# Velocity - String(Pulley) constraints

• phoenixXL
In summary: The distance from bead to pulley is the length of string joining them. The rate at which the string passes over the pulley is the rate at which that length decreases.Therefore, if the distance between the bead and the pulley is r, then the velocity of the block (or string) is v_b.
phoenixXL

## Homework Statement

A bead C can move freely on a horizontal rod. The bead is connected by blocks B and D by a string as shown in the figure. If the velocity of B is v. Find the velocity of block D.

## Homework Equations

As the string is inextensible the velocity of the string along the length is const.

## The Attempt at a Solution

The doubt I have is that, the following should be true
$$v_c\ =\ v_b.cos53°\ \ \ \ ...(1)$$

But in the solution from the book I get that
$$v_b\ =\ v_c.cos53°\ \ \ \ ...(2)$$

I used relation 1 and got wrong results, I'm just confused how do we get relation 2.
Kindly help me out.

phoenixXL said:
[
The doubt I have is that, the following should be true
$$v_c\ =\ v_b.cos53°\ \ \ \ ...(1)$$

But in the solution from the book I get that
$$v_b\ =\ v_c.cos53°\ \ \ \ ...(2)$$

I used relation 1 and got wrong results, I'm just confused how do we get relation 2.
Kindly help me out.

How is the tip of the string connected to bead C moving ? What is its velocity ?

Its velocity is $v_b$,along the length of the string, the same as that of block B.

phoenixXL said:
Its velocity is $v_b$,along the length of the string, the same as that of block B.

That means the tip of the string and bead C have different velocities . Is that so ?

I think that so (may be my misconception). The bead is confined to the horizontal bar, and can only have velocity along the horizontal bar.

Furthermore, is the velocity of TIP and the whole of the string, not the same.

The red spot in the picture depicts the tip of the string .Forget about all the pulleys and blocks for a moment.Just focus on the red spot .How does it move ?

#### Attachments

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It should have the velocity same as that of the bead $v_c$.

It hasn't been specified anywhere that I can see, but I presume the bead is tied to the string, so the string cannot slide through it.

phoenixXL, if the bead moves to the right at velocity vb, what is the component of that towards the top-right pulley?

haruspex,
As you are pointing, the following would be the diagram
( assuming $v_b$ to be the velocity of the block( or string) and $v_c$ the velocity of the bead. )

But, I got confused and tried to solve the problem using the following diagram,

So, I will be grateful to know what misconception I do have, and how can I further prevent from getting errors.

phoenixXL said:
So, I will be grateful to know what misconception I do have, and how can I further prevent from getting errors.
The question of which vector equals a component of the other comes up in a few guises. In the present case, you just have to remember that it's the string length that's constant, so the rate at which the string passes over the pulley equals the rate at which the bead gets closer to that pulley.
I find it can also help to imagine the process happening. If you pull down steadily on the string hanging from the pulley, do you expect the bead to get faster or slower?

haruspex said:
so the rate at which the string passes over the pulley equals the rate at which the bead gets closer to that pulley
How can we conclude this?

haruspex said:
I find it can also help to imagine the process happening. If you pull down steadily on the string hanging from the pulley, do you expect the bead to get faster or slower?
Faster, of course.

phoenixXL said:
haruspex,
As you are pointing, the following would be the diagram
( assuming $v_b$ to be the velocity of the block( or string) and $v_c$ the velocity of the bead. )

But, I got confused and tried to solve the problem using the following diagram,

So, I will be grateful to know what misconception I do have, and how can I further prevent from getting errors.

Your misconception is the belief that the opposite ends of the string must have the same velocities. They don't. Only the component of the motion parallel to the string is constrained. The ends of the string are completely free to move in the direction perpendicular to the string.

phoenixXL said:
How can we conclude this?
The distance from bead to pulley is the length of string joining them. The rate at which the string passes over the pulley is the rate at which that length decreases.

haruspex said:
The distance from bead to pulley is the length of string joining them. The rate at which the string passes over the pulley is the rate at which that length decreases.
Got it.

For anyone with similar problem

Let the distance of the bead from the pulley be r. Then $$-\frac{dr}{dt}\ =\ v_b\\ \implies\ -\frac{d\sqrt{x^2\ +\ y^2}}{dt}\ =\ v_b\\ \implies\ -\frac{1}{2\sqrt{x^2\ +\ y^2}}.\frac{dx^2}{dt}\ =\ v_b\\ \implies\ -\frac{1}{2\sqrt{x^2\ +\ y^2}}.2x.v_c\ =\ v_b\\ \implies\ -v_c.cosθ\ =\ v_b\\$$

Thank you so much when I derive it myself, meager confidence builds up.
Thanks you so much

## 1. What is a velocity-string constraint?

A velocity-string constraint is a mathematical relationship that describes the motion of two objects connected by a string or rope. It states that the velocities of the two objects must be equal in magnitude and opposite in direction.

## 2. How does a velocity-string constraint affect the motion of objects?

A velocity-string constraint restricts the motion of objects connected by a string, causing them to move in a coordinated manner. This means that the objects will have the same speed and direction of motion at all times.

## 3. What is a pulley in a velocity-string constraint?

A pulley is a simple machine that consists of a wheel with a groove around its circumference. It is used to redirect the direction of a force, such as the tension in a string, to make it easier to lift or move heavy objects. In a velocity-string constraint, a pulley is often used to connect two objects with a string.

## 4. How is the tension in a string calculated in a velocity-string constraint?

The tension in a string can be calculated using the formula T = m*a, where T is the tension, m is the mass of the object, and a is the acceleration of the object. In a velocity-string constraint, the tension in the string will be the same for both objects connected by the string.

## 5. What are the applications of velocity-string constraints in science and engineering?

Velocity-string constraints are commonly used in physics and engineering to analyze and predict the motion of objects connected by strings, such as in simple machines like pulleys and in more complex systems like robotic arms. They are also used in the study of fluid dynamics and in the design of mechanical systems. Understanding velocity-string constraints is crucial for solving problems involving the motion of connected objects and for designing efficient and effective machines.

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