# Time for Bead to Lose Contact: Solving for Friction and Tension Forces

• decentfellow
In summary, the conversation discusses a problem involving a bead and a string, with friction acting on the bead and a tension in the string. The relationship between the acceleration of the block and the string is also discussed. The main point of contention is whether tension is acting on the bead, with the conclusion that tension does not act on the bead, but friction does. The conversation ends with a reminder that there is a direct relationship between friction on the bead and tension in the string.
decentfellow

## Homework Statement

In the figure shown, friction force between the bead and the light string is ##\dfrac{mg}{4}##. Find the time in which the bead looses contact with the string after the system is released from rest.

## Homework Equations

Relation between the acceleration of the block ##(m_2)## and the string going over the non-movable pulley
$$2a_2=a_{string}$$

## The Attempt at a Solution

FBD of ##m_2##

From the above FBD we get the following equation:-
$$mg-2T=ma_2$$

I was having trouble figuring out whether tension would be acting or not on the bead, so I thought about the need for the string to be taught(as the dtring is ideal) hence there should be tension acting on the bead.
From the above FBD of the bead we get the following equation:-
$$ma_{string}+mg-\dfrac{mg}{4}-T=ma_{1/string}$$

We, see that there are only two equations and there are ##3## variables, so the equations are not solvable. What is wrong with the solution above, if there is anything please point out.

Last edited:
decentfellow said:
Relation between the acceleration of the block ##(m_2)## and the string going over the non-movable pulley
$$a_2=2a_{string}$$

Are you sure ?

decentfellow said:
I was having trouble figuring out whether tension would be acting or not on the bead, so I thought about the need for the string to be taught(as the dtring is ideal) hence there should be tension acting on the bead.

Tension does not act on the bead . Friction does . Also note that there is a nice relationship between friction acting on the bead and tension in the string .

decentfellow said:
From the above FBD of the bead we get the following equation:-
$$ma_{string}+mg-\dfrac{mg}{4}-T=ma_{1/string}$$

We, see that there are only two equations and there are ##3## variables, so the equations are not solvable. What is wrong with the solution above, if there is anything please point out.

The acceleration of the bead is independent of the acceleration of the string . The acceleration of the bead is quite simple to calculate as all the forces acting on the bead are given in the problem .

conscience said:
Are you sure ?
From what I have learned till now about constraint motion the acceleration of any point on the string(the string which goes over the non-moveable pulley) would be as stated in my solution.

decentfellow said:
From what I have learned till now about constraint motion the acceleration of any point on the string(the string which goes over the non-moveable pulley) would be as stated in my solution.

Think carefully , if left block moves up by distance 'x' , then how much length of string (left and middle string combined) loosen up ? How much string on the right side needs to go down in order for the string to remain taut ?

decentfellow
conscience said:
Think carefully , if left block moves up by distance 'x' , then how much length of string (left and middle string combined) loosen up ? How much string on the right side needs to go down in order for the string to remain taut ?

If the block goes up by a distance ##x## then a length ##2x## of the string remains slacken so to make the string taut again a length of ##2x## string gets shifted to the rightmost string. Is that correct and btw that's how I had made the constraints.

decentfellow said:
If the block goes up by a distance ##x## then a length ##2x## of the string remains slacken so to make the string taut again a length of ##2x## string gets shifted to the rightmost string. Is that correct

Good

decentfellow said:

Your constraint relation says that if the block goes up by 'x' , then right string moves down by 'x/2' .

conscience said:
Your constraint relation says that if the block goes up by 'x' , then right string moves down by 'x/2' .

Oh my God, now that I see it was a blunder on my part, it took me sometime to figure out that despite the right reasoning I ended up with the wrong constraint. And I am very sorry if I sounded even a bit rude in my last reply, I had no such intention

decentfellow said:
Oh my God, now that I see it was a blunder on my part, it took me sometime to figure out that despite the right reasoning I ended up with the wrong constraint. And I am very sorry if I sounded even a bit rude in my last reply, I had no such intention

This is quite a common mistake .

Last edited:
conscience said:
Tension does not act on the bead . Friction does . Also note that there is a nice relationship between friction acting on the bead and tension in the string .
Why is the tension not acting? According to me the tension is acting because if there is friction present between the bead and the string this means that there is contact between them and what I thought after thinking this much was that as tension is a force which follows Newtons third law then as the bead pulls at the string with its weight hence the ring also pulls via the tension.

decentfellow said:
Why is the tension not acting? According to me the tension is acting because if there is friction present between the bead and the string this means that there is contact between them and what I thought after thinking this much was that as tension is a force which follows Newtons third law then as the bead pulls at the string with its weight hence the ring also pulls via the tension.

Which of the two ( Tension or friction ) is a contact force between the string and the bead ? As the bead slips , friction acts on it . And because of friction , tension exists in the string . Would there be any tension in the string if there were no friction between the string and the bead ?

I repeat what I said in post#2 - There is a nice straightforward relationship between friction acting on the bead and tension in the string .

decentfellow
conscience said:
Which of the two ( Tension or friction ) is a contact force between the string and the bead ? As the bead slips , friction acts on it . And because of friction , tension exists in the string . Would there be any tension in the string if there were no friction between the string and the bead ?

I repeat what I said in post#2 - There is a nice straightforward relationship between friction acting on the bead and tension in the string .
Okay, so as the bead is not fixed to the string as it can slide on it, so the tension doesn't act on it instead the friction acts on both the string and the bead, so the straightforward relation ship you were talking about is ##T=f=\dfrac{mg}{4}##. Am I right.

decentfellow said:
Okay, so as the bead is not fixed to the string as it can slide on it, so the tension doesn't act on it instead the friction acts on both the string and the bead, so the straightforward relation ship you were talking about is ##T=f=\dfrac{mg}{4}##. Am I right.
Yes.

decentfellow said:
Okay, so as the bead is not fixed to the string as it can slide on it, so the tension doesn't act on it instead the friction acts on both the string and the bead, so the straightforward relation ship you were talking about is ##T=f=\dfrac{mg}{4}##. Am I right.

Right .

## 1. What is "Time for the bead to slide"?

"Time for the bead to slide" is a scientific term used to describe the amount of time it takes for a bead to travel down a wire or string under the influence of gravity.

## 2. How is "Time for the bead to slide" measured?

"Time for the bead to slide" is typically measured using a stopwatch or timer. The timer is started when the bead is released and stopped when it reaches the end of the wire or string.

## 3. What factors affect the "Time for the bead to slide"?

The main factors that affect the "Time for the bead to slide" are the length and angle of the wire or string, the size and weight of the bead, and the force of gravity.

## 4. Can "Time for the bead to slide" be used to measure the strength of gravity?

Yes, "Time for the bead to slide" can be used to indirectly measure the strength of gravity. By keeping all other factors constant and varying the location (e.g. different planets or heights) where the experiment is conducted, the differences in "Time for the bead to slide" can be used to calculate the strength of gravity at those locations.

## 5. Why is "Time for the bead to slide" an important concept in science?

"Time for the bead to slide" is important because it demonstrates the relationship between gravity and the motion of objects. It also allows scientists to conduct experiments and make calculations related to gravity and the laws of motion.

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