# Two blocks -which will hit in lesser time?

## Homework Statement

Two blocks A and B ,each of same mass are attached by a thin inextensible string through an ideal pulley.Initially block B is held in position as shown in figure.Now the block B is released .Block A will slide to right and hit the pulley in time t1.

Block B will swing and hit the surface in time t2 . Assume the surface as frictionless,then which block will hit in lesser time

i.e t1 or t2 ? Please explain mathematically??

## The Attempt at a Solution

Intuitively it looks block A will hit first.But mathematically I am not able to relate.Block A does a linear motion covering distance L.Block B doing a circular motion with varying radius(as it falls the initial radius length L will increase ? How to relate displacement , accelaration ?

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TSny
Homework Helper
Gold Member
Think about what causes the horizontal component of acceleration of block B.

As block B falls ,at an instant if we assume

1 the string makes angle θ with the vertical
2 T is tension in the string

then along the radial T and mgcosθ acts,whereas tangentially mgsinθ acts

TSny
Homework Helper
Gold Member
For an arbitrary θ, what is the net horizontal force acting on B? What is the net horizontal force on A?

Forces in Vertical direction = Tcosθ - mg
Forces in horizontal direction =Tsinθ

net horizontal force on A=T
net horizontal force on B=Tsinθ

TSny
Homework Helper
Gold Member
So, which block will have the greater horizontal component of acceleration?

Okay...A will have greater accelaration...thank u very much....can we mathematically calculate the time taken by block B to hit the wall in terms of given variables.
For A accelaration is T/m and distance to be covered is L .Here L=(1/2)(T/m)(t1)
from here we can calculate t1

But for B, it will be undergoing circular motion with varying radius as when it falls,the string length acting as the circular radius increases

Forces in Vertical direction = Tcosθ - mg
Forces in horizontal direction =Tsinθ

I think the horizontal force of the right mass is-
Fr_to-left=Cosθ(mgSinθ)
As at the top, the body has no horizontal force only downward force.
At the bottom too, only downward force.

TSny
Homework Helper
Gold Member
can we mathematically calculate the time taken by block B to hit the wall in terms of given variables.
For A accelaration is T/m and distance to be covered is L .Here L=(1/2)(T/m)(t1)
from here we can calculate t1
Since the tension is not constant during the motion, the acceleration of block A will not be constant. So, L = (1/2)(T/m)t12 will not be valid.
But for B, it will be undergoing circular motion with varying radius as when it falls,the string length acting as the circular radius increases

Forces in Vertical direction = Tcosθ - mg
Forces in horizontal direction =Tsinθ