextensible string, $L$, is a constant. Therefore, its time derivative is always zero:
$\frac{dL}{dt}=0$

Let's consider the general constraint equation we derived in the last lesson for a simple system: $x+2y=C$. Here, $x$ and $y$ are the positions of the blocks and $C$ is a constant. If we differentiate this entire equation with respect to time, we get:

$\frac{d}{dt}(x+2y)=\frac{d}{dt}(C)$

Using the sum and chain rules for differentiation:

$\frac{dx}{dt}+2\frac{dy}{dt}=0$

Recognizing the derivatives as velocities, we arrive at the velocity constraint equation:

$v_x+2v_y=0$

This equation mathematically relates the velocities of the two blocks. If we differentiate again with respect to time:

$\frac{d}{dt}(v_x+2v_y)=\frac{d}{dt}(0)$
$\frac{d{v}_x}{dt}+2\frac{d{v}_y}{dt}=0$

This gives us the acceleration constraint equation:

$a_x+2a_y=0$

This simple, powerful procedure allows us to determine the kinematic relationships for any system for which we can write a length equation.

2. A Formal Procedure and a Rigorous Example

The process we've just outlined can be summarized into a formal, three-step procedure that works for any constrained motion problem.

The resource you just reviewed applies the procedure to a pulley system. Now, let's watch a detailed, rigorous derivation from an MIT OpenCourseWare lecture that tackles a slightly more complex system involving two separate strings. This will demonstrate how to handle multiple constraints.

Let's consolidate the derivation from the video.

• System Analysis: The system has two masses, $m_1$ and $m_2$, and two strings. One mass ($m_2$) is attached to a movable pulley (B).
• Coordinates: A single origin is chosen at the ceiling, with the y-axis pointing downwards. The positions of the masses are $y_1(t)$ and $y_2(t)$, and the position of the movable pulley is $y_B(t)$.
• String 2 Constraint: The second string connects mass 2 to pulley B. Its length is $L_2=y_2-y_B$. Since $L_2$ is constant:
  $\frac{d^2{L}_2}{d{t}^2}=0⟹\frac{d^2{y}_2}{d{t}^2}-\frac{d^2{y}_B}{d{t}^2}=0⟹a_2=a_B$
  This confirms the intuitive fact that mass 2 and pulley B accelerate together.
• String 1 Constraint: The first string goes from a fixed point, under pulley B, and up to mass 1. Its length is expressed as the sum of its changing segments:
  $L_1=y_B+y_B+y_1+\text{constants}=2y_B+y_1+C$
• Differentiating String 1: Differentiating twice with respect to time:
  $\frac{d^2{L}_1}{d{t}^2}=0⟹2\frac{d^2{y}_B}{d{t}^2}+\frac{d^2{y}_1}{d{t}^2}=0⟹2a_B+a_1=0$
• Final Result: Substituting $a_B=a_2$ into the second result gives the final constraint relation:
  $2a_2+a_1=0\text{or}{a}_1=-2a_2$
  The negative sign indicates that if $m_2$ accelerates downwards (positive $a_2$), $m_1$ accelerates upwards (negative $a_1$), which matches our physical intuition. The factor of 2 is the crucial kinematic link we sought.

3. Connecting Math to Physical Intuition

The result $a_1=-2a_2$ is not just a mathematical abstraction. It has a clear physical meaning. For every meter that block 2 and pulley B move down, two meters of slack must be gathered—one from the segment on the left of pulley B and one from the right. This two meters of slack must be pulled up from block 1. Therefore, block 1 must move upwards by two meters. The ratio of displacements, velocities, and accelerations must be 2:1.

The following video includes a wonderful physical demonstration of this exact 2:1 ratio.

4. Practice with a Complex System

Now it's time to apply this systematic method to a more involved problem. The following example involves three objects connected by a single string.

Now, let's use the result you just studied to solve a problem.

Conclusion

In this lesson, we have bridged the gap between the static geometry of a pulley system and its dynamic motion. By applying time differentiation to the algebraic constraint equations, we can systematically derive the relationships between the velocities and accelerations of all connected parts.

Key Takeaways:

• Differentiating a position constraint equation with respect to time yields a velocity constraint equation.
• Differentiating the velocity constraint equation yields an acceleration constraint equation.
• The core mathematical principle is that the derivative of a constant (the total string length) is zero.
• The resulting equations, such as $a_1=-2a_2$, provide precise, quantitative relationships that are essential for solving the full dynamics of the system.
• Careful definition of a coordinate system and consistent use of signs are crucial for correctly interpreting the results.

Preview of the Next Lesson:

We have now thoroughly covered constraints due to inextensible strings. However, this is not the only type of constraint in mechanics. In our next lesson, we will broaden our scope to derive geometric constraint equations for systems involving surfaces in contact, such as wedge-block systems. This will introduce new geometric techniques for analyzing how objects sliding on one another have their motions linked.