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Tips And Tricks to Solve The Mechanics Problems Using Free Body Diagram: Part: II

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Free Body Diagram on Pulley An ideal pulley have three characteristics: (i) Mass-less  (ii) Smooth/Friction-less   (iii)Circular. A ...

NEET-JEE Essential Theory: Units and Measurement
Tips and Tricks to solve the Mechanics Problem using Free Body Diagram Part: 1
Motion in One Dimension (বস্তুর একমাত্রিক গতি) Part: I
Free Body Diagram on Pulley
An ideal pulley have three characteristics: (i) Mass-less 
(ii) Smooth/Friction-less  
(iii)Circular.
A pulley has three arm during its motion.
(1) Central arm is known as main arm.
(2) Handing two arm are known as side arm.
The pulleys can be studied under three possible cases. 

Case: I

Main arm is fixed while side arms are free to move. In this case net force acting on ideal pulley will always be zero. Because F=ma, for ideal pulley m=0 and hence F=0
Thus if the tension in the two side string is equal to T, the tension in the main string will be equal to 2T.
Say, m2>m1
Since the length of the string is fixed, so if m2 comes down by a distance xm1 will go up by x only. Therefore, acceleration of both the blocks are equal in magnitude and opposite direction.
Note: If a comes out positive then assumed direction of a is correct. If a comes out negative, then actual direction is opposite to the direction assumed.
Free Body Diagram, Mechanics, Newton's Law

Motion of a Connected Block Over a Mass-less and Friction-less Pulley Type: I

Two Body System:
Free Body Diagram, Mechanics, Newton's Law
Two Body System in Pulley Problems
FBD of First Mass
FBD of Second Mass
FBD of Pulley
Let two body A and B of mass m1 and m2 are connected at two ends of a mass-less in-extensible string passing over a friction-less pulley P. Since mass of the pulley is zero then, tension of the string is equal for both side of the pulley. From free body diagram we get,
For mass m1:
T1m1g=m1a
For mass m2:
m2gT1=m2a
For pulley P:
T2=2T1
From these equation we get,
m2g(m1g+m1a)=m2a
or, (m1a+m2a)=(m2gm1g)
or, (m2+m1)a=(m2m1)g
or, a=(m2m1)(m2+m1)g
And, tension of the string (same for both side of the pulley) T1=m1a+m1g=m1(m2m1)(m2+m1)g+m1g=2m1m2(m1+m2)g
And tension or thrust on the pulley is T2=2T1=4m1m2(m1+m2)g



Three Body System:
Let a body of mass m1 and other two body B and C of masses m2 and m3 are connected of a mass-less string passing over a friction-less pulley P. From free body diagram we can write,
Free Body Diagram, Mechanics, Pulley
Free Body Diagram, Mechanics, Pulley
FBD of Mass m1
FBD of Mass m2
Free Body Diagram, Mechanics, Pulley
FBD of Mass m3
For mass m1:
T1m1g=m1a
For mass m2:
m2g+T2T1=m2a
For mass m3:
m3gT2=m3a
For pulley P:
T3=2T1
From these equation we get, 
a=[(m2+m3)m1](m1+m2+m3)g
And tension of the string T1=2m1(m2+m3)(m1+m2+m3)g
And tension of the string T2=2m1m3(m1+m2+m3)g
And tension or thrust on the pulley T3=2T1=4m1(m2+m3)(m1+m2+m3)g


When a body A of mass m1 rests on a friction-less horizontal surface. Let a string passing over a pulley connect m1 with a body B of mass m2 as shown below:

Free Body Diagram, Mechanics, Pulley
Free Body Diagram, Mechanics, Pulley
Free Body Diagram, Mechanics, Pulley
Free Body Diagram, Mechanics, Pulley
From free body diagram we get,
For mass m1:
T=m1a
For mass m2:
m2gT=m2a
From these two equation we get,
Acceleration a=m2(m1+m2)g
Tension of the string T=m1m2(m1+m2)g And Tension or thrust on pulley P=T2+T2=2T




When a body A of mass m1 rests on a horizontal surface with co-efficient of friction μ between body and the table. Let a string passing over a pulley connect m1 with a body B of mass m2 as shown in below:

free body diagram, mechanics, pulley
     
  
free body diagram, mechanics, pulley
free body diagram, mechanics, pulley
free body diagram, mechanics, pulley
From free body diagram we get,
For mass m1:
R=mg and TμR=m1a
or, Tμm1g=m1a
For mass m2:
From these two equation,
Acceleration a=(m2μm1)(m1+m2)g And tension of the string T=m1m2(1+μ)(m1+m2)g  


Two masses are suspended over a pulley on a friction-less inclined plane as shown below in the figure. The mass m2 descends with an acceleration a. The mass m1 s on inclined plane:

Free Body Diagram, Mechanics Pulley
Free Body Diagram, Mechanics Pulley
Free Body Diagram, Mechanics Pulley
From free body diagram we get,
For mass m1:
Tm1gsinθ=m1a
For mass m2:
m2gT=m2a
From these equation we get,
Acceleration a=(m2m1sinθ)(m1+m2)g and,
Tension of the string T=m1m2(1+sinθ)(m1+m2)g




Two masses are suspended over a pulley on a inclined plane with coefficient of friction between the surface and the body is μas shown in figure below. The mass m2 descends with an acceleration a. The mass m1 is on inclined plane:
Free Body Diagram, Mechanics, Pulley

Free Body Diagram, Mechanics, Pulley

Free Body Diagram, Mechanics, Pulley



From free body diagram we get,
For mass m1:
R=m1gcosθ and, TμRm1gsinθ=m1a
For mass m2:
m2gT=m2a
From these two equation we get,
Acceleration a=[m1m1(sinθ+μcosθ)](m1+m2)g and,
Tension of the string T=m1m2(1+sinθ+μcosθ)(m1+m2)g


Two body A and B of masses m1 and m2 are connected by a string passing over a friction-less pulley such that m2>m1: 

Free Body diagram, Mechanics, Pulley
Free Body diagram, Mechanics, Pulley
Free Body diagram, Mechanics, Pulley
From free body diagram we get, 
For mass m1:
Tm1gsinα=m1a
For mass m2:
m2gsinβT=m2a
From these two equation we get,
Acceleration a=(m2sinβm1sinα)(m1+m2)g and,
Tension of the string T=m1m2(sinα+sinβ)(m1+m2)g




The mass M is connected to m1 and m2 via string passing over alight pulleys. Let m1>m2. Obviously m1 moves down an acceleration a. The block on the horizontal table moves towards right with acceleration a:

Free Body Diagram, Mechanics, Pulley
Free Body Diagram, Mechanics, Pulley
Free Body Diagram, Mechanics, Pulley
Free Body Diagram, Mechanics, Pulley
From free body Diagram we get,
For mass m1:
m1gT1=m1a
For mass m2:
T2m2g=m2a
For mass M:
T1T2=Ma
From these equation we get,
Acceleration a=(m1m2)(m1+m2+M)g and,
Tension of the string T1=m1(2m2+M)(m1+m2+M)g and,
Tension of the string T2=m2(2m1+M)(m1+m2+M)g 
    


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Old Millennium Physics: Tips And Tricks to Solve The Mechanics Problems Using Free Body Diagram: Part: II
Tips And Tricks to Solve The Mechanics Problems Using Free Body Diagram: Part: II
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